Distributional Itô's Formula and Regularization of Generalized

ALEA, Lat. Am. J. Probab. Math. Stat. 15, 703–753 (2018) DOI: 10.30757/ALEA.v15-27

Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals

Takafumi Amaba and Yoshihiro Ryu Fukuoka University, 8-19-1 Nanakuma, Jˆonan-ku, Fukuoka, 814-0180, Japan. E-mail address: [email protected] Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan. E-mail address: [email protected]

Abstract. We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an Itôformula for a diffusion in a distributional setting, and apply it to investigate differentiability-index s and integrability-index p > 2 for Ds which the Bochner integral belongs to p.

1. Introduction

T In this paper, we justify the symbol “ 0 δy(Xt)dt” denoting a quantity relating to the local time of a d-dimensional diffusionR process X = (Xt)t>0 with X0 being deterministic (in multi-dimensional case, we assume X0 = y), or more generally, the T 6 object “ 0 Λ(Xt)dt” where Λ is a distribution. Our diffusion process X = (Xt)t>0 is assumedR to satisfy a d-dimensional stochastic differential equation dX = σ(X )dw(t)+ b(X )dt, X = x Rd, t t t 0 ∈ 1 d where w = (w (t), , w (t))t>0 is a d-dimensional Wiener process with w(0) = 0. ··· i i The main conditions on σ = (σj )16i,j6d and b = (b )16i6d under which we will work are combinations from the following.

Hypothesis 1.1. (H1) the coeﬃcients σ and b are C∞, and have bounded derivatives in all orders > 1. (H2) (σσ∗)(x) is strictly positive, where x = X0 and σ∗ is the transposed matrix of σ.

Received by the editors April 26th, 2017; accepted June 3rd, 2018. 2010 Mathematics Subject Classification. Primary 60H07; Secondary 60J60, 60J35. Key words and phrases. Malliavin calculus, Hölder continuity of diffusion local times, Distri- butional Itô’s formula, Smoothing effect brought by time-integral. The research of the first author was supported by JSPS KAKENHI Grant Number 15K17562. 703 704 T. Amaba and Y. Ryu

(H3) σσ∗ is uniformly positive definite, i.e., there exists λ> 0 such that 2 d λ ξ Rd 6 ξ, (σσ∗)(y)ξ Rd for all ξ,y R , | | h i ∈ d where , Rd is the standard inner product on R , and Rd = is the corre- spondingh• norm.•i | • | | • | (H4) σ and b are bounded. We further formulate stochastic integrals and an Itôformula in this distributional Ds setting and investigate when the local time belongs to 2 (the Sobolev space of integrability-index 2 and differentiability-index s R with respect to the Malliavin derivative). ∈ T In fact, we will formulate 0 δy(Xt)dt as a Bochner integral in the space of generalized Wiener functional.R We remark here the Bochner integrability seems nontrivial when y = X0, since δy(Xt) no longer makes sense at t = 0. On the other hand, the local time is usually formulated as a classical Wiener functional. Hence, once the Bochner integrability is proved, a “smoothing effect” should occur T T in the Bochner integral 0 δy(Xt)dt, i.e., the differentiability-index for 0 δy(Xt)dt, should be greater thanR that of δy(Xt). R In the case of Brownian motion Xt = w(t), everything can be explicitly computed and we can exhibit this phenomenon. Namely, the following is the prototype of this study. d d Let S ′(R ) denote the space of all Schwartz distributions on R . S R R Ds Theorem 1.1. Assume d = 1. Let Λ ′( ) and s . If Λ(w(T )) 2 then the mapping ∈ ∈ ∈ T T Ds (0,T ] t Λ w(t) 2 ∋ 7→ r t r t ∈ Ds is Bochner integrable in 2 and we have T T T Λ w(t) dt Ds+1. r r 2 Z0 t t ∈ D( 1/2) Since it is known that δ0(w(t)) 2− − (see Watanabe, 1991), we obtain T (1/2) ∈ δ (w(t))dt D −, which agrees with the result by Nualart and Vives (1992) 0 0 ∈ 2 Rand Watanabe (1994a). The proof of this theorem is due to the chaos computations (which is essentially the same as Nualart and Vives, 1992 but with no approximations of the integrand). When b = σσ′/2, this computation brings the Ds Hölder-continuity of the local time with respect to space variable. The norm on p will be denoted by . k•kp,s Theorem 1.2. Let d = 1. Assume (H1), (H3) and that the drift-coefficient is 1 1 given by b = σσ′/2. Then for each s< 2 and β (0, min 2 s, 1 ), there exists a constant c = c(s,β) > 0 such that ∈ { − } 1 1 β σ(y) δy(Xt)dt σ(z) δz(Xt)dt 6 c y z Z − Z 2,s | − | 0 0 for every y,z R. ∈ The proof of this theorem seems interesting in its own right. The study of Hölder continuity of local times had been initiated by Trotter (1958, inequalities (2.1) and (2.3)), in which the almost-sure Hölder-continuity of the Brownian local Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 705 time l(t, x): t > 0, x R in time-space variable (t, x) was proved (see also Boufoussi{ and Roynette,∈1993}). There are a lot of such studies (see, e.g., Liang, 2006, Ait Ouahra et al., 2014, Lou and Ouyang, 2017 and references therein). Theorem 1.2 implies immediately the following.

Corollary 1.3. Under the conditions in Theorem 1.2, let pt(x, y) be the transition density of Xt. Then the mapping 1 R y σ(y) pt(x, y)dt R ∋ 7→ Z0 ∈ is (globally) β-Hölder continuous for every β < 1. The latter half of this paper concerns with an Itôformula in a distributional setting. The classical Itô-Tanaka formula had been extended with several formula- tions (see Föllmer et al., 1995, Bouleau and Yor, 1981, Wang, 1977/78, Kubo, 1983 and so on). In particular, according to results in Wang (1977/78) and Bouleau and Yor (1981), the Itô-Tanaka formula for f(Xt) is valid in the case where f is just a convex function. In our case, we obtained the following.

d j Theorem 1.4. Assume (H1), (H2) and (H4). Let Ai = j=1 σi ∂/(∂xj ) and L be the generator of the diffusion process X. Suppose that f P: Rd R is a measurable function such that → (i) f is continuous at x, (ii) f has at most exponential growth, T 2 (iii) 0 (Aif)(Xt) 2, kdt< + for i =1, 2, , d, T k k − ∞ ··· (iv) R (Lf)(Xt) 2, kdt< + 0 k k − ∞ for someR k N. Then we have ∈ d T T i f(XT ) f(x)= (Aif)(Xt)dw (t)+ (Lf)(Xt)dt in D−∞. − Z Z Xi=1 0 0 We can drop the assumption (H4) if f has at most polynomial growth. The definition of stochastic integral will be given in Section 4.1 and the time- T D k integral 0 (Lf)(Xt)dt is understood in the sense of Bochner integral in 2− . Kubo (1983) alsoR obtained an Itôformula for Brownian motion in a distributional setting. However, his formula does not need to consider the Bochner integrability because the time-interval of integration is a closed interval excluding zero. A generalization to the case of one-dimensional fractional Brownian motion was done by Bender (2003) (and see references therein), in which, even the case where the time-interval of integration is such as (0,T ] is considered (Bender, 2003, Theorem 4.4), though the first distributional derivative of f is assumed to be a regular distribution. But he did not give a systematic treatment of Bochner integrability. Theorem 1.4 will be proved in Section 4 and it will be established in Section 4.2 even the case where f itself is a distribution of exponential-type and furthermore the time-interval of integration is (0,T ]. d A distribution Λ S ′(R ) is said to be positive if Λ,f > 0 for every nonnegative test function f∈ S (Rd). To include local timesh fori diffusions in our scope, we prepare the following∈ 706 T. Amaba and Y. Ryu

Theorem 1.5. Assume d = 1, (H1) and (H2). Let Λ S ′(R) be positive. Then T ∈ there exists k Z>0 such that we have Λ(Xt) p, 2kdt < + for every p ∈ 0 k k − ∞ ∈ (1, ). R ∞ D 2k Hence the mapping (0,T ] t δy(Xt) p− is Bochner integrable in the case of d = 1. For multi-dimensional∋ 7→ cases, it is∈ suﬃcient to assume x = y in order to guarantee the Bochner integrability (Proposition 3.12). 6 s d s/2 d Finally, let H (R ) := (1 )− L (R , dz) for p (1, ), s R, which are p −△ p ∈ ∞ ∈ called the Bessel potential spaces (see Abels, 2012 or Krylov, 2008 for details). We will then apply the Itˆoformula (Theorem 4.9) to derive the following.

Corollary 1.6. Assume (H1), (H3) and (H4). Let p (1, ) and s R. Then for each Λ Hs(Rd), we have ∈ ∞ ∈ ∈ p s (i) Λ(X ) D ′ for t> 0 and p′ (1,p); t ∈ p ∈ T Ds+1 (ii) if p> 2, we further have Λ(Xt)dt ′ for t0 (0,T ] and p′ [2,p). t0 ∈ p ∈ ∈ R T It might be natural to ask about the class to which Λ(Xt)dt belongs when t0 t0 = 0. Some examples are included in Section 4.3. R The organization of the current paper is as follows: We first review the classical Malliavin calculus in Section 2 to introduce several notations. In particular, the mapping of Watanabe’s pull-back will be extended to the space of distributions of exponential-type. Section 3 is devoted to investigate Bochner integrability of the mapping (0,T ] t Λ(Xt) where Λ is a distribution. We will illustrate the Brow- nian case with detailed∋ 7→ computations. The methods there bring a Hölder continuity in the space variable of the local time in the case where the stochastic differential equation is written in a Fisk-Stratonovich symmetric form. In Section 4, we give a definition of stochastic integrals and formulate an Itôformula in this distributional setting. Corollary 1.6 and some examples will be presented in Section 4.3. Several estimates necessary for these examples are wrapped up in Appendix A.

2. Review of Malliavin Calculus

First, we make a brief review of the classical Malliavin calculus on the d-dimensional classical Wiener space to introduce notations. Let (W, , P) be the d-dimensional Wiener space on [0,T ], that is, W is the space of allF continuous functions [0,T ] Rd, is the σ-ﬁeld generated by the canonical process W w w(t) Rd,→ 0 6 t 6FT , and P is the Wiener measure with P(w(0) = 0) =∋ 1.7→ The expectation∈ under P will be denoted by E. The space W contains the subspace H, consisting of all absolutely continuous h W with h(0) = 0 and the square-integrable derivative. The subspace H is called∈ the Cameron-Martin subspace and forms a real Hilbert space with the inner product

T h1,h2 H := h˙ 1(t), h˙ 2(t) Rd dt, h1,h2 H. h i Z0 h i ∈

It is known that L2 := L2(W, , P) has the following orthogonal decomposition, called the Wiener-Itˆochaos expansionF :

L = R , 2 ⊕C1 ⊕C2 ⊕ · · · Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 707

where each k is a closed linear subspace of L2 spanned by multiple stochastic integrals C ˙ ˙ h1(t1) hk(tk), dw(t1) dw(tk) (Rd)⊗k , Z06t < n F L2(E): JmF = P ∪ ∩s/2 { ∈ 0 under the norm p,s deﬁned by F p,s = (I ) F p for F , where is} the Ornstein-Uhlenbeckk·k operator onk thek Wienerk − space. L Itk is known∈P that L

∞ (I )s/2F = (1 + k)s/2J F, F . (2.1) − L k ∈P Xk=0 Note that D0 = L for p (1, ), and p p ∈ ∞ ∞ F 2 = (1 + k)s J F 2, F Ds. (2.2) k k2,s k k k2 ∈ 2 kX=0 We further deﬁne D Ds D Ds ∞(E) := p(E) and −∞(E) := p(E). s>\0 1

d Let S (R ) be the real Schwartz space of rapidly decreasing C∞-functions on Rd. We denote by S (Rd), k Z the completion of S (Rd) by the norm 2k ∈ 2 k d φ 2k := 1+ x /2 φ , φ = φ(x) S (R ), | | | −△ |∞ ∈ d 2 2 where = i=1 ∂ /(∂xi) and φ = supx Rd φ(x) . △ | |∞ ∈ | | P 1 d d Deﬁnition 2.1. (i) A Wiener functional F = (F , , F ) D∞(R ) is said i j 1 ··· ∈ to be non-degenerate if det( DF , DF H )i,j− p < for any p (1, ). k h i 1 k d∞ D Rd∈ ∞ (ii) A family of Wiener functionals Fα = (Fα, , Fα ) ∞( ), α I, where I is an index set, is said to be uniformly··· non-degenerate∈ if for∈ any i j 1 p (1, ), it holds that supα I det( DFα, DFα H )i,j− p < . ∈ ∞ ∈ k h i k ∞ d d If F D∞(R ) is non-degenerate, then the mapping S (R ) φ φ(F ) D∞ ∈ S Rd ∋ 7→D s ∈ extends uniquely to a mapping ′( ) Λ Λ(F ) s>0 1

2.1. Slight extension to exponential-type distributions. It will be convenient to ex- tend the pull-back procedure from Schwartz distribution space to the space of all distributions of exponential-type. Let ∂ := ∂/(∂x ), i =1, 2, , d. i i ··· d d Deﬁnition 2.3 (Hasumi, 1961). We say φ C∞(R ) belongs to E (R ) if for any Z Z ∈ k1 kd p >0 and k1, , kd >0, we have supx R exp(p x )∂1 ∂d φ(x) < + . ∈ ··· ∈ ∈ | | | ··· | ∞ Semi-norms on E (Rd), deﬁned by

k1 kd φ p := sup sup exp(p x )∂1 ∂d φ(x) , p =0, 1, 2, d | | k1, ,kd Z>0: x R | | | ··· | ··· ··· ∈ ∈ 06k + +kd6p 1 ··· make E (Rd) a locally convex metrizable space and induces continuous inclusions d d d d d d ( D(R ) ֒ E (R ) ֒ S (R ) and S ′(R ) ֒ E ′(R ) ֒ D ′(R → → → → d d where D ′(R ) is the space of all distributions on R with the test function space d d d d D(R ), and E ′(R ) is the continuous dual of E (R ). Elements in E ′(R ) are referred as distributions of exponential-type. The following is known (see Hasumi, 1961, Proposition 3). Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 709

d Theorem 2.4. For any Λ E ′(R ), there exist k Z> and a bounded continuous ∈ ∈ 0 function f : Rd R such that → ∂kd Λ= [exp(k x )f(x)]. ∂xk ∂xk | | 1 ··· d Here, the derivatives are understood in the sense of distributional derivatives. d d Remark 2.5. The spaces E (R ) and E ′(R ) are denoted by H and Λ respectively d ∞ in Hasumi (1961). The space E ′(R )=Λ is introduced by Sebasti˜ao e Silva (1958) in the study of the Fourier transform∞ of Λ , called ultra-distributions (see Sebasti˜ao e Silva, 1958, Hasumi, 1961 and Yoshinaga∞ , 1960).

d d Deﬁne e (x) := cosh(kx ) for k Z> and x = (x , , x ) R . We k i=1 i ∈ 0 1 ··· d ∈ postpone the proofQ of the next proposition after Deﬁnition 2.7. 1 d D Rd Proposition 2.6. Suppose that Fα = (Fα, , Fα ) ∞( ), α I, where I is an index set, are uniformly non-degenerate and··· satisfy∈ ∈

Mr := sup E[exp(r Fα Rd )] < + for each r> 0. (2.4) α I | | ∞ ∈ Then for any p (1, ), k Z> and r > 0, there exists a constant c = ∈ ∞ ∈ 0 c(q,k,r,Mr) > 0, where 1/p +1/q =1, such that ∂kd(e φ) r 6 k k (Fα) p, kd c sup φ(x) ∂x ∂x Rd | | 1 d − x ··· ∈ for all φ S (Rd) and α I. ∈ ∈ D Rd k k Now, let F ∞( ) be non-degenerate. Take Λ = ∂1 ∂d [exp(k x )f(x)] d ∈ ··· | | E ′(R ) (where f is the one associated to Λ in Theorem 2.4) and assume k > 1. Let ∈ d ε (0, 1) be arbitrary and put r := k + ε> 0. Define φ C0(R ) (the space of con- ∈ Rd ∈k x d tinuous functions on vanishing at infinity) by φ(x):=(e | |/ i=1 cosh(rxi))f(x), k k k k d so that now we have Λ = ∂1 ∂d (erφ) = ∂1 ∂d [( i=1 cosh(Q rxi))φ(x)]. Take d ··· ··· any sequence φn S (R ), n N such that φn φ Q 0. Then Proposition 2.6 ∈ k k∈ | − D|∞kd→ tells us that limn [∂1 ∂d (erφn)](F ) exists in p− for each p (1, ). The →∞ ··· ∈ d∞ limit does not depend on the choice of ε> 0 and the sequence φn S (R ). Under these notations, we put the following. ∈ d Definition 2.7. We denote the limit by Λ(F ) and call the pull-back of Λ E ′(R ) by F . ∈ d Proof of Proposition 2.6: Let p (1, ), k Z> , r > 0, α I, φ S (R ) and ∈ ∞ ∈ 0 ∈ ∈ J D∞ be arbitrary. Then ∈ d E[ ∂k ∂k(e φ) (F )J]= E[ cosh(rF i ) φ(F )l (J)] 1 ··· d r α α α α Yi=1 where l (J) D∞ is of the form α ∈ kd j l (J)= P (w),D J ⊗j α h j iH Xj=0 j for some Pj (w) D∞(H⊗ ), j = 0, 1, , kd which are polynomials in Fα, its ∈ ··· i j 1 derivatives up to the order kd, and det( DF , DF )− . Take q′ (1, q), where h α αiH ij ∈ 710 T. Amaba and Y. Ryu

1/p +1/q = 1. Since Fα α I is uniformly non-degenerate, there exists c0 > 0 such that { } ∈

l (J) ′ 6 c J for all α I and J D∞. k α kq 0k kq,k ∈ ∈ Therefore by taking p′ (1, ) such that 1/p′ +1/q′ = 1, we have ∈ ∞ d i E[ cosh(rFα) φ(Fα)lα(J)] 6 c0′ φ exp r Fα p′ J q,k, | |∞k | | k k k Yi=1 for some constant c0′ > 0, which implies (k) (ekφ) (Fα) p, k 6 c0′ φ exp(r Fα ) p′ 6 c0′ sup exp(r Fα ) p′ φ . k k − | |∞k | | k α I k | | k | |∞ ∈

d Corollary 2.8. Suppose that Fα D∞(R ) for α I R, where I is an index set, satisfy ∈ ∈ ⊂

(i) Fα α I is uniformly non-degenerate; { } ∈ (ii) supα I E[exp(r Fα Rd )] < + for each r> 0; ∈ | | ∞ d (iii) the mapping I α F D∞(R ) is continuous. ∋ 7→ α ∈ k k E Rd Then for any p (1, ) and Λ = ∂1 ∂d [exp(k x )f(x)] ′( ) (f is the one associated to Λ in∈ Theorem∞ 2.4), the mapping··· | | ∈

kd I α Λ(F ) D− ∋ 7→ α ∈ p is continuous. Proof : Let p (1, ), α I and ε > 0 be arbitrary. Suppose that Λ = k k d ∈ ∞ ∈ Rd ∂1 ∂d [( i=1 cosh(rxi))φ(x)] where r > k > 1 and φ C0( ). Then by Proposi- ··· S Rd k∈ k tion 2.6, thereQ exists ψ ( ) such that Λ(Fβ) [∂1 ∂d (erψ)](Fβ) p, k < ε/3 for every β I. Furthermore,∈ by the conditionk (ii)− and··· (iii), there existskδ− > 0 such k ∈ k k k that [∂1 ∂d (erψ)](Fβ ) [∂1 ∂d (erψ)](Fα) p < ε/3 if β α <δ. Hence, for each kβ I···with β α <δ−, ··· k | − | ∈ | − | Λ(Fα) Λ(Fβ) p, k k − k − k k 6 Λ(Fα) [∂1 ∂d (erψ)](Fα) p, k k − ··· k − + [∂k ∂k(e ψ)](F ) [∂k ∂k(e ψ)](F ) k 1 ··· d r α − 1 ··· d r β kp k k + [∂1 ∂d (erψ)](Fβ ) Λ(Fβ ) p, k k ··· − k − < ε. The case of k = 0 is clear.

