MHF Preprint Series Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality

The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces

H. Kawabi & T. Miyokawa

MHF 2006-27

( Received September 11, 2006 )

Faculty of Mathematics Kyushu University Fukuoka, JAPAN The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces

Hiroshi KAWABI Faculty of Mathematics Kyushu University 6-10-1, Hakozaki, Higashi-ku, Fukuoka 812-8581, JAPAN e-mail: [email protected] and Tomohiro MIYOKAWA Department of Mathematics, Graduate School of Science Kyoto University Sakyo-ku, Kyoto 606-8502, JAPAN e-mail: [email protected]

Abstract: In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces. We show this inequality under a weaker condition than the lower bounded- ness of Bakry-Emery’s Γ2. We also discuss Riesz transforms. As examples, we deal with diffusion processes on a path space associated with stochastic partial differential equations (SPDEs in short) and a class of superprocesses with immigration. Mathematics Subject Classifications (2000): 42B25, 60J60, 60H15. Keywords: Littlewood-Paley-Stein inequality, gradient estimate condition, Riesz trans- forms, SPDEs, superprocesses.

1 Framework and Results

In this paper, we discuss the Littlewood-Paley-Stein inequality. After the Meyer’s cele- brated work [16], many authors studied this inequality by a probabilistic approach. Espe- cially, Shigekawa-Yoshida [20] studied to symmetric diffusion processes on a general state space. In [20], they assumed that Bakry-Emery’s Γ2 is bounded from below. To define Γ2, they also assumed the existence of a suitable core A which is not only a ring but also stable under the operation of the semigroup and the infinitesimal generator. However,

1 in general, it is very difficult to check the existence of A having good properties de- noted above. Hence their assumption is serious when we face several infinite dimensional diffusion processes. In this paper, we show that the Littlewood-Paley-Stein inequality also holds on gen- eral metric spaces under the gradient estimate condition (G) even if we do not assume the existence such a core A. Our condition seems somewhat weaker than the lower boundedness of Γ2. We mention that Coulhon-Duong [5] and Li [14] also discussed the Littlewood-Paley-Stein inequality under similar conditions on finite dimensional Rieman- nian manifolds. Contrary to these papers, we work on a more general framework to handle certain infinite dimensional diffusion processes in Section 4. We introduce the framework that we work in this paper. Let X be a complete sep- arable metric space. Suppose we are given a Borel probability measure µ on X and a local µ-symmetric quasi-regular Dirichlet form E in L2(µ) with the domain D(E). See Ma-R¨ockner [15] for the terminologies of quasi-regular Dirichlet forms. Then by Theorem 1.1 of Chapter V in [15], there exists a µ-symmetric diffusion process M := (Xt, {Px}x∈X ) associated with (E, D(E)). We denote the infinitesimal generator and the transition semi- group by L and {Pt}t≥0, respectively. Since {Pt}t≥0 is µ-symmetric, it can be extended p to the semigroup on L (µ), p ≥ 1. We denote it by {Pt}t≥0 again. We also denote its p generator in L (µ) by Lp and the domain by Dom(Lp), respectively if we have to specify the acting space. We assume that 1 ∈ Dom(Lp) and Lp1 = 0 for all p ≥ 1, where 1 denotes the function that is identically equal to 1. In particular, the diffusion process M is conservative. Throughout this paper, we impose the following condition:

(A): There exists a subspace A of Dom(L2) consisting of bounded continuous function 2 which is dense in D(E) and f ∈ Dom(L1) holds for any f ∈ A. Under this condition, the form E admits a carr´edu champ, namely, there exists a unique positive symmetric and continuous bilinear form Γ from D(E)×D(E) into L1(µ) such that Z E(fh, g) + E(gh, f) − E(h, fg) = 2 hΓ(f, g) dµ X

∞ holds for any f, g, h ∈ D(E) ∩ L (µ). In particular, for f, g ∈ Dom(L2), fg ∈ Dom(L1) and 1© ª Γ(f, g) = L (fg) − (L f)g − f(L g) 2 1 2 2 hold. For further information, see Theorem 4.2.2 of Chapter I in Bouleau-Hirsch [4]. In the sequel, we also use the notation Γ(f) := Γ(f, f) for the simplicity. The following gradient estimate condition is crucial in this paper. (G): There exist constants K > 0 and R ∈ R such that the following inequality holds for any f ∈ A and t ≥ 0: 2Rt © ª Γ(Ptf) ≤ Ke Pt Γ(f) . (1.1)

