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 function- als. We further formulate an Itˆoformula 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, H¨older continuity of diffusion local times, Distri- butional Itˆo’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 Itˆoformula 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 compu- tations (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 H¨older-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 H¨older continuity of local times had been initiated by Trotter (1958, inequalities (2.1) and (2.3)), in which the almost-sure H¨older-continuity of the Brownian local Distributional Itˆo’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-H¨older continuous for every β < 1. The latter half of this paper concerns with an Itˆoformula in a distributional setting. The classical Itˆo-Tanaka formula had been extended with several formula- tions (see F¨ollmer 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 Itˆo-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 Itˆoformula 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 nonneg- ative 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.