If we assume (H1), (H2) and (H4), then we have the Gaussian estimate for the transition density of X (the special case n = 0 in Lemma 3.10–(ii)) and

sup E[exp(r X(t, x, w) )] < + (2.5) t K | | ∞ ∈ for all r > 0 and compact set K (0, ). Hence by Corollary 2.8, for any Λ = k k d ⊂ ∞ kd ∂ ∂ [exp(k x )f(x)] E ′(R ), the mapping (0, ) t Λ(X(t, x, w)) D− 1 ··· d | | ∈ ∞ ∋ 7→ ∈ p is continuous for every p (1, ). ∈ ∞ Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 711

3. Bochner Integrability of Pull-Backs by Diﬀusions

Let w = (w(t))t>0 be a d-dimensional Wiener process with w(0) = 0. Let σ : Rd Rd Rd, b : Rd Rd. We consider the following stochastic diﬀerential equation→ ⊗ → dX = σ(X )dw(t)+ b(X )dt, X = x Rd. (3.1) t t t 0 ∈ In this section, we assume conditions (H1) and (H2). Under these conditions, the equation (3.1) admits a unique strong solution. We denote by X(t, x, w) > { }t 0 the unique strong solution X = (Xt)t>0 to (3.1). Furthermore, for each t> 0 and d x R, we have X(t, x, w) D∞(R ) and is non-degenerate. Henceforward for ∈ ∈ d t> 0, one can deﬁne the pull-back Λ(X(t, x, w)) of Λ S ′(R ) by X(t, x, w) as an D s ∈ element of s>0 1 0 be arbitrary. By the condition (H2), we further know that i j 1 sup det( DXt ,DXt H )ij− p < + for 1 0, S Rd ⊂ ∈ D 2k p (1, ) and Λ 2k( ), the mapping (0,T ] t Λ(X(t, x, w)) p− ∈ ∞ ∈ − ∋ 7→ ∈ is continuous (see e.g. Watanabe, 1987, Remark 2.2). In particular, the mapping 2k [t ,T ] t Λ(X(t, x, w)) D− is Bochner integrable for each t > 0, and hence 0 ∋ 7→ ∈ p 0 T 2k the Bochner integral Λ(X(t, x, w))dt makes sense as an element in D− for t0 p each t0 > 0. If we assumeR (H4) additionally, then analogous results follow also for d Λ E ′(R ) by virtue of the Gaussian estimates for the transition density function ∈ of (Xt)t>0 (or one can refer Lemma 3.10 below) and Corollary 2.8. Now our problem is the Bochner integrability on (0,T ], i.e., whether it holds T that Λ(X(t, x, w)) p, 2kdt < + or not. The main results of this section 0 k k − ∞ are TheoremR 3.4 and Proposition 3.12. Before seeing this, we shall start with the Brownian case, which would be an introductory example.

3.1. Bochner integrability of δ0(w(t)). S R R Ds Proposition 3.1. Let d = 1, Λ ′( ) and s . If Λ(w(T )) 2 then the mapping ∈ ∈ ∈ T T Ds (0,T ] t Λ w(t) 2 ∋ 7→ r t r t ∈ Ds is Bochner integrable in 2 and we have T T T Λ w(t) dt Ds+1. r r 2 Z0 t t ∈ Remark 3.2. Hence the Bochner integral poses a sort of “smoothing eﬀect”, in the sense of raising the regularity-index s, which might be a common understanding for most of us. Proof : Recall the integration by parts formula

w(t) 1/2 w(t) E[Λ′(w(t))Hn ]= t− E[Λ(w(t))Hn+1 ] √t √t n where Λ′ = ∂Λ is the distributional derivative of Λ, Hn := ∂∗ 1 is the n-th Her- 1 mite polynomial and ∂∗ = ∂ + x. The family H ∞ forms a complete − { √n! n}n=0 1/2 x2/2 orthonormal base of L2(R, (2π)− e− dx) (see Lemma A.1 in Appendix A). 712 T. Amaba and Y. Ryu

Then Λ((T/t)1/2w(t)) has the Fourier expansion (the Wiener-Itˆochaos expansion) T ∞ 1 T w(t) w(t) Λ w(t) = E[Λ w(t) Hn ]Hn , r t n! r t √t √t nX=0 and hence ∞ s T 2 (1 + n) T w(t) 2 Λ w(t) = E[Λ w(t) Hn ] k r t k2,s n! r t √t nX=0 Here we have T w(t) w(T ) E[Λ w(t) Hn ]= E[Λ(w(T ))Hn ]. r t √t √ T Therefore T T Λ w(t) 2 kr t r t k2,s T ∞ (1 + n)s w(T ) T = E[Λ(w(T ))H ]2 = Λ(w(T )) 2 , t n! n √ t k k2,s nX=0 T that is, we get (T/t)1/2Λ((T/t)1/2w(t)) = (T/t)1/2 Λ(w(T )) . Since k k2,s k k2,s T T T T 1/2 1/2 Λ w(t) dt = T Λ(w(T )) 2,s t− dt< + , Z r t r t 2,s k k Z ∞ 0 0 the function (0,T ] t (T/t )1/2Λ((T/t)1/2w(t)) Ds is Bochner integrable and ∋ 7→ ∈ 2 T (T/t)1/2Λ((T/t)1/2w(t))dt Ds. 0 ∈ 2 R Next we show T T T Λ w(t) dt Ds+1. (3.2) Z r t r t ∈ 2 0 For this, we note that T T T Λ w(t) dt Z r t r t 0 T ∞ 1 w(T ) T w(t) = E[Λ(w(T ))Hn ] Hn dt n! √ Z r t √ nX=0 T 0 t T 1/2 1/2 is the chaos expansion for 0 (T/t) Λ (T/t) w(t) dt (more precisely, we have used Corollary 3.6 below).R We shall focus on the L2-norm of the last factor: T 1 w(t) 2 E[ Hn dt ] n Z0 √t √t o 1 w(t) w(s) =2 E[Hn Hn ]dtds Z0

In the third equality, we have used the integration by parts formula and Hn′ = nHn 1. Hence we have − T T T Λ w(t) dt 2 k Z r t r t k2,s+1 0 ∞ s+1 T 2 (1 + n) w(T ) 2 1 w(t) = T E[Λ(w(T ))Hn ] E[ Hn dt ] (n!)2 √ Z √ √ nX=0 T n 0 t t o ∞ (1 + n)s w(T ) =4T 2 E[Λ(w(T ))H ]2 =4T 2 Λ(w(T )) 2 < + , n! n √ k k2,s ∞ nX=0 T which proves (3.2).

By Nualart and Vives (1992, Section 2) and Watanabe (1994a), it is known that

( 1/2) 1/2 δ (w(t)) D − − but δ (w(t)) / D− for t> 0, 0 ∈ 2 0 ∈ 2 Ds Dα where 2− := α

~~T D(1/2) Corollary 3.3. If d =1, we have δ0(w(t))dt 2 −. Z0 ∈~~

3.2. Bochner integrability of Λ(Xt) where Λ is a distribution. A distribution Λ d d ∈ D ′(R ) is said to be positive if Λ,f > 0 for all nonnegative f D(R ). It is h i d ∈ known that for a positive distribution Λ D ′(R ), there exists a Radon measure µ on Rd such that ∈ d Λ,f = δy,f µ(dy), f D(R ). h i ZRd h i ∈ The main objective in this section is to prove

Theorem 3.4. Let d =1 and x R. Suppose that Λ D ′(R) is positive. ∈ ∈ (i) If (H1), (H2) and Λ S 2k(R) where k Z>0, then for any p (1, + ), T ∈ − ∈ ∈ ∞ Λ(X(t, x, w)) p, 2kdt< + . 0 k k − ∞ R (ii) If (H1), (H2), (H4) and Λ E ′(R) hold, then ∈ T δy(X(t, x, w)) p, 2µ(dy)dt< + , (3.3) Z0 ZR k k − ∞ where µ is the Radon measure associated to Λ as above.

Remark 3.5. (a) Therefore if we assume (H1), (H2) and Λ S 2k(R) is pos- D 2k ∈ − itive, then (0,T ] t Λ(Xt) p− is Bochner integrable. If we have ∋ 7→ ∈ 2 (H4) additionally, the mapping (0,T ] R (t,y) δ (X ) D− is also × ∋ 7→ y t ∈ p Bochner integrable with respect to dt µ(dy) (the measurability can be ⊗ checked from the construction of δy(Xt)), and we have T T Λ(Xt)dt = δy(Xt)µ(dy)dt Z0 Z0 ZR T D 2 = δy(Xt)dtµ(dy) in p− . ZR Z0 714 T. Amaba and Y. Ryu

(b) Under (H1) and (H2), Lemma 2.2 and Theorem 3.4–(i) assures T δy(Xt) p, 2dt is ﬁnite for any y R. 0 k k − ∈ R Corollary 3.6. Let d =1. Assume (H1) and (H2). For each y R and n Z>0, the mapping (0,T ] t J [δ (X )] L is Bochner integrable and∈ ∈ ∋ 7→ n y t ∈ 2 T T Jn δy(Xt)dt = Jn[δy(Xt)]dt. Z Z 0 0 Proof of Corollary 3.6: By Theorem 3.4, we have T T

~~Jn[δy(Xt)] L2 dt = (1+ n) Jn[δy(Xt)] 2, 2dt Z0 k k Z0 k k − T 6 (1 + n) δy(Xt) 2, 2dt< + . Z0 k k − ∞~~

This shows the Bochner integrability of the mapping (0,T ] t Jn[δy(Xt)] L2, T ∋ 7→ ∈ and hence 0 Jn[δy(Xt)]dt L2. D∈ Ds R Second,R for each F ∞, the mapping 2 G E[GF ] is linear and bounded for each s R.∈ Therefore Bochner integrals∋ and7→ (generalized)∈ expectations may be interchanged,∈ and accordingly we have T T E Jn[ δy(Xt)dt]F = E δy(Xt)dtJnF Z Z 0 0 T = E[δy(Xt)JnF ]dt Z0 T T = E Jn[δy(Xt)]F dt = E Jn[δy(Xt)]dtF , Z Z 0 0 which proves the second assertion.

To prove Theorem 3.4, we need several implements. For each ε > 0, we consider the following d-dimensional stochastic diﬀerential equation 2 dXt = εσ(Xt)dw(t)+ ε b(Xt)dt. (3.4) ε Similarly to the equation (3.1), we denote by X (t, x, w) t>0 a unique strong ε ε ε { Rd } solution X = (Xt )t>0 to (3.4) such that X0 = x . Then it holds that for each 2 ε ∈ ε> 0, X(ε t, x, w) t>0 is equivalent to X (t, x, w) t>0 in law. A more tricky fact which we{ need is the} following. { }

d Proposition 3.7. Let Λ S ′(R ). Then for every p (1, ), s R and t> 0, we have Λ(X(ε2t, x, w)) ∈ = Λ(Xε(t, x, w)) . ∈ ∞ ∈ k kp,s k kp,s Proof : For simplicity, we give a proof in the case of d = 1. Let p (1, ), 2 ∈ ∞ s R and t > 0 be arbitrary. It is enough to prove that f(X(ε t, x, w)) p,s = ∈ ε k k f(X (t, x, w)) p,s for each f S (R). By the Veretennikov-Krylov formula (see kVeretennikov andk Krylov, 1976∈, p.279, Theorem 4), we have ε Jn[f(X (t, x, w))]

~~t r2 ε ε ε ε = (Pr1 Qr2 r1 Qrn rn−1 Qt rn f)(x)dw(r1) dw(rn), Z0 ··· Z0 − ··· − − ··· Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 715~~

ε ε R ε ε ε ε d where (Pr f)(z) := E[f(X (r, z, w))] for z , Qrf := A (Pr f) and A := εσ dx . 1 ∈ 1 1 In the case ε = 1, we will write Pr := Pr , A := A and Qr := Qr for simplicity. Therefore, we have to show that 2 2 ε t ε r2 n 2 (Pr1 Qr2 r1 Qrk rk−1 Qε t rk f)(x)dw(r1) dw(rk ) n Z0 ··· Z0 − ··· − − ··· ok=0 t r2 n ε ε ε ε = (Pr1 Qr2 r1 Qrk rk−1 Qt rk f)(x)dw(r1) dw(rk) n Z0 ··· Z0 − ··· − − ··· ok=0 in law for each n N, and hence by the scaling property of Brownian motion: 1 2 ∈ (ε− w(ε t))t>0 = (w(t))t>0 in law, it suﬃces to show that

2 2 2 2 2 2 2 (Pε r Qε r ε r Qε rn ε rn Qε t ε rn f)(x) 1 2− 1 ··· − −1 − n ε ε ε ε (3.5) = ε− (Pr Qr r Qrn rn Qt rn f)(x) 1 2− 1 ··· − −1 − ε for each 0 < r1 < < rn < t. But this is clear since we have Pε2r = Pr , 1 ε ··· 1 ε A = ε− A and Qε2r = ε− Qr for each r> 0. For each ε> 0 and x Rd, we set ∈ Xε(1, x, w) x F (ε, x, w) := − . ε Then the following two conditions are equivalent (see Watanabe, 1987, Theorem 3.4): (H2), i.e., there exists λ> 0 such that ◦ 2 d λ ξ 6 ξ, (σσ∗)(x)ξ Rd , for all ξ R . | | h i ∈ the family F (ε, x, w) is uniformly non-degenerate. ◦ { }ε>0 Proposition 3.8. Let d = 1 and x R. Suppose (H1) and σ(x)2 > 0. Then for ∈ any p (1, ), we have sup0<ε61 δ0(F (ε, x, w)) p, 2 < + . ∈ ∞ k k − ∞ 2 1 Proof : Let φ(z) := (1+z )− δ (z) S and take q (1, )so that 1/p+1/q = −△ 0 ∈ 0 ∈ ∞ 1. Then for each J D∞, we have ∈ E[δ (F (ε, x, w))J]= E[ (1 + z2 )φ (F (ε, x, w))J] 0 −△ = E[φ (F (ε, x, w))lε(J)] where l (J) D∞ is of the form ε ∈ 2 lε(J)= P0(ε, w),DJ H + P1(ε, w),D J H H h i h i ⊗ i for some Pi(ε, w) D∞(H⊗ ), i =1, 2, both of which are polynomials in F (ε, x, w), ∈ ε 2 its derivatives up to the second order and DX (1, x, w) H− (see e.g., Watanabe, 1987, equation (2.20)). k k Take q′ (1, q). Since F (ε, x, w) is uniformly non-degenerate, there exists ∈ { }ε>0 c0 > 0 such that

l (J) ′ 6 c J for all ε (0, 1] and J D∞. k ε kq 0k kq,2 ∈ ∈ Therefore we have for each J D∞, ∈ E[δ0(F (ε, x, w))J] 6 φ lε(J) q′ 6 c0 φ J q,2 k k∞k k k k∞k k which implies sup0<ε61 δ0(F (ε, x, w)) p, 2 6 c0 φ < + . k k − k k∞ ∞ Second, we recall the next fact (see Ikeda and Watanabe, 1989, Chapter V, Section 8, p.384) to prove Theorem 3.4. 716 T. Amaba and Y. Ryu