2 Remark 1.1 If we can see A is stable under the operations of {Pt} and L,

Γ2(f) ≥ −RΓ(f), f ∈ A (1.2) ¡ ¢ 1 implies (1.1) with K = 1, where Γ2(f) := 2 L1Γ(f) − 2Γ(L2f, f) . Especially, (1.2) means that the Ricci curvature is bounded by −R from below in the case where X is a finite dimensional complete Riemannian manifold. See Proposition 2.3 in Bakry [2] for details. Hence our condition (G) is weaker than (1.2). Let us introduce the Littlewood-Paley G-functions. To do this, we recall the sub- ordination of a semigroup. For t ≥ 0, we define a probability measure λt on [0, +∞) by t 2 λ (ds) := √ e−t /4ss−3/2ds. t 2 π In terms of the Laplace transform, this measure is characterized as Z ∞ √ −γs − γt e λt(ds) = e , γ > 0. 0 (α) For α ≥ 0, we define the subordination {Qt }t≥0 of {Pt}t≥0 by Z ∞ (α) −αs p Qt f := e Psf λt(ds), f ∈ L (µ). 0 Then we can easily see that Z ∞ (α) −αs kQt fkLp(µ) ≤ e kPsfkLp(µ)λt(ds) 0Z ³ ∞ ´ √ −αs − αt ≤ e λt(ds) kfkLp(µ) = e kfkLp(µ), (1.3) 0 and hence {Q(α)} is a strongly continuous contraction semigroup on Lp(µ). The in- t t≥0 √ (α) 2 p finitesimal generator of {Qt p}t≥0 in L (µ) is − α − L. In the case of L (µ), this operator will be clearly denoted by − α − Lp when the dependence of p is significant. For f ∈ L2 ∩ Lp(µ) and α > 0, we define Littlewood-Paley’s G-functions by ¯ ¯ µZ ¶ ¯ ¯ ∞ 1/2 → ¯ ∂ (α) ¯ → → 2 gf (x, t) := ¯ (Qt f)(x)¯ ,Gf (x) := tgf (x, t) dt , ∂t 0 µZ ∞ ¶1/2 ↑ ¡ (α) ¢1/2 ↑ ↑ 2 gf (x, t) := Γ(Qt f) (x),Gf (x) := tgf (x, t) dt , 0 q µZ ∞ ¶1/2 → 2 ↑ 2 2 gf (x, t) := (gf (x, t)) + (gf (x, t)) ,Gf (x) := tgf (x, t) dt . 0 Now we present the Littlewood-Paley-Stein inequality. In what follows, the notation kukLp(µ) . kvkLp(µ) stands for kukLp(µ) ≤ CkvkLp(µ), where C is a positive constant depending only on K and p.

3 Theorem 1.2 For any 1 < p < ∞ and α > R ∨ 0, the following inequalities hold for f ∈ L2 ∩ Lp(µ):

kGf kLp(µ) . kfkLp(µ), (1.4) → kfkLp(µ) . kGf kLp(µ). (1.5)

Before closing this section, we give an application of Theorem 1.2. It plays an impor- tant role in the regularity theory of parabolic PDEs on general metric spaces.

Theorem 1.3 Let 1 < p < ∞, q ≥ 1 and α > R ∨ 0. We define p (q) ¡ −q ¢1/2 p Rα (L)f := Γ ( α − Lp) f , f ∈ L (µ).

Then we have the following statements: (q) p (1) For any p ≥ 2 and q > 1, Rα (L) is bounded on L (µ). Moreover there exists a (q) positive constant kRα (L)kp,p depending only on K, p, q and αR := (α − R) ∧ α such that ° ° (q) (q) p ° ° p Rα (L)f Lp(µ) ≤ kRα (L)kp,pkfkL (µ), f ∈ L (µ). (1.6)

This implies the inclusion p ¡ q¢ 1,p © p 1/2 p ª Dom ( 1 − Lp) ⊂ W (µ) := f ∈ L (µ) ∩ D(E) | Γ(f) ∈ L (µ) .

(2) For any p ≥ 2 and 1 < q < 2, there exists a positive constant Cp,q such that ° ° 1/2 (q) ¡ q/2 −q/2¢ p ° ° p Γ(Ptf) Lp(µ) ≤ Cp,qkRα kp,p α + t kfkL (µ), t > 0, f ∈ L (µ). (1.7)

Remark 1.4 We do not know whether our gradient estimate condition (G) is sufficient or not to establish (1.6) for q = 1, i.e., so-called the boundedness of the Riesz trans- (1) p form Rα(L) := Rα (L) on L (µ), Recently, Shigekawa [18] discussed the boundedness of Rα(L) under the intertwining condition for the diffusion semigroup in a general frame- work. We remark that the intertwining condition implies (G). Hence one way to establish the boundedness of Rα(L) is to show the intertwining condition for each concrete problem.

2 Proof of Theorem 1.2

In this section, we prove Theorem 1.2 by a probabilistic method. The original idea is due to Meyer [16]. The reader is referred to see also Bakry [1], Shigekawa-Yoshida [20] and (α) p Yoshida [24]. In these papers, they expanded L(Qt f) , f ∈ A, by employing the usual functional analytic approach for the proof of the Littlewood-Paley-Stein inequality. Note that this approach is valid because they imposed the existence of a good core A described in Section 1. On the other hand, in this paper, A in condition (A) does not have such good properties. So we cannot draw their proof directly. To overcome this difficulty,

4 we need more delicate probabilistic arguments based on Itˆo’sformula. We give details and prove Theorem 1.2 for 1 < p < 2 in the second subsection. In the third subsection, we introduce the notion of H-functions to prove Theorem 1.2 for p > 2. Our gradient estimate condition (G) plays a crucial role when we compare between G-functions and H-functions. For the case p = 2, (1.4) is proved as equality by using spectral resolution of L. See Proposition 3.1 in [20] for the proof. We note that (1.5) is derived from (1.4) by using the standard duality argument. See Theorem 4.4 in [20] for the detail.

2.1 Preparations In this subsection, we make some preparations. We recall the diffusion process M = ↑ (Xt, {Px}x∈X ) associated with the Dirichlet form (E, D(E)). From now on, we write Px → in place of Px. Let (Bt,Pa ) be one-dimensional Brownian motion starting at a ∈ R with ∂2 ↑ → ˜ the generator ∂a2 . We set Yt := (Xt,Bt), t ≥ 0, and P(x,a) := Px ⊗ Pa . Then M := (Y , {P }) is a µ ⊗ m-symmetric diffusion process on X × R with the (formal) generator t (x,a) R L + ∂2 , where m is one-dimensional Lebesgue measure. We put P ↑ := P ↑µ(dx), ∂a2 R µ X x ↑ → Pµ⊗δa := X P(x,a)µ(dx) and denote the integration with respect to Px ,Pa , P(x,a) and ↑ → Pµ⊗δa by Ex, Ea , E(x,a) and Eµ⊗δa , respectively. We denote the semigroup on Lp(X × R; µ ⊗ m) associated with the diffusion process ˆ ˆ 2 {Yt}t≥0 by {Pt}t≥0 and its generator by Lp. We also denote the Dirichlet form on L (X × ˆ ˆ ˆ R; µ ⊗ m) associated with L2 by (E, D(E)). That is, © ¯ ¡ ¢ ª ˆ 2 ¯ 1 ˆ D(E) = u ∈ L (X × R; µ ⊗ m) lim u − Ptu, u 2 < +∞ , t&0 t L (X×R;µ⊗m) ¡ ¢ ˆ 1 ˆ ˆ E(u, v) = lim u − Ptu, v 2 for u, v ∈ D(E). t&0 t L (X×R;µ⊗m)

ˆ ∞ We denote by C := A ⊗ C0 (R) the totality of all linear combinations of f ⊗ ϕ, f ∈ A, ϕ ∈ ∞ 2 2 C0 (R), where (f ⊗ ϕ)(x, a) := f(x)ϕ(a). Meanwhile, the spaces L (µ) ⊗ L (m) and D(E) ⊗ H1,2(R) are usual tensor products of Hilbert spaces. Then we have

Lemma 2.1 Cˆ is dense in D(Eˆ). Moreover for u, v ∈ D(E) ⊗ H1,2(R), we have Z Z Z ¡ ¢ ∂u ∂v Eˆ(u, v) = E u(·, a), v(·, a) m(da) + µ(dx) (x, a) (x, a)m(da). (2.1) R X R ∂a ∂a

→ Proof. We denote by {Tt}t≥0 the transition semigroup associated with (Bt, {Pa }a∈R). We regard that it acts on L2(m). First, we note that the following identity holds:

ˆ 2 2 Pt(f ⊗ ϕ) = (Ptf) ⊗ (Ttϕ), f ∈ L (µ), ϕ ∈ L (m). (2.2)

By (2.2), we can see Cˆ ⊂ D(E) ⊗ H1,2(R) ⊂ D(Eˆ) and the identity (2.1). We also have ¡ ¢ ˆ 2 2 0 2 2 E1(f ⊗ ϕ, f ⊗ ϕ) ≤ E1(f, f)kϕkL2(m) + kfkL2(µ) kϕ kL2(m) + kϕkL2(m) (2.3)

5 holds for f ∈ D(E), ϕ ∈ H1,2(R). By (2.3), we see that Cˆ is dense in D(E) ⊗ H1,2(R) with ˆ ∞ 1,2 respect to E1-topology, because A and C0 (R) are dense in D(E) and H (R), respectively. Hence it is sufficient to show D(E) ⊗ H1,2(R) is dense in D(Eˆ). Since L2(µ) ⊗ L2(m) S ¡ ¢ 2 ˆ 2 2 ˆ is dense in L (X × R; µ ⊗ m), t>0 Pt L (µ) ⊗ L (m) is dense in D(E). On the other hand, (2.2) also leads us to [ ¡ ¢ [ ¡ ¢ ¡ ¢ ˆ 2 2 2 2 1,2 ˆ Pt L (µ) ⊗ L (m) = Pt(L (µ)) ⊗ Pt(L (m)) ⊂ D(E) ⊗ H (R) ⊂ D(E). t>0 t>0 Therefore the proof is complete. Here we note that, due to Fitzsimmons [6], the Dirichlet form (Eˆ, D(Eˆ)) is quasi- regular. Thus we can apply the general theory of quasi-regular Dirichlet forms in [15]. (α) Now we fix a function f ∈ A. We set u(x, a) := Qa f(x), a ≥ 0. Then it holds that µ ¶ ∂2 + L − α u(·, a) = 0 in L2(µ). ∂a2 (α) Furthermore for a ∈ R, we consider v(x, a) := u(x, |a|) = Q|a| f(x). Then by (1.3), we have Z ³ √ ´1/2 −2 α|a| 2 −1/4 kvkL2(X×R;µ⊗m) ≤ e kfkL2(µ)da = α kfkL2(µ). (2.4) R The main purpose of this subsection is to discuss the semi-martingale decomposition of v(Xt∧τ ,Bt∧τ ), t ≥ 0, where τ := inf{t > 0 | Bt = 0}. As the first step, we give the following fundamental lemma: Lemma 2.2 v ∈ D(Eˆ) holds. Proof. At the beginning, we note L2(X × R; µ ⊗ m) ∼= L2(R,L2(X; µ); m). According to Fubini’s theorem, we have £ ¤ h £ ¤ i ˆ ↑ → Ptv(x, a) = E(x,a) u(Xt, |Bt|) = Ex Ea u(·, |Bt|) (Xt) . (2.5) We recall Tanaka’s formula Z t → |Bt| = |B0| + sgn(Bs)dBs + Lt(0), t ≥ 0, Pa -a.s., 0 where {Lt(0)}t≥0 is the local time of one-dimensional Brownian motion {Bt}t≥0 at the origin. Then by using Itˆo’sformula, we have Z t ∂u u(·, |B |) = u(·, |B |) + (·, |B |)sgn(B )dB t 0 ∂a s s s Z 0 Z t ∂u t ∂2u + (·, |B |)dL (0) + (·, |B |)ds ∂a s s ∂a2 s 0 Z 0 t √ = u(·, |B0|) − α − Lu(·, |Bs|)sgn(Bs)dBs Z 0 Z t √ t − α − Lu(·, |Bs|)dLs(0) + (α − L)u(·, |Bs|)ds. (2.6) 0 0