Proposition 3.9. Let x Rd and suppose that (H1) and (H2). Then for any ∈ p (1, ) and k Z> , there exists C > 0 such that ∈ ∞ ∈ 0 d φ F (ε, x, w) p, 2k 6 C φ 2k for φ S (R ), ε (0,T ]. k ◦ k − k k− ∈ ∈ The last tool we need is the following. Lemma 3.10. Let x Rd. ∈ (i) If (H1) and (H2) hold, then there exist K > 0, T (0,T ] with the following 0 ∈ property: For each k , , k Z> and p (1, ), there exist constants 1 ··· d ∈ 0 ∈ ∞ ν0,c1,c2 > 0 such that

k1 kd ν0 x y ∂1 ∂d δy(X(t, x, w)) p, (n+2) 6 c1t− exp c2 | − | k ··· k − n − t o for each t (0,T ] and y R such that y > K. ∈ 0 ∈ | | (ii) If (H1), (H2) and (H4) hold, then for each k1, , kd Z>0, K > x and p (1, ), there exist constants ν ,c ,c > 0 such··· that∈ | | ∈ ∞ 0 1 2 2 k1 kd ν0 x y ∂1 ∂d δy(X(t, x, w)) p, (n+2) 6 c1t− exp c2 | − | k ··· k − n − t o for each t (0,T ] and y R such that y > K. ∈ ∈ | | Here, n := k + + k in both cases. 1 ··· d Proof : (i) Assume (H1) and (H2). For simplicity of notation, we prove the case d = 1. The case d > 2 can be proved by a similar argument. It suﬃces to prove that there exist K > 0 and T (0,T ] with the following property: 0 ∈ For each n Z>0 and p (1, ), there exist constants ν0,c1,c2 > 0 such that ∈ ∈ ∞

(n) ν0 x y E[Jδy (Xt)] 6 c1t− exp c2 | − | J p,n+2, | | n − t ok k for each J D∞, t (0,T0] and y R with y > K. ∈ ∈ ∈ 2 | | 1 Let y R be arbitrary and ϕ(z) := (1 + z )− δy(z) S0. Take a C∞- function φ∈: R R such that −△ ∈ → 1 if ξ 6 1/3, φ(ξ)= 0 if ξ > 2/3 z y and then we set ψy(z) := φ x−y . Let p (1, ) be arbitrary and let q (1, ) | − | ∈ ∞ ∈ ∞ be such that 1/p +1/q = 1. Further take q′ (1, q). Since ψ δ = δ , we have for ∈ y y y each J D∞ that ∈ n (n) d 2 E[Jδy (Xt)] = E[Jψy(Xt) n (1 + z )ϕ (Xt)] dz −△ n+2 (3.6) (j) = E[ϕ(Xt) ψy (Xt)lj,t(J) ], n Xj=0 o (j) (0) where ψy is the j-th derivative of ψy (with convention that ψy = ψy) and each lj,t(J) is of the form n+2 P (t, w)J + P (t, w),DJ + + P (t, w),D J ⊗(n+2) (3.7) 0 h 1 iH ··· h n+2 iH i for some Pi(t, w) D∞(H⊗ ), i = 0, 1, ,n + 2, which is a polynomial in Xt = ∈ ··· 2 X(t, x, w), its derivatives and DX(t, x, w) − , but does not depend on ψ . k kH y Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 717

~~y x Note that ψy(z) 0 for z x < | −3 | , and hence the last term in (3.6) equals to ≡ | − | n+2 (j) E[ (ϕψy )(Xt)lj,t(J) 1 y x ]. {|Xt − x| > | − | } Xj=0 3~~

Therefore, by taking p′ (1, ) such that 1/p′ +1/q′ = 1, we have ∈ ∞ n+2 1/p′ (n) (j) y x E[Jδy (Xt)] 6 ϕψy lj,t(J) q′ P Xt x > | − | . (3.8) | | k k∞k k | − | 3 Xj=0 Henceforth we shall focus on each factor in the last equation. (j) j (j) First, since ψy (z) = x y − φ ((z y)/ x y ) for j = 0, 1, ,n + 2, and by Lemma 2.2 we have | − | − | − | ··· (j) sup ϕψy < + for K > 0 and j =0, 1, ,n + 2. (3.9) y R k k∞ ∞ ··· y ∈>K | | Second, it is well known that for each r (1, ), there exist ν,c1′ = c1′ (r) > 0 such that ∈ ∞ 2 ν DX − 6 c′ t− for any t (0,T ] k tkH r 1 ∈

(see Kusuoka and Stroock , 1985 , (3.25) Corollary p.22 or Ikeda and Watanabe, 1989, Chapter V, Section 10, Theorem 10.2). Now, bearing in mind the form (3.7), we see that there exist ν0,c1′′ > 0 such that

ν0 l (J) ′ 6 c′′t− J for j =0, 1, ,n + 2 and t (0,T ]. (3.10) k j,t kq 1 k kp,2 ··· ∈ Third, we shall prove that there exist K > 0 and T (0,T ] with the following 0 ∈ property: there exist c1′′′,c2 > 0 such that y x x y P Xt x > | − | 6 c1′′′ exp c2 | − | (3.11) | − | 3 n − t o for all t (0,T0] and y R with y > K. For this, we recall the following general fact (see∈Ikeda and Watanabe∈ , 1989| |, Chapter V, section 10, Lemma 10.5):

Let κ > 0 and X = (Xt)06t6T be a one-dimensional continuous semimartingale with its Doob-Mayer decomposition Xt = X0 +Mt+ A such that M = t α(s)ds, A = t β(s)ds and t h it 0 t 0 supR max α(t) , βR(t) 6 κ. 06t6T {| | | |} Then for any a> 0 and t (0, min a ,T ], it holds that ∈ { 2κ } 4 a2 P(τa 0 : X X >a . a { | t − 0| } Since we have assumed (H1), σ2 and b are Lipschitz continuous. Thus there exists α, β > 0 such that max σ2(y) , b(y) 6 α y x + β for any y R. Put {| | | |} | − | ∈ κ(z) := αz + β and ﬁx y R arbitrarily. Let T0 (0,T ] so that 1 6αT0 > 0. ∈ y ∈x − Deﬁne τ :=: τ y x /3 := inf t > 0 : Xt x > | −3 | . Then the stopped process τ | − | { | − | } X = (Xt τ )t>0 is a continuous semi-martingale satisfying ∧ t τ t τ ∧ ∧ Xt τ = x + σy(Xs τ )dw(s)+ by(Xs τ )ds, ∧ Z0 ∧ Z0 ∧ 718 T. Amaba and Y. Ryu

where σy(z) := min σ(z),κ( y x ) and by(z) := min b(z),κ( y x ) . y{ x | − | } { | − | }6βT0 By setting a = | − | in the above, we see that if y x > ξ0 := max , 1 > 3 | − | { 1 6αT0 } ξ0 ξ − 0 and 0 T0 since ξ is a nondecreas- 6(αξ0+β) 7→ 6(αξ+β) ing function) then

~~y x τa y x P Xt X0 > | − | 6 P Xt X0 > | − | | − | 3 | − | 3 6 P(τa~~

4√3 1 Thus (3.11) is satisﬁed if we set K := x + ξ0, c1′′′ := and c2 := (72(α + β))− . | | √πξ0 Now, combining (3.8), (3.9), (3.10) and (3.11), we obtain the result. (ii) Assume (H1), (H2) and (H4). In this case, σ and b are bounded, and hence the following well-known estimate is available: There exist c1′′′,c2 > 0 such that y x x y 2 P Xt x > | − | 6 c1′′′ exp c2 | − | (3.12) | − | 3 n − t o for all t (0,T ] and x, y R. By using this instead of (3.11), and combining with (3.8), (3.9∈) and (3.10), we∈ obtain the result.

~~We are now in a position to prove Theorem 3.4.~~

Proof of Theorem 3.4: (i) Let K > 0 and T0 (0,T ] be as in Lemma 3.10–(i). Since ∈ x y 2 ( x y )2 y2 2 xy + x2 | − | > | | − | | = − | | 1 as y + , y2 y2 y2 → | | → ∞ there exists K′ > 0 such that y x y > | |, for any y >K′. (3.13) | − | 2 | | d Let K′′ := max K,K′ . Let k N be such that Λ S 2k(R ). Let p (1, ) be arbitrary. We have{ } ∈ ∈ − ∈ ∞

T T0 T Λ(Xt) p, 2kdt = Λ(Xt) p, 2kdt + Λ(Xt) p, 2kdt, − − − Z0 k k Z0 k k ZT0 k k D 2k here, the last term is ﬁnite since (0,T ] t Λ(Xt) p− is continuous. The other term is estimated as ∋ 7→ ∈

~~T0 T0 Λ(Xt) p, 2kdt 6 δy(Xt) p, 2µ(dy)dt 6 I1 + I2, Z0 k k − Z0 ZR k k − Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 719 where~~

T0 I1 := δy(Xt) p, 2µ(dy)dt, Z0 Z y >K′′ k k − | | T0 I2 := δy(Xt) p, 2µ(dy)dt. Z0 Z y 6K′′ k k − | | We shall look at the integral I1. By Lemma 3.10–(i) and (3.13), there exist ν0,c1,c2 > 0 such that we have x y ν0 −c2 | − | ν0 y δy(Xt) p, 2 6 c1t− e t 6 c1t− exp c2 | | (3.14) k k − − 2t for y >K′′ and t (0,T ]. To dominate the last quantity, we shall prove that for | | ∈ 0 some c3 > 0, it holds that

ν0 y t− exp( c2 | | ) y K′′ − 2t ν0 6 ν0 y = t− exp c2 | | t− exp c2 0 exp( c2 | | ) − 4t − 4t → − 4t as t 0, which proves (3.15). Combining (3.14) and (3.15), we obtain ↓ T0 y I1 6 c1c3 exp c2 | | µ(dy)dt Z0 Z y >K′′ − 4t | | n o y 6 c1c3T exp c2 | | µ(dy)= c1c3T Λ,f < + Z y >K′′ − 4T h i ∞ | | n o c y /(4T ) where f S (R) is given by f(y) := φ(y)e− 2| | , y R and φ is a C∞-function ∈ ∈ such that φ = 0 on a neighbourhood of 0 and φ(y) = 1 if y > K′′. | | Next we turn to I2. By Proposition 3.9 and with noting that supa R δa 2 < + (Proposition 2.2), we have ∈ k k− ∞ 1/2 1/2 √t t sup δy(X(t, x, w)) p, 2 = t sup δy(X (1, x, w)) p, 2 y R k k − y R k k − ∈ ∈ √ = sup δ(y x)/√t(F ( t, x, w)) p, 2 6 C sup δa 2 y R k − k − a R k k− ∈ ∈ for each t> 0. Hence we can conclude that there exists C′ > 0 such that 1/2 δy(X(t, x, w)) p, 2 6 C′t− for y 6 K′′, t (0,T ]. k k − | | ∈ Now, it is easy to deduce that T 1/2 I2 6 C′µ( y R : y 6 K′′ ) t− dt< + . { ∈ | | } Z0 ∞ (ii) is proved similarly by using Lemma 3.10–(ii) instead of Lemma 3.10–(i).

d Recall that the support of Λ D ′(R ) is deﬁned as the complement of ∈ there exists an open neighbourhood y R : U of y such that for any f D(Rd),  .  ∈ suppf U Λ,f =∈ 0.  ⊂ ⇒ h i The proof of the following lemma will be given after Proposition 3.12 below. 720 T. Amaba and Y. Ryu

~~d d Lemma 3.11. Let x R and Λ D ′(R ) be such that suppΛ x. Then there ∈ ∈ 6∋ exists k Z> such that ∈ 0~~

lim Λ(X(t, x, w)) p, k =0 for each p (1, ), t 0 k k − ∈ ∞ ↓ if either one of the following holds: d (i) (H1), (H2) and Λ S ′(R ). ∈ d (ii) (H1), (H2), (H4) and Λ E ′(R ). ∈ From this, the following is immediate.

d d Proposition 3.12. Let x R and Λ D ′(R ) be such that suppΛ x. Under ∈ ∈ 6∋ the assumption in Lemma 3.11, there exists k Z> such that ∈ 0 T p Λ(X(t, x, w)) p, kdt< + for each p (1, ). Z0 k k − ∞ ∈ ∞ Proof of Lemma 3.11: We shall prove under the assumption (ii). The proof in the case (i) is omitted since one can prove similarly. We assume d = 1 just for simplicity. Let p (1, ) be arbitrary. By Propo- dk ∈ ∞ sition 2.4, we can write Λ = dxk [exp(k x )f(x)], where k is a nonnegative integer and f : R R is a bounded continuous| function.| Since x→ / suppΛ, we have r := inf x y : y suppΛ > 0. Let Ω := ∈ 0 {| − | ∈ } y suppΛBr /2(y) (i.e., the (r0/2)-neighbourhood of suppΛ), where Br(y) is the ∪ ∈ 0 open ball with center y and radius r. Then, we have Λ D ′(Ω), and the function f can be rearranged so that suppf Ω. ∈ ⊂ Now let ε > 0 and J D∞ be arbitrary. Putting ek(x) = ek( x ) = exp(k x ), we have ∈ | | | | e ε (k) ε E[Λ(X (1, x, w))J]= E[(ekf) (X (1, x, w))J] = E[exp(k Xε(1, x, w) )f(Xε(1, x, w))l (J)] | | e ε where l (J) D∞ is of the form ε ∈ k j l (J)= P (ε, w),D J ⊗j ε h j iH Xj=0

j for some Pj (ε, w) D∞(H⊗ ), j =0, 1, , k, which are polynomials in F (ε, x, w), ∈ ··· ε 2 its derivatives up to the order k, and DX (t, x, w) H− . Take q′ (1, q). Since F (ε, x, w) is uniformly non-degenerate,| there exists| c ,ν > 0 such∈ that { }ε>0 0 ν l (J) ′ 6 c ε− J for all ε (0,T ] and J D∞. k ε kq 0 k kq,k ∈ ∈

Therefore by taking p′ (1, ) such that 1/p′ +1/q′ = 1, we have ∈ ∞ E[Λ(Xε(1, x, w))J] ε ε 6 exp(k X (1, x, w ) )f(X (1, x, w)) ′ l (J) ′ k | | kp k ε kq ν ε ε 6 c ε− exp(k X (1, x, w) )f(X (1, x, w)) ′ J , 0 k | | kp k kq,k Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 721

~~ε ν ε ε which implies Λ(X (1, x, w)) p, k 6 c0ε− exp(k X (1, x, w) )f(X (1, x, w)) p′ for any ε> 0. Byk Proposition k3.7−, we obtain k | | k~~

~~√t Λ(X(t, x, w)) p, k = Λ(X (1, x, w)) p, k k k − k k − ν/2 √t √t 6 c t− exp(k X (1, x, w) )f(X (1, x, w)) ′ 0 k | | kp ν/2 r0 = c0t− exp(k Xt )f(Xt)1{|Xt − x| > } ′ . | | 2 p~~

ν′ By Lemma 3.10–(ii), we ﬁnd that for any r> 0, E[exp(r X(t, x, w) )] = O(t− ) as 2 r0| r| /(c t) t 0 for some ν′ > 0. However we have P( X x > )= O(e− 0 1 ) as t 0 ↓ | t − | 2 ↓ for some constant c1 > 0, so that the last quantity converges to zero as t 0, and hence get the conclusion. ↓

3.3. Hölder continuity of local time in space variable: a special case. In this section, we assume d = 1, (H1) and (H3). Let X = (Xt)t>0 be a unique strong solution to the following one-dimensional stochastic differential equation 1 dX = σ(X )dw(t)+ σ(X )σ′(X )dt, X = x R, (3.16) t t 2 t t 0 ∈ or equivalently, dX = σ(X ) dw(t), X = x. t t ◦ 0 The main purpose in this section is to prove Theorem 1.2. 2 t Note that the object σ(y) 0 δy(Xu)du in Theorem 1.2 is identified with the symmetric local time associatedR to the diffusion process (Xt)t>0. See Remark 4.15. The Hermite polynomials Hn, n Z>0 are defined by H0(x) = 1 and Hn(x) := n ∈ ∂∗ 1(x) for n N and x R, where the operator ∂∗ is given by ∈ ∈ ∂∗f(x) := f ′(x)+ xf(x), x R − ∈ for any differentiable function f : R R. The proof of Theorem 1.2 starts from→ this paragraph. Let y,z R be arbitrary. d ∈ Let A = Az := σ(z) dz and pt(z1,z2) be the transition-density function of X. For each t > 0, the Krylov-Veretennikov formula tells us that δa(Xt)= n∞=0 Jn[δa(Xt)] (the convergence is in D−∞) for every a R, where P 2 ∈

~~Jn[δa(Xt)] = Πn(x; t,a)[t1, ,tn]dw(t1) dw(tn) (3.17) Z06t <~~

= pt1 (x, z1)Az1 [pt2 t1 (z1,z2)] Azn [pt tn (zn,a)]dz1 dzn. ZR ··· ZR − ··· − ··· We remark that the stochastic integral in (3.17) is well deﬁned since the square- integrability of each component (t , ,t ) Π (x; t,a)[t , ,t ] of the chaos 1 ··· n 7→ n 1 ··· n kernel Πn(x; t,a) n∞=0 is established in Watanabe (1994b, Corollary 4.5), in a more { } 1 2 general situation. Since the generator L = 2 A commutes with A, we have

t L (t t )L (tn tn )L (t tn)L Π (x; t,a)[t , ,t ]=(e 1 Ae 2− 1 Ae − −1 Ae − )δ n 1 ··· n ··· a n n tL = [A exp( t1 + (t2 t1)+ + (tn tn 1) + (t tn) L)]δa = (A e )(δa) { − ··· − − − } 722 T. Amaba and Y. Ryu

~~tL in the distributional sense. By using (e δa)(x) = pt(x, a), the formula (3.17) reduces to~~

~~n Jn[δa(Xt)] = [Axpt(x, a)] dw(t1) dw(tn). Z06t <~~

~~1 1 ∞ n σ(a) δa(Xs)ds = [σ(a)Ax pt(x, a)] dw(t1) dw(tn)dt. Z0 Z0 Z06t <~~

~~Hence we have~~

~~1 1 2 σ(y) δy(Xt)dt σ(z) δz(Xt)dt Z − Z 2,s 0 0 ∞ 1 s n n = (1 + n) E [σ(y)A pt(x, y) σ(z)A pt(x, z)] Z x − x nX=0 n 0 2 dw(t1) dw(tn)dt , × Z06t <~~

~~1 2 n n In := E [σ(y)Ax pt(x, y) σ(z)Axpt(x, z)] dw(t1) dw(tn)dt Z0 − Z06t <~~

~~E dw(t1) dw(tn) dw(t1) dw(tn) × Z06t <~~

Since σ is Lipschitz continuous (which is because of (H1)), the vector field A = σ(z)(d/dz) is complete, so that one can associate the one-parameter group sA sA of diffeomorphisms e s R. Note that e (x) for each x R is defined by { } ∈ ∈ d esA(x)= σ(esA(x)) and esA(x) = x. ds |s=0 Lemma 3.13. d euA(x)= σ(x) ∂ euA(x) for every u R and x R. du ∂x ∈ ∈ Proof : By the homomorphism property: e(s+u)A =esA euA, we have d esA(x)= ◦ ds d (s+u)A d sA uA ∂ sA du u=0 e (x)= du u=0 e (e (x)) = σ(x) ∂x e (x).

To continue the calculation of In, we shall look at Axpt(x, a). The unique strong w(t)A solution X = (Xt)t>0 is now expressed by Xt =e (x) (To check this, just apply Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 723 the Itôformula for ew(t)A(x)). Therefore by using Lemma 3.13, we have w(t)A Axpt(x, a)= AxE[δa(e (x))] u2/2 ∂ √tuA e− = σ(x) δa(e (x)) du ∂x ZR √2π u2 /2 ∂ √tuA √tuA e− = σ(x) e (x) δa′ (e (x)) du ZR ∂x √2π u2 /2 1 d √tuA √tuA e− = e (x) δa′ (e (x)) du ZR √t du √2π u2/2 1 d √tuA e− = δa(e (x)) du √t ZR du √2π u2/2 √tuA 1 d e− = δa(e (x)) − du, ZR √t du √2π where the integral is understood as the coupling of the Schwartz distribution and the test function. By the repetition of the above procedure, we obtain n n u2/2 n √tuA ( 1) d e− Axpt(x, a)= δa(e (x)) −n/2 n du ZR n t du √2π o u2/2 √tuA 1 e− = δa(e (x)) Hn(u) du. ZR tn/2 √2π For each t > 0, define a mapping ϕ : R R by ϕ(u) := e√tuA(x). Since ϕ is → 1/2 continuously differentiable and ϕ′(u) = √t σ(ϕ(u)) > √tλ for every u, where λ> 0 is the constant appeared| in (H3),| we see| that |ϕ is a diffeomorphism. Hence 1 1 we can apply the change of variables b = ϕ(u) (and then du/db = (√tσ(b))− ) in the above integral to get (ϕ−1(b))2/2 n 1 1 e− Axpt(x, a)= δa(b) Hn(ϕ− (b)) db ZR t(n+1)/2 √2πσ(b) (ϕ−1(a))2/2 1 1 e− = Hn(ϕ− (a)) . t(n+1)/2 √2πσ(a) By using Lemma A.1–(i), we obtain the formula n (n+1)/2 n ( 1) t− ∞ n ξ2/2 iξϕ−1(a) A pt(x, a)= − (iξ) e− e dξ. (3.19) x 2πσ(a) Z −∞ Therefore by using (3.19), σ(y)Anp (x, y) σ(z)Anp (x, z) x t − x t n (n+1)/2 ( 1) t− ∞ n ξ2/2 iξϕ−1(y) iξϕ−1(z) = − (iξ) e− e e dξ (3.20) 2π Z − −∞ =: IIn. Lemma 3.14. For any α [0, 1] and θ R, we have eiθ 1 6 2 θ α with the convention 00 := 1. ∈ ∈ | − | | |

1 1 z da 1 when t = 1, we see that ϕ− (z) = Rx σ(a) and the stochastic process ϕ− (Xs), which is nothing but the Wiener process w(s), is known as the Lamperti transformation of X. 724 T. Amaba and Y. Ryu

Proof : Let α [0, 1]. If θ R ( 1, 1), then we see that eiθ 1 6 2 6 2 θ α. On the other hand,∈ for θ ( ∈1, 1),\ −eiθ 1 6 θ 6 θ α 6 |2 θ α− because| α | | [0, 1]. Thus we obtain eiθ ∈1 6−2 θ α for| all− θ| R|. | | | | | ∈ | − | | | ∈ Let β [0, 1) be arbitrary. By Lemma 3.14, we have ∈ (n+1)/2 t− 1 1 β ∞ n+β ξ2/2 IIn 6 ϕ− (y) ϕ− (z) ξ e− dξ | | π | − | Z | | −∞ (n+1)/2 (n+β 1)/2 t− 2 − n + β +1 1 1 β = Γ ϕ− (y) ϕ− (z) . π 2 | − | d 1 1 1 1 1/2 z 1 Since ϕ− (b)= , we have ϕ− (y) ϕ− (z) = t− σ(b)− db , so that db √tσ(b) | − | | y | R t (n+β+1)/22(n+β 1)/2 n + β +1 6 − − β (3.21) IIn β/2 Γ y z , | | πλ 2 | − | where it is recalled again that λ> 0 is what appeared in (H3). Substituting (3.21) into (3.20), σ(y)Anp (x, y) σ(z)Anp (x, z) | x t − x t | (n+β+1)/2 (n+β 1)/2 n + β +1 β 6 c2t− 2 − Γ y z 2 | − | β/2 1 where c2 := (πλ )− . Note that c2 does not depend on y and z. With putting n + β +1 c3(n) := Γ , 2 we have obtained

n n (n+β+1)/2 (n+β 1)/2 β σ(y)A p (x, y) σ(z)A p (x, z) 6 c t− 2 − c (n) y z . (3.22) | x t − x t | 2 3 | − | Now, by substituting (3.22) into (3.18), we have 2 1 1 (c2) n+β 2 2β n (n+β+1)/2 (n+β+1)/2 In 6 2 c3(n) y z s s− ds t− dt. n! | − | Z0 Zs Note that the last iterated integral is ﬁnite because β < 1, and gives 1 1 n (n+β+1)/2 (n+β+1)/2 2 s s− ds t− dt = . Z Z (1 β)(n β + 1) 0 s − − Hence we have (c )2 2n+β+1c (n)2 I 6 2 y z 2β 3 . n (1 β) | − | n!(n β + 1) − − Finally we have 1 1 ∞ 2 s σ(y) δy(Xt)dt σ(z) δz(Xt)dt = (1 + n) In Z − Z 2,s 0 0 nX=0

(c )2 ∞ 2n+β+1c (n)2 6 2 y z 2β (1 + n)s 3 . (1 β) | − | n!(n β + 1) − nX=0 − By Stirling’s formula, we see that the quantity 2n+β+1c (n)2 2n+β+1 n + β +1 2 (1 + n)s 3 =(1+ n)s Γ n!(n β + 1) n!(n β + 1) 2 − − Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 725 behaves like n+β+1 n+β−1 s 2 n + β 1 2 2 (1 + n) n π(n + β 1) − (n β + 1)√2πn( )n × − 2e − e p s n+β+1 (1 + n) 2 n + β 1 n+β 1 3 − s+β 2 = n π(n + β 1) − = O(n − ) (n β + 1)√2πn( )n − 2e − e 3 1 as n for each β. Hence the sum converges if s + β 2 < 1, i.e., s + β < 2 . The proof→ ∞ of Theorem 1.2 ﬁnishes. − − Proof of Corollary 1.3: This is clear from Theorem 1.2 and the inequality 1 1 σ(y) pt(x, y)dt σ(z) pt(x, z)dt Z − Z 0 0 1 1 = E[σ(y) δy(Xt)dt σ(z) δz(Xt)dt] Z − Z 0 0 1 1 6 σ(y) δy(Xt)dt σ(z) δz(Xt)dt 2, 1/2. k Z0 − Z0 k −

~~4. Itˆo’s Formula for Generalized Wiener Functionals~~

Let w = (w(t))t>0 be the d-dimensional Wiener process with w(0) = 0 and w w let ( ) > be the filtration generated by w: := σ(w(s):0 6 s 6 t), for Ft t 0 Ft t > 0. Similarly to the previous section, we fix x Rd and consider the following d-dimensional stochastic differential equation ∈ dX = σ(X )dw(t)+ b(X )dt, X = x Rd. (4.1) t t t 0 ∈ Also in this section, we assume the conditions (H1) and (H2). We denote by X(t, x, w) > a unique strong solution X = (X ) > to (4.1). { }t 0 t t 0

4.1. Stochastic integrals of pull-backs by diffusion. For each J D∞, we will de- i ∈ 1 d note by (DJ) the i-th component of DJ D∞(H): DJ = ((DJ) , , (DJ) ) (Recall H is the Cameron-Martin subspace∈ of the d-dimensional Wiener··· space). d For each t [0,T ], the evaluation map evt : H h h(t) R naturally ∈ ∋ 7→ d ∈ induces a map id evt : L2(H) = L2 H L2 R and then we write 1 ⊗ d d ∼ ⊗ → ⊗ DtJ :=: ((DtJ) , , (DtJ) ) := dt (id evt)(DJ) for a.a. t [0,T ]. d··· ⊗ ∈ Let Λ D ′(R ). Throughout this section, we assume either one of ∈ d (H1), (H2) and Λ S ′(R ); ◦ ∈ d (H1), (H2), (H4) and Λ E ′(R ). ◦ ∈ Then Λ(X(t, x, w)) is defined as a generalized Wiener functional. If T 2 N 0 Λ(X(t, x, w)) 2, kdt is finite for some k , we define the stochastic inte- k k − ∈ T i gralsR Λ(X(t, x, w))dw (t), i =1, , d as elements in D−∞ via the pairing 0 ··· R T T i i E[ Λ(X(t, x, w))dw (t) J]= E[Λ(X(t, x, w))(DtJ) ]dt, (4.2) Z0 Z0 for J D∞ and i = 1, 2, , d. We define the stochastic integral T ∈ ··· i 6 s Λ(X(t, x, w))dw (t) similarly for each 0

This pairing is well deﬁned because of the following: Proposition 4.1. For each k N and i =1, 2, , d, there exists a constant C > 0 such that ∈ ··· T T 1/2 i 2 E[Λ(X(t, x, w))(DtJ) ] dt 6 C Λ(X(t, x, w)) 2, kdt J 2,k+1 Z0 | | n Z0 k k − o k k for all J D∞. ∈ Proof : For simplicity of notation, we prove in the case d = 1. For each k N and ∈ J D∞, ∈ T T E[Λ(Xt)DtJ] dt 6 Λ(Xt) 2, k DtJ 2,kdt Z | | Z k k − k k 0 0 (4.3) T 1/2 T 1/2 2 2 6 Λ(Xt) 2, kdt DtJ 2,kdt . n Z0 k k − o n Z0 k k o By Meyer’s inequality, there exist constants c′, C′ > 0 such that k′ k′ l k′ c′ D F 6 F ′ 6 C′ D F , for all F D k k2 k k2,k k k2 ∈ 2 Xl=0 for k′ =1, 2, , k + 1. Therefore we have ··· k 2 k 2 2 l 2 l 2 D J 6 (C′) D D J 6 (C′) (k + 1) D D J k t k2,k k t k2 k t k2 n Xl=0 o Xl=0 k 2 l 2 = (C′) (k + 1) E[ D D J ⊗l ] k t kH Xl=0 k T T 2 2 = C′′ E[(DtJ) ]+ E[(Ds Ds DtJ) ]ds1 dsl , Z ··· Z l ··· 1 ··· n Xl=1 0 0 o 2 where C′′ := (C′) (k + 1), so that T 2 DtJ 2,kdt Z0 k k k+1 T T 2 6 C′′ E[(Du Du J) ]du1 dul Z ··· Z l ··· 1 ··· Xl=1 0 0 k+1 l 2 2 2 6 C′′ D J 6 (c′)− C′′ J . k k2 k k2,k+1 Xl=0 Hence by substituting this into (4.3), we get the result.

T i Remark 4.2. (a) As is easily seen, the stochastic integral 0 Λ(Xt)dw (t) has an- other expression: R T i • Λ(Xt)dw (t)= D∗[(0, , Λ(Xu)du, , 0)]. Z0 ··· Z0 ··· i-th position Hence it coincides with the Skorokhod integral| as long{z as the} object standing on the D1 right of D∗ lies in the domain 2(H) and then automatically coincides with the Itô Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 727 integral because of the adaptedness. In fact, in view of the Clark-Ocone formula, every J L can be written as J = E[J]+ d T E[(D J)i w]dwi(t) and then ∈ 2 i=1 0 t |Ft (4.2) is just the Itôisometry. Therefore, ourP stochasticR integral may be a natural extension of the classical anticipative stochastic integral to this distributional setting. T (k+1) (b) By Proposition 4.1, we have Λ(X )dwi(t) D− for i = 1, 2, , d 0 t ∈ 2 ··· T 2 R provided 0 Λ(Xt) 2, kdt< + . k k − ∞ R The following is the main result in this section. This is a version of a result by Uemura (2004, Proposition 1). R Ds Theorem 4.3. Let s , p > 2 and assume that (0,T ] t Λ(X(t, x, w)) p is continuous. Then we∈ have ∋ 7→ ∈ T i Ds Λ(X(t, x, w))dw (t) p, for i =1, 2, , d Z0 ∈ ··· provided either one of the following

(i) limt 0 Λ(X(t, x, w)) p,s =0. ↓ k k (ii) s > 0 and T Λ(X(t, x, w)) 2 dt< . 0 k kp,s ∞ R Remark 4.4. (a) See also a remark just after Lemma 4.12 for verification of the continuity assumption. (b) From the proof of Theorem 4.3, we would find that T i Ds Λ(Xt)dw (t) p for any t0 > 0 Zt0 ∈ and i =1, 2, , d if (0,T ] t Λ(X ) Ds is continuous. ··· ∋ 7→ t ∈ p The proof of Theorem 4.3 mainly consists of the following series of Proposi- tions 4.6, 4.7 and 4.8. We will give the proof at the last of this section. w Before the next definition, we note that E[F ] D∞ for every t > 0 if |Ft ∈ F D∞. ∈ Definition 4.5. Let t > 0. We say that a generalized Wiener functional F D−∞ w w ∈ is -measurable if it holds that E[F G]= E[F E[G ]] for any G D∞. Ft |Ft ∈ Proposition 4.6. Let s R and p > 2. Then there exists c = c(p,s,T ) > 0 such that, for any mapping F ∈: (0,T ] t F (t) Ds with F (t) is w-measurable for ∋ 7→ ∈ p Ft any t (0,T ], any division 0= t0

~~w for m > 1 (here we have used the condition that F (tk 1) is t -measurable), one − F k−1 ﬁnds that the chaos expansion is given by~~

∞ i i i i F (tk 1)(w (tk) w (tk 1)) = Jm 1[F (tk 1)](w (tk) w (tk 1)), − − − − − − − mX=1 where F (tk 1)= ∞ Jm[F (tk 1)] is the chaos expansion of F (tk 1). Hence, by − m=0 − − using (4.4), we haveP n s/2 i i (I ) F (tk 1)(w (tk) w (tk 1)) − L − − − Xk=2 n ∞ s/2 i i = (1 + m) Jm 1[F (tk 1)](w (tk) w (tk 1)) − − − − Xk=2 mX=1 n ∞ s/2 (2 + m) s/2 i i = (1 + m) Jm[F (tk 1)](w (tk) w (tk 1)). (1 + m)s/2 − − − mX=0 kX=2

By Meyer’s Lp-multiplier theorem (see e.g. Ikeda and Watanabe, 1989, Chapter V, Section 8, Lemma 8.2), there exists c′ = c′(p,s) > 0 such that n i i F (tk 1)(w (tk) w (tk 1)) p,s k − − − k Xk=2 n s/2 i i = (I ) F (tk 1)(w (tk) w (tk 1)) p k − L − − − k Xk=2 n ∞ s/2 i i 6 c′ (1 + m) Jm[F (tk 1)](w (tk) w (tk 1)) p k − − − k mX=0 Xk=2 n s/2 i i = c′ (I ) F (tk 1) (w (tk) w (tk 1)) . − L − − − p Xk=2

s/2 w Note that (I ) F (tk 1) L2 and is t -measurable. Hence the last quantity −L − ∈ F k−1 in the Lp-norm can be written as T n s/2 i (I ) F (tk 1) 1(t ,t ](t)dw (t). Z − L − k−1 k 0 Xk=2 Thus by using the Burkholder-Davis-Gundy inequality, we get n i i p F (tk 1)(w (tk) w (tk 1)) p,s k − − − k Xk=2 T n p/2 s/2 2 6 c′′E (I ) F (tk 1) 1(t ,t ](t)dt Z − L − k−1 k n 0 Xk=2 o for some constant c′′ > 0. Finally, using the assumption p > 2 and the Jensen inequality, we reached n n i i p p F (tk 1)(w (tk) w (tk 1)) p,s 6 c F (tk 1) p,s(tk tk 1) k − − − k k − k − − Xk=2 Xk=2 for some constant c> 0. Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 729