6 Hence (2.6) leads us that Z h t √ i → → Ea [u(·, |Bt|)] = u(·, |a|) − Ea α − Lu(·, |Bs|)dLs(0) 0 h Z t i → +Ea (α − L)u(·, |Bs|)ds . (2.7) 0

(α) On the other hand, since f ∈ A, it holds u(·, |a|) = Q|a| f(·) ∈ Dom(L2). Hence

Z t [u(·,|a|)] (α) (α) (α) Mt := (Q|a| f)(Xt) − (Q|a| f)(X0) − L(Q|a| f)(Xs)ds, t ≥ 0, 0

2 ↑ is an L (Pµ )-martingale. Then we have

Z t ↑£ ¤ (α) (α) Ex u(Xt, |a|) = (Q|a| f)(x) + Ps(LQ|a| f)(x)ds, µ-a.e. x ∈ X. (2.8) 0 By summarizing (2.5), (2.7) and (2.8), we can proceed as ¡ ¢ 1 ˆ v − Ptv, v L2(X×R;µ⊗m) t Z Z Z 1 n t o = − da P (LQ(α)f)(x)ds · Q(α)f(x)µ(dx) t s |a| |a| ZR ZX 0 Z 1 h £ t √ ¤ i + da E↑ E→ α − Lu(·, |B |)dL (0) (X ) · Q(α)f(x)µ(dx) t x a s s t |a| ZR ZX Z0 1 h £ t ¤ i − da E↑ E→ (α − L)u(·, |B |)ds (X ) · Q(α)f(x)µ(dx) t x a s t |a| ZR ZX 0 1 t ¡ ¢ = − da P LQ(α)f, Q(α)f ds t s |a| |a| L2(µ) ZR Z0 Z t √ 1 →£ ¤ (α) + da Ea α − Lu(x, |Bs|)dLs(0) · Pt(Q|a| f)(x)µ(dx) t R X 0 Z Z Z t 1 →£ ¤ (α) − da Ea (α − L)u(x, |Bs|)ds · Pt(Q|a| f)(x)µ(dx) t R X 0 =: −I1(t) + I2(t) − I3(t), (2.9)

2 where we used symmetry of {Pt}t≥0 on L (µ). For the term I1(t), we see the following estimate by using contractivity of {Pt}t≥0 on L2(µ) and (1.3).

Z Z t 1 (α) (α) |I (t)| ≤ da kLQ fk 2 · kQ fk 2 ds 1 t |a| L (µ) |a| L (µ) Z R 0 √ −2 α|a| 1 ≤ e kLfkL2(µ) · kfkL2(µ)da = √ kLfkL2(µ) · kfkL2(µ). (2.10) R α

7 For the term I2(t), by using same arguments as above, we have ¯ Z Z ³ ´ ¯ ¯1 √ £ ¤ (α) ¯ |I (t)| = ¯ da α − Lu(x, 0)E→ L (0) P (Q f)(x)µ(dx)¯ 2 t a t t |a| ¯ ZR X ¯ 1¯ ¡√ (α) ¢ £ ¤ ¯ = ¯ α − Lf, P Q f E→ L (0) da¯ t t |a| L2(µ) a t R Z √ ∞ √ 2 − αa →£ ¤ ≤ k α − LfkL2(µ) · kfkL2(µ) e Ea Lt(0) da. t 0 Here we recall 1 n (y + |r − a|)2 o P →(L(t, r) ∈ dy) = √ exp − dy, y > 0. a πt 4t See page 155 of Borodin-Salminen [3]. Then we can continue as 2 √ |I2(t)| ≤ k α − LfkL2(µ) · kfkL2(µ) t Z Z ∞ √ n ∞ 1 ³ (a + y)2 ´ o × e− αa y √ exp − dy da πt 4t √0 0 ≤ 8k α − Lfk 2 · kfk 2 Z L (µ) Z L (µ) ∞ 2 ∞ 2 √ 1 − a − y × √ e 2 da ye 2 dy = 4k α − LfkL2(µ) · kfkL2(µ). (2.11) 0 2π 0