N n 2n Given n , i =1, 2, , d and the dyadic division tk := kT/2 k=0 of [0,T ], we define ∈ ··· { } 2n i i Φn := Λ(X(tk 1, x, w))(w (tk) w (tk 1)). − − − Xk=2 Proposition 4.7. Let s R, p > 2 and suppose that ∈ (i) (0,T ] t Λ(X(t, x, w)) Ds is continuous and ∋ 7→ ∈ p (ii) limt 0 Λ(X(t, x, w)) p,s =0. ↓ k k Then we have Φ Φ 0 as n,m . k n − mkp,s → → ∞ n m Proof : Suppose that n 0. By the assumption, the mapping (0,T ] t Λ(X ) Ds ∋ 7→ t ∈ p is uniformly continuous, from which, we easily get Φ Φ 0 as n . k n − mkp,s → → ∞ Proposition 4.8. Suppose k Z>0 and ∈ k (i) (0,T ] t Λ(X(t, x, w)) D− is continuous; ∋ 7→ ∈ 2 (ii) limt 0 Λ(X(t, x, w)) 2, k =0. ↓ k k − Then we have T i Φn Λ(X(t, x, w))dw (t) 2, (k+1) 0 as n . k − Z0 k − → → ∞ Proof : The same argument in Proposition 4.1 leads us to T i Φn Λ(Xt)dw (t) 2, (k+1) k − Z0 k − 2n t1 tk 1/2 6 2 2 Λ(Xt) 2, kdt + Λ(Xtk−1 ) Λ(Xt) 2, kdt . Z k k − Z k − k − n 0 kX=2 tk−1 o D k By the assumption, we have (0,T ] t Λ(Xt) 2− is uniformly continuous, and hence the above quantity converges∋ 7→ to zero as∈n . → ∞ Proof of Theorem 4.3: (i) Suppose that limt 0 Λ(Xt) p,s = 0. In the first, suppose ↓ k k that s < 0. Let k be the largest integer, not exceeding s, i.e., k = max k′ Z : { ∈ 6 T i k′ s . By the assumption, the stochastic integral 0 Λ(Xt)dw (t) is defined as } Dk 1 an element in 2− (see Remark 4.2–(b)). On theR other hand, Proposition 4.7 Ds tells us that Φn n∞=1 is a Cauchy sequence in p and hence converges to some s { } k 1 Φ D . Hence Φ converges to Φ also in D − . Now by Proposition 4.8, it must ∈ p n 2 T i T i Ds be Λ(Xt)dw (t) = limn Φn = Φ. Thus we have Λ(Xt)dw (t) p. 0 →∞ 0 ∈ Second,R suppose that s > 0. In this case,R it is clearly satisfied that T Λ(X ) p dt< + , and hence it reduces to the case (ii). 0 k t kp,s ∞ R 730 T. Amaba and Y. Ryu

(ii) Suppose that s > 0 and T Λ(X ) p dt < + . Then by Proposition 4.6, 0 k t kp,s ∞ we have R n 2 T p 6 p p Φn p,s const. Λ(Xtk−1 ) p,s(tk tk 1) Λ(Xt) p,sdt, k k k k − − → Z k k Xk=2 0 n n as n . Here, tk = kT/2 , k = 0, 1, , 2 . Hence Φn n∞=1 forms a bounded → ∞Ds ··· { } family in p, so that by Alaoglu’s theorem (for dual spaces of separable normed Ds spaces), there exists a subsequence Φnl l∞=1 and Φ p such that Φnl Φ weakly Ds { } ∈ → in p. In particular, since s > 0 and p > 2, this convergence is still valid in the weak topology on L . On the other hand, it is clearly satisﬁed that T Λ(X ) 2dt< , 2 0 k t k2 ∞ and hence Λ(Xt) t>0 is now a square-integrable ( t)t>0-adaptedR process. The { } F classical stochastic analysis proves that Φ T Λ(X )dwi(t) in L . Therefore, it n → 0 t 2 T i R T i Ds must be Λ(Xt)dw (t) = liml Φnl = Φ, and thus Λ(Xt)dw (t) p. 0 →∞ 0 ∈ R R

~~4.2. Distributional Itô’s formula. Let Ai, i =1, 2, , d and L be the vector fields and the second-order differential operator given by···~~

d k ∂f (Aif)(z) := σi (z) (z) ∂zk kX=1 d 2 d 1 j ∂ f i ∂f (Lf)(z) := (σσ∗) (z) (z)+ b (z) (z) 2 i ∂z ∂z ∂z i,jX=1 i j Xi=1 i d d for f S (R ) and z R . In the case of d = 1, the vector ﬁeld A1 will be denoted ∈ ∈ d by A. Under the assumption (H1), these operators naturally act on S ′(R ) and d E ′(R ). Theorem 4.9 (cf. Kubo, 1983). Let x Rd and assume (H1) and (H2). Then for d ∈ each Λ S ′(R ) and t (0,T ], we have ∈ 0 ∈ Λ(X(T, x, w)) Λ(X(t , x, w)) − 0 d T T i (4.5) = (AiΛ)(X(t, x, w))dw (t)+ (LΛ)(X(t, x, w))dt in D−∞. Z Z Xi=1 t0 t0 d Similarly, we have (4.5) for Λ E ′(R ) if we further assume (H4). ∈ Proof : For simplicity of notation, we assume d = 1. The case d > 2 is similar. Fix t0 > 0. Suppose that Λ S 2k. Then there exist φn S (R), n N such that ∈ − ∈ ∈ Λ = limn φn in S 2k. By Itˆo’s formula, we clearly have →∞ − T T

φn(XT ) φn(Xt0 )= (Aφn)(Xt)dw(t)+ (Lφn)(Xt)dt − Zt0 Zt0 for each n N. What we have to prove is the following: As n , ∈ → ∞ (a) Λ(Xt) φn(Xt) 2, 2k 0 for t = t0 and T , k T − k − → (b) [AΛ(Xt) Aφn(Xt)]dw(t) 2, (2k+2) 0, − k Zt0 − k → T (c) [LΛ(Xt) Lφn(Xt)]dt 2, (2k+2) 0. − k Zt0 − k → Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 731

It is easy to show (a). In fact, for any p (1, + ), we have the inequality ∈ ∞ K Λ(Xt) φn(Xt) p, 2k 6 c0t− Λ φn 2k (4.6) k − k − | − |− where the constants c0 > 0 and K > 0 can depend on p but not on t and φn’s (see Ikeda and Watanabe, 1989, Chapter V, Section 9, Theorem 9.1 and Section 10, Theorem 10.2). We shall prove (b). By Proposition 4.1 and (4.2), we have T T 2 2 [AΛ(Xt) Aφn(Xt)]dw(t) 2, (2k+2) 6 C [A(Λ φn)](Xt) 2, (2k+1)dt. k Zt0 − k − Zt0 k − k − Next we shall show that there exists c,ν > 0 such that 2 ν 2 [A(Λ φn)](Xt) 2, (2k+1) 6 ct− (Λ φn)(Xt) 4, 2k (4.7) k − k − k − k − for every t [t ,T ], and then the above quantities converge to zero uniformly in ∈ 0 t [t0,T ] as n , and hence (b) is proved. To prove (4.7), it suﬃces to show that:∈ there exist→c,ν ∞ > 0 such that ν E[(σΨ′)(Xt)J] 6 ct− Ψ(Xt) 4, 2k J 2,2k+1 (4.8) k k − k k for each Ψ S 2k and J D∞. In fact, we have ∈ − ∈ E[(σΨ′)(Xt)J]= E Ψ(Xt) P0(t)σ(Xt)J + P1(t),D σ(Xt)J H , n h i o i for some Pi(t) D∞(H⊗ ), i = 0, 1 which are polynomials in Xt = X(t, x, w), its ∈ 2 derivatives and DX − . Hence k tkH E[(σΨ′)(Xt)J]

6 Ψ(Xt) 4, 2 k P0(t)σ(Xt)J 4/3,2k + P1(t),D σ(Xt)J H 4/3,2k . k k − nk k kh i k o 1 1 3 Noting that 2 + 4 = 4 < 1, we can make estimates ν P (t)σ(X )J 6 c′t− J , k 0 t k4/3,2k k k2,2k ν P (t),D σ(X )J 6 c′t− J kh 1 t iH k4/3,2k k k2,2k+1 for each t > 0, and for some constants c′,ν > 0 (where, c′ may depend on the derivatives of σ up to the (2k + 1)-th order, which are assumed to be bounded by (H1)). Now (4.8) follows. (c) is proved similarly. The statement for Λ E ′(R) is also proved similarly. ∈ Theorem 4.10. Let x Rd. Suppose (H1), (H2) and (H4). Let f : Rd R be a locally-integrable function∈ such that → (i) f is continuous at x, d (ii) f E ′(R ), T∈ 2 (iii) 0 (Aif)(X(t, x, w)) 2, kdt< + for i =1, 2, , d, T k k − ∞ ··· (iv) R (Lf)(X(t, x, w)) 2, kdt< + 0 k k − ∞ for someR k N. Then we have ∈ f(X(T, x, w)) f(x) − d T T i (4.9) = (Aif)(X(t, x, w))dw (t)+ (Lf)(X(t, x, w))dt in D−∞. Z Z Xi=1 0 0 d The equation (4.9) still holds without (H4) if f S ′(R ). ∈ 732 T. Amaba and Y. Ryu

Proof : By the conditions (ii), (iii), (iv) and Theorem 4.9, we have for each t0 > 0 that d T T i f(XT ) f(Xt )= (Aif)(Xt)dw (t)+ (Lf)(Xt)dt − 0 Z Z Xi=1 t0 t0 D in −∞. Letting t0 0 with using (i), we have f(Xt0 ) f(x) in probability. Moreover, again by (i),↓ we can take δ > 0 such that for any→ y Rd, y x < δ 4 4 ∈ 4 | − | implies f(y) < f(x) +1. Thus E[f(Xt) ] 6 ( f(x) +1) +E[f(Xt) ; Xt x >δ], | | | | 4 | | | − | where lim supt 0 E[f(Xt) ; Xt x > δ] < + in view of (ii) and Lemma 3.10– ↓ 2 | − | ∞ 2 (ii). Therefore f(X ) 6 is L -bounded, so that (f(X ) f(x)) 6 is { t }0

4.3. An application and examples. In the sequel, we denote by X = (Xt)t>0 the unique strong solution to the d-dimensional stochastic differential equation dX = σ(X )dw(t)+ b(X )dt, X = x Rd, (4.10) t t t 0 ∈ where w = (w(t))t>0 is a d-dimensional Wiener process. We denote by L the 1 d i d i associated generator, i.e., L = 2 i,j=1(σσ∗)j ∂i∂j + i=1 b ∂i. d Let Fε ε I D∞(R ) be a boundedP and uniformlyP non-degenerate family (see { } ∈ ⊂ Definition 2.1). Here, the boundedness is used in the sense of Fε ε I is bounded k d { } ∈ in D (R ) for each p (1, ) and k Z> . p ∈ ∞ ∈ 0 Lemma 4.12. For any s R, p (1, ) and p′ >p, there exists c = c(s,p,p′) > 0 such that ∈ ∈ ∞ s/2 Λ(Fε) p,s 6 c (1 ) Λ Rd k k k −△ kLp′ ( ,dx) d s/2 d for every ε I and Λ S ′(R ) with (1 ) Λ L ′ (R , dx). ∈ ∈ −△ ∈ p 1 2 We note that one can take p′ = p when Fε = ε− w(ε T ) as mentioned in Remark A.5. We don’t give a proof of Lemma 4.12. However, we prove a similar inequality (Lemma A.4) in Section A.3. The same techniques there are available to show Lemma 4.12 if we replace the harmonic oscillator H by the Bessel potential (1 ). (Although the step (b) in the proof of Lemma A.4 uses results by Bongioanni−△ and Torrea (2006), we don’t need for the proof of Lemma 4.12. The steps (a) and (b) for the proof of Lemma 4.12 can be established analogously by a standard integration- by-parts techniques in the Malliavin calculus. The steps (c)–(e) also run in exactly the same way.) s d s/2 d Let H (R ) := (1 )− L (R , dz), p (1, ), s R be the Bessel potential p −△ p ∈ ∞ ∈ spaces (see Abels, 2012 and Krylov, 2008 for details). By imitating the proof of Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 733

Corollary 2.8 with using Lemma 4.12 (instead of Proposition 2.6), we can show that s d s for each p (1, ), s R and Λ H (R ), the mapping (0, ) t Λ(X ) D ′ ∈ ∞ ∈ ∈ p ∞ ∋ 7→ t ∈ p is continuous for p′ (1,p) under assumptions (H1) and (H2). We are now in a∈ position to prove Corollary 1.6. Proof of Corollary 1.6: (i) is clear by Lemma 4.12. (ii) The operator L is a uniformly elliptic operator of the second order satisfying (H4). Hence the elliptic regularity theorem (see e.g., Abels, 2012, Chapter III, 1 s+2 d k Section 7.3, Theorem 7.13) assures that f := (1 L)− Λ H (R ) and σ ∂ f − ∈ p j k ∈ Hs+1(Rd) for each j, k =1, 2, , d. On the other hand, Theorem 4.9 gives p ··· d T T k j f(XT ) f(Xt )= (σ ∂kf)(Xt)dw (t)+ (Lf)(Xt)dt. − 0 Z j Z j,kX=1 t0 t0 s+2 Let p′ [2,p) be arbitrary. We ﬁnd that f(X ) D ′ by (i) and ∈ T ∈ p T k j Ds+1 (σ ∂kf)(Xt)dw ′ for T > 0 by Theorem 4.3, so that we have t0 j ∈ p R T Ds+1 Ds+2 (Lf)(Xt)dt ′ . Furthermore, since [t0,T ] t f(Xt) ′ is continu- t0 ∈ p ∋ 7→ ∈ p Rous (recall the remark just before the proof), this mapping is Bochner integrable T s+2 and f(Xt)dt D ′ . Therefore t0 ∈ p R T T T Ds+1 Λ(Xt)dt = f(Xt)dt (Lf)(Xt)dt p′ . Zt0 Zt0 − Zt0 ∈

~~Remark 4.13. By the above remark (just after Lemma 4.12), one would see that in Corollary 1.6, we can take p′ = p when σ = (identity matrix) and b = 0, i.e., Xt = w(t).~~

s T Second, we investigate the class D to which Λ(Xt)dt belongs when t0 = 0 p t0 d for several cases of Λ S ′(R ). R ∈ Example 4.14. Assume d = 1, (H1), (H3) and (H4). Let x z 2b(η) s(x) := exp dη dz, Z − Z σ(η)2 0 x n 0z o for x R. 2b(η) dz ∈ m(x) := 2 exp 2 dη 2 , Z0 n Z0 σ(η) o σ(z) The function s(x) is called the scale function of L and the measure m(dx) = m′(x)dx is called the speed measure of L. We ﬁx y R and deﬁne u : R R by ∈ → m (y) u(x) :=: u(x, y) := ′ s(x) s(y) , x R. (4.11) 2 | − | ∈ Then it is easily checked that Lu = δy and x 2b(η) σ(x) (Au)(x) = sgn(x y) exp dη , x R. − − Z σ(η)2 σ(y)2 ∈ n y o in the distributional sense. Now we shall prove 1 Ds s< 2 ; 1 X(t,x,w)

~~1 In fact, let s (0, 1/2) and p (1, s ). Take p′ > p so that sp′ < 1. Then by Proposition 3.7 and∈ Lemma A.4, there∈ exists c> 0 such that~~

~~1 X(t,x,w)1 p Xt 2b(η) σ(Xt) D exp 2 dη 2 p∞ for all p (1, ). n − Zy σ(η) o σ(y) ∈ ∈ ∞~~

From this and (4.12), we find lim supt 0 (Au)(Xt) p,s < for s < 1/2 and p ↓ k k ∞ ∈ (1, 1 ). Then by Theorem 4.3–(ii), T (Au)(X )dw(t) Ds for s< 1 and p [2, 1 ), s 0 t ∈ p 2 ∈ s and thus we reached: under (H1),R (H3) and (H4), T s 1 1 δy(Xt)dt D for s< and p 2, . Z ∈ p 2 ∈ s 0 See Airault et al. (2000) for a more general and stronger result. T Remark 4.15. In particular, we see from Example 4.14 that 0 δy(Xt)dt is a classical Wiener functional when d = 1, which is related to the localR time at y. This can be seen as follows: Classically, the symmetric local time l(y,t): y R,t > 0 is defined as a unique increasing process such that { ∈ } e t Xt y = x y + sgn(Xs y)dXs + l(y,t) | − | | − | Z0 − and equivalently given by e 1 t l(y,t) = lim 1(y ε,y+ε)(Xs)d X s ε 0 2ε − h i ↓ Z0 e 1 2 (see Revuz and Yor, 1999). Since it holds limε 0(2ε)− σ 1(y ε,y+ε) = σδy = σ(y)δy ↓ − in S ′(R), we have t 2 l(y,t)= σ(y) δy(Xu)du. Z0 e s s ε In the sequel, we denote D − := D − . 2 ∩ε>0 2 Example 4.16. Assume d = 1, (H1), (H3) and (H4). Let x, y R be such that x = y. ∈ 6 k By using Lemma 4.12, we see that the mapping (0,T ] t δ′ (X(t, x, w)) D− ∋ 7→ y ∈ 2 is continuous for some k Z> . ∈ 0 Define u(z1,z2) by (4.11), and then v(z) := (∂ u)(z,y) − z2 1 = m′(y)s′(y)sgn(z y) m′′(y) s(z) s(y) E ′(R) 2n − − | − |o ∈ Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 735

satisfies Lv = δy′ and σ(z) (Av)(z)= 2m′(y)s′(y)δy(z) m′′(y)s′(z)sgn(z y) . 2 − − Since Xt = X(t, x, w) is non-degenerate for each t > 0, it is known by Watanabe Ds 1 1 (1991) that δy(Xt) p if s ( 1, 2 ) and p (1, 1+s ). Hence by virtue of con- ∈ ∈ − − ∈ T ditions (H1) and (H3), for each t0 (0,T ) and p > 2, we find (Au)(Xt)dw(t) t0 1 ∈ ∈ ( p 1) R Dp − −, and thus T ( 1 1) D p − − δy′ (Xt)dt p for any t0 > 0 and p > 2. Zt0 ∈ Example 4.17. In the above Example 4.16, we shall try to investigate the class T > to which 0 δy′ (Xt)dt belongs. Let p 2 be arbitrary. Since δy(x + ε ) = 1/2 2 1 2 1/2 y• x lima 0(2πaR)− exp (x + ε y) /(2a) = ε− lima 0(2πε− a)− exp ( −ε 2 ↓ 2 1 {− •− } ↓ {− − ) /(2ε− a) = ε− δ(y x)/ε( ) in the distributional sense, we have • } − • 1 σ(Xt)(m′s′)(y)δy(Xt) = (σm′s′)(y)ε− δ(y x)/ε(Fε), − where ε := √t, X = X(t, x, w) and F = (X x)/ε. We shalle prove t ε t − 1 ε− δ(y x)/ε(Fε) p,s 0e as ε = √t 0 for every s< 1. (4.13) k − k → ↓ − Note that the probability densitye function p e of F has an estimate sup p e (z) Fε ε 0<ε61 Fε 1/2 z2/(2c′) 6 Cp(z) for some C > 0, where p(z) := (2πc′)− ee− and c′ > 0 is a constant. By refining the argument in the proof of Lemma A.4, one finds that for every p′ >p, s/2 δ(y x)/ε(Fε) p,s 6 const. (1 ) δ(y x)/ε L ′ (R,µ) k − k k −△ − k p for all ε (0, 1], where µe(dz) = p(z)dz. To estimate the last quantity, we apply the Fourier∈ transformation and integration by parts to get iξz 1 ∞ e y−x 2 s/2 2 iξ ε ε− (1 ) δ(y x)/ε(z)= − 2 s/2 ε− e dξ −△ − 2π Z (1 + ξ )− −∞ 2 iξz 1 ∞ d e y−x iξ ε = 2 2 2 s/2 e dξ, 2π(y x) Z ndξ (1 + ξ )− o − −∞ in which, one finds that there exists c = c(s) > 0 such that d2 eiξz 6 c(1 + z2)(1 + ξ2)s/2. dξ2 (1 + ξ2) s/2 −