For the term I3(t), we also have

Z ° Z t ° 1 ° →£ ¤° (α) |I3(t)| ≤ °Ea (α − L)u(·, |Bs|)ds ° · kQ|a| fkL2(µ)da t L2(µ) ZR Z 0 h t i ³ √ ´ 1 → (α) − α|a| ≤ E k(α − L)Q f(·)kL2(µ)ds · e kfkL2(µ) da t a |Bs| ZR Z0 h t i ³ √ ´ 1 → − α|a| ≤ Ea (αkfkL2(µ) + kLfkL2(µ))ds · e kfkL2(µ) da t R 0 √ 2 2 = 2 αkfk 2 + √ kLfk 2 · kfk 2 . (2.12) L (µ) α L (µ) L (µ) Finally, we insert estimates (2.10), (2.11) and (2.12) into (2.9). Then we can easily see ¡ ¢ ¡ ¢ 1 ˆ 1 ˆ lim v − Ptv, v L2(X×R;µ⊗m) = sup v − Ptv, v L2(X×R;µ⊗m) < +∞. t&0 t t>0 t This and (2.4) complete the proof. By Lemma 2.2, we can apply Fukushima’s decomposition theorem. That is, there exist a martingale additive functional of finite energy M [v] and a continuous additive functional of zero energy N [v] such that

[v] [v] v˜(Xt,Bt) − v˜(X0,B0) = Mt + Nt , t ≥ 0, P(x,a)-a.s. for q.e.-(x, a), (2.13)

8 wherev ˜ is an Eˆ-quasi-continuous modification of v ∈ D(Eˆ). See Theorem 5.2.2 of Fukushima-Oshima-Takeda [7]. We note that, since Eˆ has strong local property, M [v] is continuous. Due to Theorem 5.2.3 of [7], we know that Z tn ³ ´2o [v] ∂v hM it = Γ(v, v)(Xs,Bs) + (Xs,Bs) ds. (2.14) 0 ∂a See also Theorem 5.1.3 and Example 5.1.1 of [7] for details. From now, we discuss the explicit expression of N [v]. Let us define a signed measure ν on X × R by √ ν(dxda) := 2 α − Lv(x, a)µ(dx)δ0(da), where δ0 is Dirac measure on R with mass at the origin. The total variation of ν is given by ¯√ ¯ |ν|(dxda) := 2¯ α − Lv(x, a)¯µ(dx)δ0(da). Then we have Lemma 2.3 There exists a constant C > 0 such that ZZ ¯ ¯ q ¯ ¯ ˆ ∞ (g ⊗ ϕ)(x, a) · |ν|(dxda) ≤ C E1(g ⊗ ϕ, g ⊗ ϕ), g ∈ A, ϕ ∈ C0 (R). X×R That is, ν is of finite 1-order energy integral. (For the definition of measures of finite 1-order energy integral, see Sections 2.2 and 5.4 of [7].)

Proof. At the beginning, we take a positive constant a0 such that supp(ϕ) ⊂ [−a0, a0]. We first consider in the case of ϕ(0) ≤ 0. Let ε > 0. Then for µ-a.e. x ∈ X, we have Z q ¡√ ¢2 |ϕ(a)| α − Lv(x, a) + ε δ0(da) R q¡√ ¢ = −ϕ(0) α − Lv(x, 0) 2 + ε q q ¡√ ¢2 ¡√ ¢2 = ϕ(a0) α − Lv(x, a0) + ε − ϕ(0) α − Lv(x, 0) + ε Z q a0 ∂ n ¡√ ¢ o = ϕ(a) α − Lv(x, a) 2 + ε da 0 ∂a Z q Z √ a0 √ a0 0 ¡ ¢2 α − Lv(x, a) · (α − L)v(x, a) = ϕ (a) α − Lv(x, a) + εda − ϕ(a) q¡√ ¢ da 0 0 α − Lv(x, a) 2 + ε Z Z q¡√ ¢ ≤ |ϕ0(a)| α − Lv(x, a) 2 + εda + |ϕ(a)| · |(α − L)v(x, a)|da. R R By letting ε & 0 on both sides, we have Z √ |ϕ(a)| · | α − Lv(x, a)|δ0(da) R Z Z ¯√ ¯ ≤ |ϕ0(a)| · ¯ α − Lv(x, a)¯da + |ϕ(a)| · |(α − L)v(x, a)|da, µ-a.e. x ∈ X. R R

9 Therefore we can proceed as ZZ ¯ ¯ ¯(g ⊗ ϕ)(x, a)¯ · |ν|(dxda) X×R Z Z ³ ¯√ ¯ ´ ≤ 2 |g(x)| |ϕ0(a)| · ¯ α − Lv(x, a)¯da µ(dx) X Z R ³ Z ´ +2 |g(x)| |ϕ(a)| · |(α − L)v(x, a)|da µ(dx) °√ X ° R ° ° 0 ≤ 2 α − Lv 2 kϕ kL2(m)kgkL2(µ) ° L (X×°R;µ⊗m) ° ° 2 2 +2 (α − L)v L2(X×R;µ⊗m)kϕkL (m)kgkL (µ) √ ¡°√ ° ¢q −1/4 ° ° ˆ ≤ 2 2α α − Lf 2 + k(α − L)fkL2(µ) E1(g ⊗ ϕ, g ⊗ ϕ) q L (µ) ˆ =: C E1(g ⊗ ϕ, g ⊗ ϕ), where we used (2.4) and ˆ 2 2 0 2 E(g ⊗ ϕ, g ⊗ ϕ) = E(g, g)kϕkL2(m) + kgkL2(µ)kϕ kL2(m) for the last line. This is the desired result. In the case of ϕ(0) ≥ 0, we easily see Z q Z 0 n q o ¡√ ¢2 ∂ ¡√ ¢2 |ϕ(a)| α − Lv(x, a) + ε δ0(da) = ϕ(a) α − Lv(x, a) + ε da. R −a0 ∂a (2.15) By using (2.15), we can draw the same argument in the case of ϕ(0) ≤ 0. Therefore the proof is complete. Due to Lemma 2.3, ν is of finite 1-order energy integral. Then for each β > 0, there ˆ exists a unique Uβν ∈ D(E) such that the following relation holds: ZZ ˆ ∞ Eβ(Uβν, g ⊗ ϕ) = (g ⊗ ϕ)(x, a)ν(dxda), g ∈ A, ϕ ∈ C0 (R). (2.16) X×R

Lemma 2.4 (1) Uαν = v. ˆ ˆ ˆ (2) Uβν = v − (β − α)Rβv holds, where {Rβ}β>0 is the resolvent of {Pt}t≥0. Proof. (1) We need to show (2.16). By using the integration by parts formula, for µ-a.e. x ∈ X, we have Z ∂v (x, a)ϕ0(a)da ∂a R Z Z ∞ √ ∞ √ = − α − Lu(x, a)ϕ0(a)da + α − Lu(x, a)ϕ0(−a) da Z0 0 ∞ √ d ¡ ¢ = − α − Lu(x, a) ϕ(a) + ϕ(−a) da da 0 Z √ ∞ ∂ √ ¡ ¢ = 2 α − Lu(x, 0)ϕ(0) + α − Lu(x, a) ϕ(a) + ϕ(−a) da 0 ∂a

10 Z √ ∞ ¡ ¢ = 2 α − Lu(x, 0)ϕ(0) − (α − L)u(x, a) ϕ(a) + ϕ(−a) da Z0 √ = 2 α − Lv(x, 0)ϕ(0) − (α − L)v(x, a)ϕ(a)da. (2.17) R Then (2.17) leads us to our desired equality as follows: Z Z √ √ ˆ Eα(v, g ⊗ ϕ) = daϕ(a) α − Lv(x, a) α − Lg(x)µ(dx) RZ X Z ³ √ ´ + µ(dx)g(x) 2 α − Lv(x, 0)ϕ(0) − (α − L)v(x, a)ϕ(a)da Z X R √ = 2 α − Lv(x, 0)g(x)ϕ(0)µ(dx) ZZX = (g ⊗ ϕ)(x, a)ν(dxda). X×R ¡ ˆ ˆ (2) We recall Eβ(Rβv, g ⊗ ϕ) = v, g ⊗ ϕ)L2(X×R;µ⊗m). Then we have ¡ ¢ ¡ ˆ ˆ ˆ Eβ v − (β − α)Rβv, g ⊗ ϕ = Eβ(v, g ⊗ ϕ) − (β − α) · v, g ⊗ ϕ)L2(X×R;µ⊗m) = Eˆ (v, g ⊗ ϕ) ZZα = (g ⊗ ϕ)(x, a)ν(dxda), X×R where we used (1) for the last line. Hence the proof of (2) is also complete. Due to Lemma 5.4.1 of [7] and the lemma above, we have Z t [v] Nt = α v˜(Xs,Bs) ds − At, t ≥ 0, 0 wherev ˜ is an Eˆ-quasi-continuous modification of v and A is the continuous additive func- tional corresponding to ν. Since ν does not charge out of X × {0}, due to Theorem 5.1.5 of [7], At∧τ = 0 holds. Thus we get Z t∧τ [v] Nt∧τ = α v˜(Xs,Bs) ds. (2.18) 0 By summarizing (2.13), (2.14), and (2.18), we have the following proposition which plays a crucial role later. Proposition 2.5 We have the semi-martingale decomposition Z t∧τ [v] v˜(Xt∧τ ,Bt∧τ ) − v˜(X0,B0) = Mt∧τ + α v˜(Xs,Bs) ds, t ≥ 0, (2.19) 0 under P(x,a) for q.e.-(x, a). Moreover it holds Z t∧τ n ³ ´2o [v] ∂v hM it∧τ = Γ(v, v)(Xs,Bs) + (Xs,Bs) ds. (2.20) 0 ∂a

11 Since v(x, a) = u(x, a) holds for a ≥ 0, we can regard that this proposition also gives the

semi-martingale decomposition of u(Xt∧τ ,Bt∧τ ). Before closing this subsection, we need the following lemma because we will deal with the measure µ ⊗ δa as an initial distribution.

Lemma 2.6 µ ⊗ δa does not charge any set of zero capacity for m-almost all a ∈ R. ˆ Proof. Let N ⊂ X × R be a set of zero capacity with respect to E1. Then by the item (4) in Theorem 4.1 of Okura [17], Na is a set of zero capacity with respect to E1 for m-a.e. a ∈ R, where the set Na ⊂ X is defined by Na := {x ∈ X|(x, a) ∈ N}, a ∈ R. Thus we have

(µ ⊗ δa)(N) = µ(Na) ≤ CapE1 (Na) = 0. This completes the proof.