~~Hence we obtain~~

2 s/2 c 2 ∞ 2 s/2 ε− (1 ) δ(y x)/ε(z) 6 (1 + z ) (1 + ξ ) dξ. | −△ − | 2π(y x)2 Z − −∞ Note that ∞ (1 + ξ2)s/2dξ < + since s< 1. We thus have ∞ − R−∞ 2p′ s/2 p′ 2 p′ sup ε− (1 ) δ 6 c′ (1 + z ) p(z)dz < + , (y x)/ε L ′ (R,µ) ε>0 k −△ − k p ZR ∞ where the constant c′ > 0 may depend on y x, and which gives (4.13). − 736 T. Amaba and Y. Ryu

By (H1), (H3) and (H4), we see that limt 0 σ(Xt) = σ(x) in D∞. Combining ↓ T ( 1) D − − this with (4.13) and Theorem 4.3–(i), we obtain 0 (σδy)(Xt)dw(t) p . On T ( 1)∈ the other hand, we have (σs′)(X )sgn(X y)dRw(t) L Dp− −. Therefore 0 t t − ∈ 2 ⊂ T R (Au)(Xt)dw(t) Z0 T T m′′(y) D( 1) = m′(y)s′(y) (σδy)(Xt)dw(t) (σs′)(Xt)sgn(Xt y)dw(t) p− −. Z0 − 2 Z0 − ∈ Hence by Theorem 4.10, T D( 1) δy′ (Xt)dt p− − for any p > 2. Z0 ∈ Example 4.18 (Cauchy’s principal value of 1/x). Let d = 1. The tempered distri- 1 bution p.v. x is defined by 1 f(x) p.v. ,f := lim dx, f S (R). h x i ε 0 Z x >ε x ∈ ↓ | | Let Λ S ′(R) be the regular distribution (i.e., a distribution associated with a locally∈ integrable function) given by Λ = x log x x. Then the distributional 1 | |− derivatives are: Λ′ = log x and Λ′′ = p.v. x . | | 1 We shall first show the Bochner integrability of the pull-back of p.v. x by a 1 Brownian motion w(t): (0,T ] t p.v. (w(t)). For each J D∞and p,q > 1 ∋ 7→ x ∈ such that 1/p +1/q = 1, we obtain 1 E[ p.v. (w(t))J]= E[ x log x x ′′(w(t))J] x | |− = E[ w(t) log w(t) w(t) l (J)] | |− t 6 w (t) log w(t) w(t) l (J) . k | |− kpk t kq 2 i D D i Here lt(J) = i=0 Pi(t),D J H⊗i ∞ for some Pi(t) ∞(H⊗ ), i = 0, 1, 2 h i ∈ ∈2 1 which are polynomialsP in w(t), its derivatives and Dw(t) − = t− . Since we have k kH x2/2 p p e− w(t) log w(t) p = (√tx) log(√t x ) dx k | |k ZR | | √ 2π x2/2 x2/2 e p e 6 2p √t log √t p x p − dx +2ptp/2 x log x − dx, | | ZR | | √ ZR | | √ 2π 2π 1 which tends to zero as t 0, and hence lim supt 0 p.v. x (w(t)) p, 2 < + for ↓ 1 ↓ k 2 k − ∞ p > 2. This proves that (0,T ] t p.v. (w(t)) D− is Bochner integrable. ∋ 7→ x ∈ 2 On the other hand, it is clear that T log w(t) 2dt< + . 0 k | |k2 ∞ Now, by Theorem 4.10, we have R T 1 T 1 w(T ) log w(T ) = log w(t) dw(t)+ p.v. (w(t))dt. (4.14) | | Z0 | | 2 Z0 x The chaos expansion of log w(1) is given by | | x2/2 ∞ 1 1 e− log w(1) = E[log w(1) ]+ lim Hn 1(x) dxHn(w(1)). | | | | n! ε 0 Z x >ε x − √2π nX=1 ↓ | | Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 737

1/2 2 1 Put µ(dx) := (2π)− exp( x /2)dx. It is clear that limε 0 x >ε x H1(x)µ(dx)= − ↓ | | R µ(dx) =1. For n > 2, by using Lemma A.1–(ii), we have R R 1 lim Hn(x)µ(dx) ε 0 Z x >ε x ↓ | | 1 = lim xHn 1(x) (n 1)Hn 2(x) µ(dx) ε 0 Z x >ε x{ − − − − } ↓ | | 1 = Hn 1(x)µ(dx) (n 1) lim Hn 2(x)µ(dx) ZR − − − ε 0 Z x >ε x − ↓ | | 1 = (n 1) lim Hn 2(x)µ(dx). − − ε 0 Z x >ε x − ↓ | | Therefore if n =2k + 1, then 1 lim Hn(x)µ(dx) ε 0 Z x >ε x ↓ | | 1 = ( 2k)( (2k 2))( (2k 4)) ( 2) lim H1(x)µ(dx) − − − − − ··· − · ε 0 Z x >ε x ↓ | | = ( 1)k(2k)!!. − 1 Similarly, if n is even, limε 0 x− Hn(x)µ(dx) = 0. Thus we get ↓ x >ε R| | ∞ (2n)!! log w(1) = E[log w(1) ]+ ( 1)n H (w(1)), | | | | − (2n + 2)! 2n+2 nX=0 from which, we find that log w(t)/√t 2 = log w(1) 2 k | |k2,s k | |k2,s ∞ ((2n)!!)2 = E[log w(1) ]2 + (1 + n)s . | | (2n + 2)! nX=0 Noting (2n)!! = 2nn!, the Stirling formula tells us √ 1 log w(t)/ t 2,s = log w(1) 2,s < + iff s< 2 . ◦A k similar| computation|k k shows| that|k ∞ ◦ 1 1 (p.v. )(w(1)) < + iff s< . k x k2,s ∞ −2 Now, for s< 1/2, we have T T 2 2 2 log w(t) 2,sdt 6 4 (log √t) dt + T log w(1) 2,s < + , Z0 k | |k Z0 k | |k ∞ T (1/2) so that log w(t) dw(t) D − by Theorem 4.3–(ii). Hence by (4.14), 0 | | ∈ 2 R T 1 D(1/2) p.v. (w(t))dt 2 −. Z0 x ∈ Example 4.19. Let d > 2 and x, y Rd. Suppose (H1), (H3) and (H4). Since δ Hs(Rd) if s< (p 1)d/p (Lemma∈ A.6), we find from Corollary 1.6, y ∈ p − − T (p−1)d (1 p ) δy(Xt)dt Dp − − for t0 > 0 and p > 2, Zt0 ∈ 738 T. Amaba and Y. Ryu

where Xt = X(t, x, w). Assume that x = y, and then we shall further investigate the class to which T 6 1 d δ (X )dt belongs. Let f := (1 L)− δ and let φ : R R be a C∞-function 0 y t − y → Rsuch that (1) x / suppφ, (2) φ 1 on a neighbourhood of y, and (3) suppφ is compact. By Theorem∈ 4.10, we have≡ d T T i j (φf)(XT ) (φf)(x)= (σ ∂i(φf))(Xt)dw (t)+ (L(φf))(Xt)dt, − Z j Z i,jX=1 0 0 in which we note that

L(φf) = (Lφ)f + σ φ, σ f Rd + φf δ h ∇ ∇ i − y d and (Lφ)f, σ φ, σ f Rd S (R ). By Lemma A.7–(ii), h ∇ ∇ i ∈ p (1, ); limt 0 (φf)(Xt) p,s = 0; ∈d ∞ p ↓ k k 2 2, so that 0 j i t ∈ p T Rp ( p−1 d) δy(Xt)dt Dp − − for any d > 2 and p > 2. Z0 ∈ Remark 4.20. In Examples 4.16 and 4.19, the conclusions seem not the best possible 1 T T ( p 1) given p > 2, because recalling that δ′ (Xt)dt and δy(Xt)dt belong to Dp − − t0 y t0 (p−1)d (1 p ) R R and Dp − − respectively for each t0 > 0 (Note that Xt’s in each case are T diﬀerent though we are using the same symbol), it is natural to ask that 0 δy′ (Xt)dt and T δ (X )dt also do. To get further results for p (1, 2), one wouldR have to 0 y t ∈ investigateR Theorem 4.3 for p (1, 2), which we could not. ∈ Appendix A. Auxiliary Lemmas

A.1. Some knowledge of Hermite polynomials. The Hermite polynomials Hn, n n ∈ Z> is defined by H (x) = 1 and H (x) := ∂∗ 1(x) for n N and x R, where 0 0 n ∈ ∈ ∂∗f(x) := f ′(x)+ xf(x), x R − ∈ for any differentiable function f : R R. The Hermite functions are now defined by → x2 φ (x) := H (√2x)e− 2 , x R and n Z> . n n ∈ ∈ 0 Some facts about H ∞ and φ ∞ are summarized as follows. { n}n=0 { n}n=0 Lemma A.1. (i) For each n Z> , ∈ 0 x2 2 n 2 n + 2 n x d x ( 1) e 2 ∞ n ξ iξx Hn(x) = ( 1) e 2 e− 2 = − (iξ) e− 2 e dξ. − dxn √2π Z −∞ (ii) Hn′ = nHn 1 and Hn(x)= xHn 1(x) (n 1)Hn 2(x). 1 − − − − − 1/2 x2/2 (iii) H > is a complete orthonormal basis of L (R, (2π)− e− dx). { √n! n}n 0 2 Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 739

~~1/2 (iv) (√πn!)− φn n>0 is a complete orthonormal system of L2(R, dx). { } 2 2 Z d (v) (1 + x )φ = 2(n + 1)φ for n > , where = 2 . −△ n n ∈ 0 △ dx~~

A.2. Some knowledge of Heaviside function. We begin with introducing some re- 2 d sults by Bongioanni and Torrea (2006). Let H := x , A := dx + x and d − △ B := dx + x. From the point of view of Lemma A.1–(iii), (iv), A family of linear operators− (λ + H)s, s R and λ > 0 can also be deﬁned by the relation s λ+1 s ∈ (λ + H) φ = (2(n + )) φ for n Z> . n 2 n ∈ 0 Then for s< 0, the operator Hs/2 has an integral representation (see Bongioanni and Torrea, 2006, Proposition 2)

~~s/2 (H f)(x)= Ks/2(x, y)f(y)dy for f S (R), (A.1) ZR ∈ where the kernel Ks/2(x, z) has an estimate Ks/2(x, y) 6 cΦs/2( x y ) for some constant c> 0 and | − |~~

(1+s) x2/4 x − 1 x <1 +e− 1 x >1 if s> 1,  | | {| | } x2/{|4 | } − Φs/2(x)= (1 log x )1 x <1 +e− 1 x >1 if s = 1,  − | | {|x2/|4 } {| | } − 1 x <1 +e− 1 x >1 if s< 1. {| | } {| | } −  Lemma A.2. If s ( 1, 1 ) and p (1, 1 ), Hs/2δ (x) L (R, dx). ∈ − − 2 ∈ 1+s y ∈ p Proof : Since s> 1, we have − s/2 (H δy)(x)= Ks/2(x, y) 2 (1+s) |x−y| 6 c x y − 1 x y <1 +e− 4 1 x y >1 . | − | {| − | } {| − | } Hence we can easily conclude the result.

By using Bongioanni and Torrea (2006, Theorem 4, Lemma 4, Theorem 7) and Lp-multiplier theorem (see Thangavelu, 1993, Chapter 4, Section 4.2, Theorem 1 s/2 4.2.1) for operators [H− (H + 2)] , s R, we can deduce that for each s R, ∈ ∈ there exists a constant c1 = c1(s) > 0 such that

(s+1)/2 H f Lp(R,dx) k k (A.2) s/2 s/2 s/2 6 c1 H φ Lp(R,dx) + xH f Lp(R,dx) + H f Lp(R,dx) k k k k k k for all f = f(x) S (R), where φ := Af. By a standard argument, this inequality ∈ s/2 s/2 extends and is still valid for f = f(x) S ′(R) such that (H Af), (H f), ∈ (xHs/2f) L (R, dx). ∈ p Proposition A.3. Let y R. For each s< 1/2 and p (1, 1/s), we have ∈ ∈ s/2 (i) H 1( ,y) Lp(R, dx), −∞ s/∈ 2 (ii) limy H 1( ,y) L (R,dx) =0 and →−∞ p ks/2 −∞ k (iii) supy R H 1( ,y) Lp(R,dx) < . ∈ k −∞ k ∞ 740 T. Amaba and Y. Ryu

Proof : Let f(x) := 1( ,y)(x) and φ(x) := Af(x)= δy(x)+ x1( ,y)(x). Then by (A.2), we have −∞ −∞ s/2 H 1( ,y) Lp(R,dx) k −∞ k (s 1)/2 (s 1)/2 6 2c1 H − δy Lp(R,dx) + H − (x1( ,y)) Lp(R,dx) k k k −∞ k (s 1)/2 (s 1)/2 + xH − 1( ,y) Lp(R,dx) + H − 1( ,y) Lp(R,dx) , k −∞ k k −∞ k which is ﬁnite by virtue of Lemma A.2 and by a similar argument in Lemma A.2 with using the integral expression (A.1). Furthermore again by (A.1), we ﬁnd also that the last four terms converge to zero as y , and uniformly bounded in y R. → −∞ ∈

A.3. Fractional inequalities. Let Fε ε I D∞ be a bounded and uniformly non- degenerate family (see Deﬁnition{2.1).} ∈ Here,⊂ the boundedness is used in the sense Dk Z of Fε ε I is bounded in p for each p (1, ) and k >0. { } ∈ ∈ ∞ ∈ Lemma A.4. For any s R, p (1, ) and p′ > p, there exists c = ∈ ∈ ∞ c(s,p,p′, Fε ε I ) > 0 such that { } ∈ 2 s/2 Λ(F ) 6 c (x ) Λ R k ε kp,s k −△ kLp′ ( ,dx) 2 s/2 for every ε I and Λ S ′(R) with (x ) Λ L ′ (R, dx). ∈ ∈ −△ ∈ p See also Remark A.5. The following proof is based on the technique in Watanabe (1991). Proof : We show in the case of 2 6 s 6 1. Other cases are similarly proved. Deﬁne a linear operator T (ε) for− 2 6 α 6 1 and ε I by α − ∈ α/2 2 α/2 T (ε)φ := (I ) (x )− φ (F ) α − L −△ ε for φ = φ(x) S (R). Since Fε ε I is uniformly non-degenerate, the density function p (x)∈ is uniformly bounded{ } ∈ in (ε, x) I R: Fε ∈ ×

c0 := sup sup pFε (x) < . ε I x R ∞ ∈ ∈ Take p′ >p> 1 arbitrary. We divide the proof into ﬁve steps. (a) In the ﬁrst place, when α = 2, we shall show that T 2(ε): Lp′ (R, dx) D0 − − → p = Lp and is a continuous linear operator with an estimate

T 2(ε) L ′ (R,dx) Lp 6 c1 for any ε I k − k p → ∈ for some constant c > 0. Let φ S (R) be arbitrary. Then 1 ∈ 1 2 T 2(ε)φ p = (I )− (x )φ (Fε) p = (Hφ)(Fε) p, 2, k − k k − L −△ k k k − 2 where H := x , and for each J D∞ and p′′ (p,p′), we have −△ ∈ ∈ 2 E[(Hφ)(F )J]= E[((x )φ)(F )J] 6 φ(F ) ′′ l (J) ′′ , ε −△ ε k ε kp k ε kq 2 i where 1/p′′ +1/q′′ = 1 and lε(J) is of the form lε(J) = i=0 Pi(ε, w),D J H⊗i i h i for some Pi(ε, w) D∞(H⊗ ), i = 0, 1, 2, all of which areP polynomials in Fε, its ∈ 2 derivatives up to the second order and DFε − . Since Fε ε I is uniformly non- k kH { } ∈ degenerate, there exists c′ > 0 such that l (J) ′′ 6 c′ J . Hence we have 0 k ε kq 0k kq,2 E[(Hφ)(Fε)J] 6 c′ φ(Fε) p′′ J q,2 6 c0c′ φ R J q,2 0k k k k 0k kLp′ ( ,dx)k k Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 741

D for each J ∞, which implies T 2(ε)φ p 6 c1 φ L ′ (R,dx), where c1 := c0c0′ . ∈ k − k k k p Since S (R) is dense in Lp′ (R, dx), we obtain the desired estimate. (b) Next, we focus on the case of α = 1. We shall show that this operator R D0 actually deﬁnes a continuous linear mapping T1(ε): Lp′ ( , dx) p = Lp with an estimate →

T1(ε) L ′ (R,dx) Lp 6 c2 for any ε I, k k p → ∈ for some constant c > 0. Let φ = φ(x) S (R) be arbitrary. Then 2 ∈ 1/2 2 1/2 1/2 T1(ε)φ Lp = (I ) (x )− φ (Fε) = (H− φ)(Fε) p,1. k k − L −△ Lp k k

By Meyer’s inequality, there exists a positive constant c2′ > 0 such that J p,1 6 1 k k c′ ( J + DJ ) for every J D . Hence we have 2 k kLp k kLp(H) ∈ 2 1/2 6 1/2 1/2 (H− φ)(Fε) p,1 c2′ (H− φ)(Fε) Lp + (H− φ)′(Fε)DFε Lp(H) . k k k k k k 1/2 2 1/2 We easily have (H− φ)(Fε) L 6 c0 (x )− φ R . Take q′ (1, ) k k p k −△ kLp′ ( ,dx) ∈ ∞ so that 1/p′ + 1/q′ = 1/p. Then there exists c2′′ = c2′′(p′, q′) > 0 such that J J 6 c′′ J J for all (J ,J ) L ′ L ′ (H). Hence we k 1 2kLp(H) 2 k 1kLp′ k 2kLq′ (H) 1 2 ∈ p × q have 1/2 (H− φ)′(F )DF k ε εkLp(H) 2 1/2 6 c′′ [(x )− φ]′(F ) DF 2 k −△ ε kLp′ k εkLq′ (H) 2 1/2 6 R c2′′c0 sup DFε Lq′ (H) [(x )− φ]′ Lp′ ( ,dx). ε I k k k −△ k ∈ By a result by Bongioanni and Torrea (2006, Lemma 3 and Theorem 4), there exists a constant c2′′′ = c2′′′(p′) > 0 such that 2 1/2 2 1/2 [(x )− φ]′ R + (x )− φ R k −△ kLp′ ( ,dx) k −△ kLp′ ( ,dx) d 2 1/2 2 1/2 6 R R + x (x )− φ Lp′ ( ,dx) + (x )− φ Lp′ ( ,dx) kdx −△ k k −△ k 2 1/2 + x(x )− φ R k −△ kLp′ ( ,dx) 6 c′′′ φ R , 2 k kLp′ ( ,dx) and hence we have obtained

T (ε)φ 6 c φ R for all ε I k 1 kLp 2k kLp′ ( ,dx) ∈ as desired, where c2 := c0c2′ (1 + c2′′ supε I DFε L ′ (H))c2′′′. ∈ k k q (c) For each ε I, z C and φ S (R), we define ∈ ∈ ∈ e z/2 2 z/2 T (ε)φ := (I ) [(x )− φ](F ), φ S (R) z − L −△ ε ∈ z/2 2 z/2 (the operators (I ) and (x )− are defined by the same definition − L −△ formulae for the real-exponent case). For any φ S (R) and ψ D∞, we shall show z E[(T (ε)φ)ψ] is analytic. For this, we note∈ that ∈ 7→ z z/2 z/2 E[(T (ε)φ)ψ]= E[(H− φ)(F )(I ) ψ] z ε − L ∞ z/2 z/2 (A.3) = (1 + m) E (H− φ)(Fε)Jm[ψ] . mX=0 742 T. Amaba and Y. Ryu

z/2 z/2 Letting φn n Z be as in Lemma A.1, we have H− φn = (2n + 1)− φn for ∈ >0 { } 1/2 n Z>0 and φn := (√πn!)− φn n Z is a complete orthonormal system of ∈ { } ∈ >0 L (R, dx). Then we have φ = ∞ φ, φ R φ in L (R, dx). Therefore 2 e n=0h niL2( ,dx) n 2 P e e z/2 ∞ z/2 E[(H− φ)(F )J [ψ]] = (2n + 1)− φ, φ R E[φ (F )J [ψ]] ε m h niL2( ,dx) n ε m nX=0 e e is an inﬁnite sum of analytic functions in z. Furthermore, this series converges absolutely and locally-uniformly in z. In fact, with recalling that c0 = supε I supx R pFε (x) < where pFε is the probability density function of Fε, and∈ by using∈ the Cauchy-Schwartz∞ inequality,

∞ z/2 (2n + 1)− φ, φ R E[φ (F )J [ψ]] h niL2( ,dx) n ε m nX=0 e e ∞ 1 1 Re(z/2) 6 c J [ψ] (2n + 1)− φ, H − φ R 0k m kL2 h niL2( ,dx) nX=0 e 1/2 ∞ 2 1 Re(z/2) 6 c J [ψ] (2n + 1)− H − φ R , 0k m kL2 k kL2( ,dx) nX=0

z/2 is ﬁnite because φ S (R). Therefore, E (H− φ)(F )J [ψ] is indeed an analytic ∈ ε m function in z. Thus each term in the summation in (A.3) is analytic in z. We see further that the each term has the estimate z/2 E (H− φ)(F )J [ψ] | ε m | z/2 6 (H− φ)(F ) J [ψ] k ε kL2 k m kL2 z/2 6 c H− φ R J [ψ] . 0k kL2( ,dy)k m kL2 Therefore

∞ z/2 z/2 z/2 (1 + m) E (H− φ)(F )J [ψ] 6 c H− φ R ψ . | ε m | 0k kL2( ,dy)k k2,Re(z) mX=0 z/2 Here, the two quantities H− φ R and ψ are continuously depend- k kL2( ,dy) k k2,Re(z) ing on z because φ S (R) and ψ D∞. Hence the inﬁnite sum in (A.3) converges ∈ ∈ absolutely and locally-uniformly in z, so that E[(Tz(ε)φ)ψ] is analytic in z. (d) For φ S (R) and ψ D∞, Φ(z) := E[(T (ε)φ)ψ] is analytic on C. By the ∈ ∈ z Marcinkiewicz Lp-multiplier theorem (see e.g. Thangavelu, 1993, Chapter 4, Sec- 2 iτ tion 4.2, Theorem 4.2.1), one gets supτ R (x ) L ′ (R,dx) L ′ (R C;dx) < + , ∈ k −△ k p → p → ∞ where L ′ (R C; dx) is the space of all complex-valued measurable functions that p → are p′-th order integrable. On the other hand, by Meyer’s Lp-multiplier theorem (see e.g. Ikeda and Watanabe, 1989, Chapter V, Section 8, Lemma 8.2) gives iτ 0 supτ R (I ) Lp D (C) < + . By using these, we can conclude the following: ∈ k − L k → p ∞ (1) the complex function Φ has the estimate sup sup Φ(α + iτ) < + τ R 26α61 | | ∞ ∈ − for each φ and ψ (see Hirschman’s Lemma in Stein, 1956). Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 743

(2) for τ R, the operators T 2+iτ (ε) and T1+iτ (ε) uniquely extends to a ∈ − 0 bounded linear operators L ′ (R, dx) D (C) with estimates p → p R D0 C sup T 2+iτ (ε) Lp′ ( ,dx) p( ) < + , τ R k − k → ∞ ∈ R D0 C sup T1+iτ (ε) Lp′ ( ,dx) p( ) < + τ R k k → ∞ ∈ (Note that one can take these upper bounds independent of ε since the bounds c1 and c2 obtained in steps (a) and (b) are independent of ε). Then by Stein’s interpolation theorem2 (see Stein, 1956, Theorem 1), we can conclude that the operators Tz(ε) for 2 6 Re(z) 6 1 uniquely extends to a bounded linear operator 0 − L ′ (R, dx) D (C) and p → p R sup sup Tα(ε) Lp′ ( ,dx) Lp < + . ε I 26α61 k k → ∞ ∈ − (e) Now, for each φ S (R), we have ∈ s/2 2 s/2 2 s/2 φ(F ) = (I ) [(x )− (x ) φ](F ) k ε kp,s k − L −△ −△ ε kLp = T (ε)[(x2 )s/2φ] k s −△ kLp 2 s/2 6 Ts(ε) L ′ (R,dx) Lp [(x ) φ] L ′ (R,dx). k k p → k −△ k p Again by denseness of S (R), this inequality extends to Λ S ′(R) such that 2 s/2 ∈ (x ) Λ L ′ (R, dx). −△ ∈ p

Remark A.5. In Lemma A.4 and Lemma 4.12, one can take p′ = p when Fε = 1 2 1 d ε− w(ε T ), where ε (0,T ] =: I, T > 0 and w = (w , , w ) is the canonical process, i.e., the d-dimensional∈ Wiener process starting at··· zero (assume d = 1 if consid- 1 2 1 ering Lemma A.4), because then DF = ε− Dw(ε T ) = ε− (1 2 , , 1 2 ) ε [0,ε T ] ··· [0,ε T ] and DF i, DF j = δ T are non-random. h ε ε iH ij

A.4. Regularity of something related to resolvent kernel associated with elliptic operators. Suppose (H1), (H3) and (H4). We set, for f S (Rd) and x Rd, ∈ ∈ d 2 d 1 i ∂ f i ∂f Lf(x)= (σσ∗) (x) (x)+ b (x) (x). 2 j ∂x ∂x ∂x i,jX=1 i j Xi=1 i The fractional power (1 )s/2 is deﬁned as a pseudo-diﬀerential operator: −△ s/2 2 s/2 i ξ,x d (1 ) φ(x)= (1 + ξ ) φ(ξ)e h idξ, φ S (R ), −△ ZRd | | ∈ b d/2 i ξ,y where φ(ξ) =(2π)− Rd φ(y)e− h idy is the Fourier transform of φ, and is also given by R b s/2 1 ∞ s 1 t t d (1 ) φ(x)= t− 2 − e− (e △φ)(x)dt, φ S (R ) −△ Γ( s/2) Z ∈ − 0

2Strictly speaking, this theorem was stated in the case where φ and ψ are simple functions in Stein (1956, Theorem 1). This is because just that his argument is based on general σ-ﬁnite measure spaces where we can not speak about ‘smoothness’. The same arguments apply to our case. 744 T. Amaba and Y. Ryu

~~t for d~~

s x − dx< + iff s < d, Z x <1 | | ∞ | | d where the integral is over R and x = x Rd . | | | | Lemma A.6. For every p (1, ) and s ( d, (p 1)d/p), we have (1 ∈ ∞ ∈ − − − − )s/2δ L (Rd, dx) for each y Rd. △ y ∈ p ∈ Proof : The transition density associated to the semigroup generated by is given d/2 2 △ by p (x, y)=(4πt)− exp( y x /(4t)). Hence we have t −| − | s/2 1 ∞ s 1 t (1 ) δy(x)= t− 2 − e− pt(x, y)dt −△ Γ( s/2) Z − 0 d/2 2 (4π)− ∞ s+d 1 t x y = t− 2 − e− exp | − | dt Γ( s/2) Z − 4t − 0 n o d/2 2 (4π)− ∞ d+s x y (d+s) 2 1 u = s+d x y − u − e− exp | − | du, 4− 2 Γ( s/2)| − | Z0 n − 4u o − (d+s) which behaves, up to a multiplicative constant, as x y − when x y 0 and rapidly decreasing as x y . In fact, by| using− | the identity | − | → | − | → ∞ 2 2 u x y ( x y 2u) x y e− exp | − | = exp | − |− e−| − |, n − 4u o n − 4u o and putting a := x y , the last integral can be written as | − | 2 ∞ d+s 1 u a a u 2 − e− exp du =e− (I + II) Z0 n − 4uo where a 2 d+s 1 (a 2u) I := u 2 − exp − du, Z0 n − 4u o 2 ∞ d+s 1 (a 2u) II := u 2 − exp − du. Za n − 4u o a d+s 1 2 d+s 6 2 2 Since d+s> 0, we have I 0 u − du = d+s a . On the other hand, by using 2 a d+s + d+s 1 (1 v) R 2 ∞ 2 − the change of variable u = 2 v, one has II = (a/2) 2 v − exp( 2v a)dv. 2 − > > (1 v) R a a 6 For a 1 and v 2, we have exp( −2v a) = exp( 2 (v 2)) exp( 2v ) a 1 − − − − d+s 6 6 2 exp( 2 (v 2)) exp( 2 (v 2)) = e exp( v/2). Thus we obtain II c0e(a/2) − − +− d−+s 1 · − > ∞ 2 if a 1, where c0 := 2 v − exp( v/2)dv< + . The arguments above shows d+s 1 u a2 − ∞ that ∞ u 2 − e− exp(R )du decreases rapidly when a + , as claimed. 0 − 4u → ∞ Therefore,R (1 )s/2δ L (Rd, dx) if (d + s)p < d, i.e., s< (p 1)d/p. −△ y ∈ p − − d Let x R . Under the conditions (H1), (H3) and (H4), we denote by Xt = X(t, x, w)∈ a unique strong solution to the stochastic differential equation (4.10). In the sequel, we fix y Rd such that y = x and define ∈ 6 1 d f (z) := (1 L)− δ (z), z R . y − y ∈ Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 745

~~d Let φ : R R be a C∞-function such that (1) x / suppφ, (2) φ 1 on a neighbourhood→ of y, and (3) suppφ is compact. ∈ ≡~~

Lemma A.7. Assume (H1), (H3) and (H4). Let p (1, ) be arbitrary. ∈ ∞ (p 1)d s d (i) For each s < min 1 − , 0 and t > 0, we have ( f )(X ) D (R ) { − p } ∇ y t ∈ p and (0,T ] t ( f )(X ) Ds (Rd) is continuous. ∋ 7→ ∇ y t ∈ p > d p (ii) Assume d 2. Then for each 2

Proof : Let p (1, ) and t0 > 0 be arbitrary. In the following, we write f := fy. ∈ ∞ (p′ 1)d (i) Take p′ > p such that s < min 1 −′ , 0 . Since X 6 6 is uni- { − p } { t}t0 t T formly non-degenerate, the assertion follows by Lemma 4.12 once we show (1 s/2 d − ) (∂f/∂z ) L ′ (R , dz). △ k ∈ p Denote by pt(z,z′) the transition density associated to L. By a standard estimate (see e.g., Friedman, 1964, Chapter 9, Section 6, Theorem 7), there exist c,C > 0 such that 2 d/2 z z′ pt(z,z′) 6 C(2πct)− exp | − | , n − 2ct o 2 ∂pt 1/2 d/2 z z′ (z,z′) 6 Ct− (2πct)− exp | − | ∂z − 2ct k n o

~~d for every k = 1, 2, , d and z,z′ R . We may assume c > 2 by rearranging C > 0. Then we have··· ∈~~

∂f 1 ∞ t ∂pt (z) 6 e− (z,y) dt ∂zk Γ(1) Z ∂zk 0 (A.4) 2 ∞ 1/2 t d/2 z y 6 C t− e− (2πct)− exp | − | dt, Z0 n − 2ct o so that ∂f (1 )s/2 (z) −△ ∂z k |z−z′|2 t 1 ∞ s 1 t e− 4 ∂f 6 2 t− − e− d (z′) dz′dt Γ( s/2) Z ZRd ∂z − 0 √4πt k |z−y|2 2c(t+u) C′ ∞ ∞ s 1 1/2 (t+u) e− 6 t− 2 − u− e− dudt Γ( s/2)Γ(1/2) Z Z (2πc(t + u))d/2 − 0 0 |z−y|2 C′ ∞ 1−s e− 2cv 2 1 v = 1 s v − e− d/2 dv, Γ( −2 ) Z0 (2πcv)

s/2 d for some constant C′ > 0. Hence (1 ) (∂kf)(z) Lp′ (R , dz). p −△ ∈ Rd (ii) Suppose that s< min p 1 d, 0 . Take δ > 0 so that z : z x <δ c { − − } { ∈ | − | }⊂ (suppφ) . Note that ∂k(φfy) = (∂kφ)fy +φ(∂kfy), (∂kφ)(x)fy (x)=0and(∂kφ)fy d ∈ S (R ). Therefore by bounded convergence theorem, limt 0 [(∂kφ)fy](Xt) p,s 6 ↓ k k limt 0 [(∂kφ)fy](Xt) p = 0. So in the following, we investigate the behaviour of ↓ k k (φ(∂kfy))(Xt) and (φfy)(Xt). For this, we divide the proof into four steps. 746 T. Amaba and Y. Ryu

~~(a) Since s 6 0, we notice that~~

s/2 1 ∞ s 1 u (I ) F = u− 2 − e− TuF du | − L | Γ( s ) Z 2 0 − 1 ∞ s 1 u s/2 6 u− 2 − e− Tu F du = (I ) F Γ( s ) Z | | − L | | − 2 0 for any F L2, where Tu = exp(u ), u > 0 is the Ornstein-Uhlenbeck semigroup ∈ L 1 1 1 on the Wiener space. Then, taking p′,q,r > 1 so that p′ + q + r < 1 and with putting F := (φ∂kf)(Xt) = (φ∂kf)(Xt)1 Xt x >δ , we have {| − | } (φ∂ f)(X ) p = (I )s/2F p k k t kp,s k − L kp p 1 6 E (I )s/2F − (I )s/2 F − L − L | | s/2 s/2 p 1 = E (φ∂kf)(Xt) (I ) (I ) F − 1 X x > δ | | − L − L {| t − | } n o s/2 p 1 1/r 6 c (φ∂ f)(X ) ′ (I ) F − P X x >δ , 0k k t kp | − L | q,s | t − | where c0 = c0(p′,q,r) > 0 is a constant independent of t. We easily have (φ∂kf)(Xt) p′ 6 φ (∂kf)(Xt) p′ and k k | |∞k k s/2 p 1 (I ) F − | − L | q,s s/2 p 1 6 (I ) F − k{ − L } kq s/2 p 1 s/2 p 1 = (I ) F q(−p 1) = (I ) (φ∂kf)(Xt) p−′′ , k − L k − k − L k where p′′ := q(p 1). Thus we have obtained − (φ∂ f)(X ) p k k t kp,s s/2 p 1 6 c0 φ (∂kf)(Xt) p′ (I ) (φ∂kf)(Xt) −′′ (A.5) | |∞k k k − L kp 1/r P X x >δ . × | t − | Similarly, we have (φf)(X ) p k t kp,s s/2 p 1 6 c0′ f(Xt) p′ (I ) (φf)(Xt) −′′ (A.6) k k k − L kp 1/r P X x >δ × | t − | for some constant c0′ > 0, independent of t. (b) We will write ε := √t in the sequel. We shall give estimates for each factors in (A.5) and (A.6), though the proof is presented in the next step. Note that for the last factors in (A.5) and (A.6), there exists c3,c3′ ,K > 0 such that δ2 P X x >δ 6 c exp for t = ε2 (0,K]. (A.7) | t − | 3 − c ε2 ∈ 3′ Let p (1, ) anew be arbitrary. To see (A.5), we shall prove ∈ ∞ d p d lim ε (∂kf)(Xt) p = 0 if p< , (A.8) ε 0 k k d 1 ↓ − d sp s/2 p d lim ε − (I ) (φ∂kf)(Xt) p = 0 if p< . (A.9) ε 0 k − L k d + s 1 ↓ − Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 747

~~On the other hand, for (A.6), we shall prove~~

d p d lim ε f(Xt) p = 0 if p< , (A.10) ε 0 k k d 1 ↓ − d sp s/2 p lim ε − (I ) (φf)(Xt) p =0 ε 0 k − L k ↓ d 2d d (A.11) if p< min , = , d + s 1 d 2 d + s 1 − − − 2d d where d 2 is understood as + if d = 2 and the last equality follows from s> 2 . − ∞ − (c) Take p′ > p arbitrary. First we shall prove (A.8). By using Proposition 3.7 ε and Lemma 4.12 (Recall that ε = √t and Xt = x + εFε, where Fε = (X (1, x, w) x)/ε is uniformly non-degenerate), we have −

p′ p′ p′ (∂kf)(Xt) p = (∂kf)(x + εFε) p 6 c1 (∂kf)(x + εz) Rd k k k k k kLp′ ( ,dz) 2 p′ ∞ 1/2 u d/2 εz (y x) 6 c1′ u− e− (2πcu)− exp | − − | du dz ZRd n Z0 n − 2cu o o ′ z (y x) 2 p ∞ d−1 | − − | d (1 d) 2 1 v − cv = c1′ ε− z (y x) − v − e− e 2 dv dz. ZRd n| − − | Z0 o for some constants c1,c1′ > 0 independent of ε (In the last equality, we have applied εz (y x) 2 the change of variables v = | −2cu− | ). The last factor can be further computed as follows:

z p′ (1 d) √(2/c) z | | d−1 1 I := z − e− | | v 2 − dv dz, Z z >1 | | Z0 | | n o ′ ( z √2cv)2 p (1 d) √(2/c) z ∞ d−1 1 − | |− II := z − e− | | v 2 − e 2cv dv dz Z z >1 | | Z z | | n | | o 2 p′ d−1 (2/c) z p′ 2 √ 6 The term I is estimated as I = ( d 1 ) z >1 z − e− | | dz − | | {| | } 2 p′ p′√(2/c) z R ( ) e− | |dz < + . On the other hand, by applying the change of d 1 z >1 ∞ − R| | 748 T. Amaba and Y. Ryu variable v = z u and noting that z > 1, | | | |

~~(1 √2cu)2 1 √2/c √2/c Since exp( − )=e− 2cu e exp( u) 6 e exp( u), we get − 2cu − −~~

p′ p′√2/c p′√2/c z ∞ d−1 1 u II 6 e e− | |dz u 2 − e− du < + Z z >1 Z1 ∞ | | (the arguments so far will be used repeatedly below). Putting all together, we have d d p′ obtained that for p′ (p, d 1 ), lim supε 0 ε (∂kf)(Xt) p < + , which implies ∈ ↓ k k ∞ (A.8). − For (A.9), with assuming ε (0, 1], we have ∈ (1 )s/2[(φ∂ f)(x + ε )](z) | −△ k • | z z′ 2 − | − | (A.12) 1 ∞ s e 4u 2 1 u = s u− − e− d/2 φ∂kf (x + εz′)dz′du Γ( 2 ) Z0 ZRd (4πu) −

~~t /2 By using (A.4), change of variables εz′ = z′′ and the semigroup property of e △ (recall that c > 2), we have~~

z z′ 2 − | − | e 4u d/2 φ∂kf (x + εz′) dz′ ZRd (4πu) | | z z′ 2 (x+εz′) y 2 + − | − | − | − | ∞ e 4u e 2cv 6 1/2 v C v− e− dv d/2 d/2 dz′ Z0 ZRd (4πu) (2πcv) εz εz′ 2 (x+εz′) y 2 | − | + − ε2u − | cv− | d ∞ 1/2 v e 4 e 2 = Cε v− e− dv 2 d/2 d/2 dz′ Z0 ZRd (4πε u) (2πcv) εz z′′ 2 (x+z′′) y 2 | − | + − ε2u − | cv− | ∞ 1/2 v e 4 e 2 = C v− e− dv 2 d/2 d/2 dz′′ Z0 ZRd (4πε u) (2πcv) εz (y x) 2 | − − | + − 2(2ε2 u+cv) ∞ 1/2 v e = C v− e− 2 d/2 dv Z0 (2π(2ε u + cv)) εz (y x) 2 + − | −2 − | ∞ e 2(cε u+cv) 6 1 d/2 1/2 v C(2− c) v− e− 2 d/2 dv, Z0 (2π(cε u + cv)) where in the last inequality, we have used that

~~(cε2 + cv)d/2 (cε2 + cv)d/2 6 1 d/2 sup 2 d/2 sup 2 d/2 = (2− c) . u,v>0 (2ε + cv) u,v>0 (2ε +2v) Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals 749~~

Substituting this estimate into (A.12), we get (1 )s/2[(φ∂ f)(x + ε )](z) | −△ k • | εz (y x) 2 | − − | − 2c(ε2u+v) ∞ ∞ s 1 u 1/2 v e 6 2 c2 u− − e− v− e− 2 d/2 dudv Z0 Z0 (2πc(ε u + v)) εz (y x) 2 | − − | − 2c(u+v) ∞ ∞ s ε 2u v e s 2 1 1/2 −( − + ) = c2ε u− − v− e d/2 dudv Z0 Z0 (2πc(u + v)) εz (y x) 2 | − − | − 2c(u+v) s ∞ ∞ s 1 1/2 (u+v) e 6 2 c2ε u− − v− e− d/2 dudv Z0 Z0 (2πc(u + v)) εz (y x) 2 − | − − | ∞ 1−s e 2cu s 2 1 u = c2′ ε u − e− d/2 du, Z0 (2πcu) for some constants c2,c2′ > 0 independent of ε, and so that, by using Lemma 4.12, ′ (I )s/2(φ∂ f)(X ) p k − L k t kp s/2 p′ 6 c2′′ (1 ) [(φ∂kf)(x + ε )](z) Rd k −△ • kLp′ ( ,dz) εz (y x) 2 ′ − | − cu− | p sp′ ∞ 1−s 1 u e 2 6 2 c2′′′ε u − e− d/2 du dz ZRd n Z0 (2πcu) o sp′ d (d (1 s)) = c2′′′ε − z (y x) − − − ZRd n| − − | 2 ′ d−(1−s) z (y x) p ∞ 1 u − | − − | u 2 − e− e 2cu du dz. × Z0 o for some constants c2′′,c2′′′ > 0. Here, note that d (1 s) > 0 because d > 2 and d d−(1−s) − − ∞ 2 1 u s> 2 . Hence 0 u − e− du< + and by repeating the above argument, − ∞ ′ ′ R d (sp d) s/2 p we see that if p

p′ f(x + εz) Rd k kLp′ ( ,dz) ′ z (x y) 2 p d 1 d ∞ d 1 u − | − − | 6 c3ε− z (x y) − u 2 − e− e 2cu du dz h ZRd n| − − | Z0 o ′ z (x y) 2 p (d 1)/2 ∞ 1/2 u − | − − | + z (x y) − − u− e− e 2cu du dz ZRd n| − − | Z0 o i (A.13) for some constant c3 > 0, independent of ε. Actually, then we have d p′ d lim supε 0 ε f(Xt) p < + if p

~~To prove (A.13), we begin with the inequality~~

~~d/2 y (x+εz) 2 C(2πc)− ∞ d/2 u − | − | f(x + εz) 6 u− e− e 2cu du. Γ(1) Z0 We divide the integral as~~

y (x+εz) 2 ∞ d/2 u − | − | u− e− e 2cu du Z0 y (x+εz) y (x+εz) 2 y (x+εz) 2 | − | d/2 u − | − | ∞ d/2 u − | − | = u− e− e 2cu du + u− e− e 2cu du. Z0 Z y (x+εz) | − | The ﬁrst term in the last equation is estimated as

y (x+εz) εz (x y) 2 | − | d/2 u − | − − | u− e− e 2cu du Z0 εz (x y) 2 d 1 2 d ∞ d 2 u − | − − | = (2c) 2 − εz (x y) − u 2 − e− e 2cu du | − − | Z |y−(x+εz)| 2c εz (x y) 2 d 1 d ∞ d 1 u − | − − | 6 (2c) 2 εz (x y) − u 2 − e− e 2cu du | − − | Z |y−(x+εz)| 2c εz (x y) 2 d 1 d ∞ d 1 u − | − − | 6 (2c) 2 εz (x y) − u 2 − e− e 2cu du. | − − | Z0 On the other hand, the second term is

y (x+εz) 2 ∞ d/2 u − | − | u− e− e 2cu du Z y (x+εz) | − | y (x+εz) 2 (d 1)/2 ∞ 1/2 u − | − | 6 εz (x y) − − u− e− e 2cu du. | − − | Z0 Now a change of variable leads us to (A.13) and thus (A.10) is proved. For (A.11), it is suﬃcient to prove

s/2 p′ (1 ) [(φf)(x + ε )](z) Rd k −△ • kLp′ ( ,dz) 2 p′ sp′ d 1 (s+d) ∞ d+s 1 u |z−(x−y)| 6 c4ε − z (x y) − u 2 − e− e− 2cu du dz (A.14) h ZRd n| − − | Z0 o 2 p′ 2−d ∞ s u |z−(x−y)| + z (x y) 2 u− 2 e− e− 2cu du dz ZRd n| − − | Z0 o i where c4 > 0 is a constant independent of ε. Note that d+s> 0 and s> 0, so that d+s s − ∞ 2 1 u ∞ 2 u it is assured that 0 u − e− du< + and 0 u− e− du< + , respectively. ∞ d sp′ s/∞2 p′ Note also that 1 (Rs+d) < 0. These imply lim supRε 0 ε − (I ) (φf)(Xt) p < − d 2d ↓ k −L k + if p

~~We divide the integral as~~

2 2−(s+d) εz (x y) ∞ 1 u − | − − | u 2 − e− e 2cu du Z0 εz (x y) 2 2−(s+d) εz (x y) | − − | 1 u − | − − | = u 2 − e− e 2cu du Z0 2 2−(s+d) εz (x y) ∞ 1 u − | − − | + u 2 − e− e 2cu du. Z εz (x y) | − − | We estimate the ﬁrst term as follows.

εz (x y) 2 2−(s+d) εz (x y) | − − | 1 u − | − − | u 2 − e− e 2cu du Z0 εz (x y) 2 1 d+s−2 2 (s+d) ∞ d+s−2 1 u − | − − | = (2c) − 2 εz (x y) − u 2 − e− e 2cu du | − − | Z |εz−(x−y)| 2c εz (x y) 2 1 d+s−3 1 (s+d) ∞ d+s 1 u − | − − | 6 (2c) − 2 εz (x y) − u 2 − e− e 2cu du | − − | Z |εz−(x−y)| 2c εz (x y) 2 1 d+s−3 1 (s+d) ∞ d+s 1 u − | − − | 6 (2c) − 2 εz (x y) − u 2 − e− e 2cu du. | − − | Z0 The second term is estimated as

2 2−(s+d) εz (x y) ∞ 1 u − | − − | u 2 − e− e 2cu du Z εz (x y) | − − | εz (x y) 2 2−d ∞ s u − | − − | 6 εz (x y) 2 u− 2 e− e 2cu du. | − − | Z0 Combining these, and a change of variable, we reach the estimate (A.14), and hence (A.11) is proved. (d) In view of (A.8), (A.9), (A.10) and (A.11), what we have to do now is to ﬁnd p′,q,r > 1 such that 1 1 1 p′ + q + r < 1; • d d p′ < d 1 and p′′ := q(p 1) < d (1 s) . • − − − − p 1 p d p In fact, since s< p 1 d, we can take ε (0, d ) such that s< p 1 1 εd < p 1 d. − − ∈ − − − − − Then take p′,q> 1 so that d 1 1 d 1 1 1 1 − < < − + ε and = ε < . d p′ d q d − d d 1 1 These conditions imply 1/p′ +1/q < −d + d = 1 and hence one can take r > 0 (p′′ 1)d such that 1/p′ +1/q +1/r < 1. Finally, we have if p′′ 1 > 0, then 1 −′′ = − − p (p′′ 1)d (p′′ 1)d p′′ 1 p p d 1 (p −1)q > 1 (p −1)d = 1 p −1 = p 1 q = p 1 1 εd > s, which − − − − − − − ′′ − − − − d 6 (p 1)d implies p′′ < d (1 s) . If p′′ 1 0, then 1 p−′′ > 0 > s, which also implies d − − − − p′′ < d (1 s) . − − Therefore, by taking p′,q,r > 1 as above, we conclude from (A.5), (A.6) and 2 (A.7)–(A.11) that (φ∂kf)(Xt) p,s and (φf)(Xt) p,s converge to zero as t = ε 0. k k k k ↓ 752 T. Amaba and Y. Ryu

~~Acknowledgements~~

~~The author would like to thank an anonymous referee for pointing out several mistakes in a preliminary version.~~

~~References~~

H. Abels. Pseudodifferential and singular integral operators. De Gruyter Graduate Lectures. De Gruyter, Berlin (2012). ISBN 978-3-11-025030-5. MR2884718. H. Airault, J. Ren and X. Zhang. Smoothness of local times of semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 330 (8), 719–724 (2000). MR1763917. M. Ait Ouahra, A. Kissami and H. Ouahhabi. On fractional derivatives of the local time of a symmetric stable process as a doubly indexed process. Random Oper. Stoch. Equ. 22 (2), 99–107 (2014). MR3213598. C. Bender. An Itôformula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Process. Appl. 104 (1), 81– 106 (2003). MR1956473. B. Bongioanni and J. L. Torrea. Sobolev spaces associated to the harmonic oscillator. Proc. Indian Acad. Sci. Math. Sci. 116 (3), 337–360 (2006). MR2256010. B. Boufoussi and B. Roynette. Le temps local brownien appartient p.s. àl’espace 1/2 de Besov Bp, . C. R. Acad. Sci. Paris Sér. I Math. 316 (8), 843–848 (1993). ∞ MR1218273. N. Bouleau and M. Yor. Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 292 (9), 491–494 (1981). MR612544. H. Föllmer, P. Protter and A. N. Shiryayev. Quadratic covariation and an extension of Itô’s formula. Bernoulli 1 (1-2), 149–169 (1995). MR1354459. A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964). MR0181836. M. Hasumi. Note on the n-dimensional tempered ultra-distributions. Tôhoku Math. J. (2) 13, 94–104 (1961). MR0131759. N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition (1989). ISBN 0-444- 87378-3. MR1011252. N. V. Krylov. Lectures on elliptic and parabolic equations in Sobolev spaces, volume 96 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2008). ISBN 978-0-8218-4684-1. MR2435520. I. Kubo. Itôformula for generalized Brownian functionals. In Theory and application of random fields (Bangalore, 1982), volume 49 of Lect. Notes Control Inf. Sci., pages 156–166. Springer, Berlin (1983). MR799940. S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1), 1–76 (1985). MR783181. Z. Liang. Fractional smoothness for the generalized local time of the indefinite Skorohod integral. J. Funct. Anal. 239 (1), 247–267 (2006). MR2258223. S. Lou and C. Ouyang. Local times of stochastic differential equations driven by fractional Brownian motions. Stochastic Process. Appl. 127 (11), 3643–3660 (2017). MR3707240. Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals 753

D. Nualart and J. Vives. Chaos expansions and local times. Publ. Mat. 36 (2B), 827–836 (1993) (1992). MR1210022. D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition (1999). ISBN 3- 540-64325-7. MR1725357. J. Sebastião e Silva. Les fonctions analytiques comme ultra-distributions dans le calcul opérationnel. Math. Ann. 136, 58–96 (1958). MR0105615. E. M. Stein. Interpolation of linear operators. Trans. Amer. Math. Soc. 83, 482–492 (1956). MR0082586. S. Thangavelu. Lectures on Hermite and Laguerre expansions, volume 42 of Math- ematical Notes. Princeton University Press, Princeton, NJ (1993). ISBN 0-691- 00048-4. MR1215939. H. F. Trotter. A property of Brownian motion paths. Illinois J. Math. 2, 425–433 (1958). MR0096311. H. Uemura. Tanaka formula for multidimensional Brownian motions. J. Theoret. Probab. 17 (2), 347–366 (2004). MR2053708. A. Ju. Veretennikov and N. V. Krylov. Explicit formulae for the solutions of stochastic equations. Mat. Sb. (N.S.) 100(142) (2), 266–284, 336 (1976). MR0410921. A. T. Wang. Generalized Ito’s formula and additive functionals of Brownian motion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (2), 153–159 (1977/78). MR0488327. S. Watanabe. Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15 (1), 1–39 (1987). MR877589. S. Watanabe. Donsker’s δ-functions in the Malliavin calculus. In Stochastic analysis, pages 495–502. Academic Press, Boston, MA (1991). MR1119846. S. Watanabe. Fractional order Sobolev spaces on Wiener space. Probab. Theory Related Fields 95 (2), 175–198 (1993). MR1214086. S. Watanabe. Some refinements of Donsker’s delta functions. In Stochastic analysis on infinite-dimensional spaces (Baton Rouge, LA, 1994), volume 310 of Pitman Res. Notes Math. Ser., pages 308–324. Longman Sci. Tech., Harlow (1994a). MR1415679. S. Watanabe. Stochastic Levi sums. Comm. Pure Appl. Math. 47 (5), 767–786 (1994b). MR1278353. K. Yoshinaga. On spaces of distributions of exponential growth. Bull. Kyushu Inst. Tech. Math. Nat. Sci. 6, 1–16 (1960). MR0126717.