2.2 Proof of Theorem 1.2 (1 < p < 2) In this subsection, we return to the proof of Theorem 1.2 in the case of 1 < p < 2. Here we recall the following identities for our later use. See [16] for the proof. Lemma 2.7 Let η : X × [0, +∞) → [0, +∞) be a measurable function. Then h Z τ i Z Z ∞

Eµ⊗δa η(Xt,Bt)dt = µ(dx) (a ∧ t)η(x, t) dt (2.21) 0 X 0 and h Z τ ¯ i Z ∞ ¯ (0) Eµ⊗δa η(Xt,Bt)dt¯Xτ = x = (a ∧ t)Qt η(·, t)(x) dt. (2.22) 0 0

Since {Xt}t≥0 and {Bt}t≥0 are mutually independent under Pµ⊗δa and µ is the invariant measure of {Xt}t≥0, we can see the following identity holds for any bounded Borel function h on X: Z £ ¤ Eµ⊗δa h(Xτ ) = h(x)µ(dx). (2.23) X [v] Hereafter, we abbreviate Mt∧τ as Mt for simplicity. By Proposition 2.5 and Lemma 2.6, there exists a non-negative sequence {an}n∈N such that limn→∞ an = ∞, (2.19) and

(2.20) hold under Pµ⊗δan for any n ∈ N. 2 We set Vt :=v ˜(Xt∧τ ,Bt∧τ ). We apply Itˆo’sformula to Vt . Proposition 2.5 implies 2 2 d(Vt ) = 2VtdMt + 2αVt dt + dhMit ¡ 2 2¢ = 2VtdMt + 2 gf (Xt,Bt) + αVt dt. (2.24) 2 p/2 Let ε > 0. By applying Itˆo’sformula to (Vt + ε) again, we also have ¡ 2 ¢p/2 ¡ 2 ¢p/2−1 ¡ 2 ¢p/2−1¡ 2 2¢ d Vt + ε = p Vt + ε VtdMt + p Vt + ε gf (Xt,Bt) + αVt dt p(p − 2)¡ ¢ + V 2 + ε p/2−2V 2dhMi 2 t t t ¡ 2 ¢p/2−1 ¡ 2 ¢p/2−1 2 ≥ p Vt + ε VtdMt + p(p − 1) Vt + ε gf (Xt,Bt) dt,

12 where we used p < 2 for the last line. Hence by taking the expectation of the inequality above and using u(x, a) = v(x, a) for a ≥ 0, we have

h Z τ i ¡ 2 ¢p/2−1 2 Eµ⊗δan p(p − 1) Vt + ε gf (Xt,Bt) dt 0 h i 2 p/2 2 p/2 ≤ Eµ⊗δan (Vτ + ε) − (V0 + ε) h i ¡ 2 ¢p/2 ≤ Eµ⊗δan Vτ + ε h i ¡ 2 ¢p/2 = Eµ⊗δan u(Xτ ,Bτ ) + ε h i Z ¡ 2 ¢p/2 2 p/2 = Eµ⊗δan f(Xτ ) + ε = (|f(x)| + ε) µ(dx), (2.25) X where we used (2.23) for the last line. Here, by recalling (2.21), the left hand side of (2.25) is equal to

Z Z ∞ 2 p/2−1 2 p(p − 1) µ(dx) (t ∧ an)(u(x, t) + ε) gf (x, t) dt. X 0 Therefore, by letting ε → 0 and n → ∞, we have

Z Z ∞ Z p−2 2 p p(p − 1) µ(dx) t|u(x, t)| gf (x, t) dt ≤ |f(x)| µ(dx). (2.26) X 0 X Now we recall the maximal ergodic inequality ° ° ° ° p ° sup |Ptf|° ≤ kfkLp(µ), p > 1. t≥0 Lp(µ) p − 1

See Theorem 3.3 in Shigekawa [19] for details. It leads us that Z ½Z ¾ ° ° ∞ p/2 ° °p 2−p p−2 2 Gf Lp(µ) = µ(dx) t|u(x, t)| |u(x, t)| gf (x, t) dt X 0 Z ½Z ∞ ¾p/2 ¡ ¢2−p p−2 2 ≤ µ(dx) t sup |Ptf(x)| |u(x, t)| gf (x, t) dt X 0 t≥0 ½Z ¾ 2−p ¡ ¢p 2 ≤ sup |Ptf(x)| µ(dx) X t≥0 ½Z Z ∞ ¾p/2 p−2 2 × t|u(x, t)| gf (x, t) dt µ(dx) X 0 ½Z ¾ 2−p ½Z ¾p/2 2 ° ° p p ° °p . |f(x)| µ(dx) |f(x)| µ(dx) = f Lp(µ), X X where we used (2.26) for the last line. This completes the proof.

13 2.3 Proof of Theorem 1.2 (p > 2) In the case of p > 2, we need additional functions, namely H-functions defined by ½Z ∞ ¾1/2 → (0) → 2 Hf (x) := tQt (gf (·, t) )(x) dt , 0 ½Z ∞ ¾1/2 ↑ (0) ↑ 2 Hf (x) := tQt (gf (·, t) )(x) dt , 0 ½Z ∞ ¾1/2 (0) 2 Hf (x) := tQt (gf (·, t) )(x) dt . 0 We begin by the following proposition: Proposition 2.8 For p > 2, the following inequality holds for any f ∈ A:

kHf kLp(µ) . kfkLp(µ). Proof. By a slight modification, we can prove in the same way as the proof of Proposition 4.2 in Shigekawa-Yoshida [20]. However we give the proof for the reader’s convenience. Let us recall that, due to (2.24), we have Z t∧τ Z t∧τ 2 2 ¡ 2 2¢ Vt∧τ − V0 = 2 Vs dMs + 2 αVs + gf (Xs,Bs) ds. (2.27) 0 0 R ¡ ¢ t∧τ 2 2 Since At := 2 0 αVs +gf (Xs,Bs) ds, t ≥ 0, is a continuous increasing process, (2.27) 2 2 implies that Zt := Vt∧τ − V0 , t ≥ 0 is a submartingale. Now we need an inequality for submartingales. Let {Zt}t≥0 be a continuous submartin- gale with the Doob-Meyer decomposition Zt = Mt + At, where {Mt}t≥0 is a continuous martingale and {At}t≥0 is a continuous increasing process with A0 = 0. Due to Lenglart- L´epingle-Pratelli[13], it holds that h i p p p E[A∞] ≤ (2p) E sup |Zt| , p > 1. (2.28) t≥0 Then by using (2.28) and Doob’s inequality, we have h Z τ i © ¡ 2 2¢ ªp/2 Eµ⊗δan 2 αVs + gf (Xs,Bs) ds 0 h i 2 2 p/2 . Eµ⊗δan sup |Vt∧τ − V0 | h t≥0 i 2 2 p/2 . Eµ⊗δan |Vτ − V0 | h i 2 2 p/2 = Eµ⊗δan |u(Xτ ,Bτ ) − u(X0,B0) | h i (α) 2 (α) 2 p/2 = Eµ⊗δan |(Q0 f(Xτ )) − (Qan f(X0)) | h i h i (α) p (α) p . Eµ⊗δan |(Q0 f(Xτ )| + Eµ⊗δan |Qan f(X0)| p (α) p p = kfkLp(µ) + kQan fkLp(µ) . kfkLp(µ). (2.29)

14 On the other hand, by using (2.22), (2.29) and Jensen’s inequality, we have Z °n ∞ op/2° p ° (0)¡ 2¢ ° kHf k p = ° tQ gf (·, t) dt ° L (µ) t 1 0 Z L (µ) °n ∞ op/2° ° (0)¡ 2¢ ° = lim ° (an ∧ t)Qt gf (·, t) dt ° n→∞ 1 0 Z L (µ) hn ∞ op/2i (0)¡ 2¢ = lim Eµ⊗δa (an ∧ t)Qt gf (·, t) (Xτ )dt n→∞ n 0 Z h h τ ¯ ip/2i 2 = lim Eµ⊗δ Eµ⊗δ gf (Xs,Bs) ds¯Xτ n→∞ an an 0 Z h h¡ τ ¢ ¯ ii 2 p/2¯ ≤ lim inf Eµ⊗δan Eµ⊗δan gf (Xs,Bs) ds Xτ n→∞ 0 h Z τ i ¡ 2 ¢p/2 = lim inf Eµ⊗δan gf (Xs,Bs) ds n→∞ 0 h Z τ i © ¡ 2 2¢ ªp/2 p ≤ lim inf Eµ⊗δan αVs + gf (Xs,Bs) ds . kfkLp(µ). n→∞ 0 This completes the proof. Next we study the relationship between G-functions and H-functions. In the proof of this proposition, condition (G) plays a key role. Proposition 2.9 (1) For any f ∈ A and α > R ∨ 0, the following inequality holds: √ ↑ ↑ Gf ≤ 2 KHf . (2) For any f ∈ A, the following inequality holds: → → Gf ≤ 2Hf . Proof. We only give a proof of the item (1). The item (2) can be proved in the same way. By condition (G) and Schwarz’s inequality, we have the following estimate for any α > R ∨ 0 and f ∈ A: Z ³ ∞ ´2 (α) −αs 1/2 Γ(Qt f) ≤ e Γ(Psf) λt(ds) 0 ³ Z ∞ ´ ³ Z ∞ ´ −(α−R)s −(α+R)s ¡ ¢ ≤ e λt(ds) · e Γ Psf λt(ds) 0 Z 0 √ ³ ∞ ´ − α−Rt −(α−R)s ¡ ¢ ≤ Ke e Ps Γ(f) λt(ds) 0 (α−R)¡ ¢ ≤ KQt Γ(f) . (2.30) Then (2.30) yields g↑(x, 2t)2 = Γ(Q(α)f)(x) f ³ 2t ´ (α) (α) = Γ Qt (Qt f) (x) ³ ´ (α−R) (α) (0)¡ ↑ 2¢ ≤ KQt Γ(Qt f) (x) ≤ KQt gf (·, t) (x), (2.31)

15 Therefore we have Z ∞ ↑ 2 ↑ 2 (Gf (x)) = 4 tgf (x, 2t) dt 0 Z ∞ (0)¡ ↑ 2¢ ↑ 2 ≤ 4K tQt gf (·, t) (x) dt = 4K(Hf (x)) , 0 where we changed the variable t to 2t in the first line and used (2.31) for the second line. This completes the proof. It is clear that Propositions 2.8 and 2.9 conclude the desired inequality (1.4). Therefore the proof of Theorem 1.2 is completed.

3 Proof of Theorem 1.3

Before giving the proof of Theorem 1.3, we make a preparation parallel to Yoshida [24]. R ∞ Let ν be a finite signed measure on [0, ∞). We denote byν ˆ and kνk := 0 |ν|(ds) the Laplace transform and the total variation of ν, respectively. For α > 0, we define a bounded operatorν ˆ(α − L) on Lp(µ), 1 ≤ p < ∞, by Z −αs νˆ(α − L)f := e Psf ν(ds). [0,∞) Thus we easily have

p kνˆ(α − L)fkLp(µ) ≤ kνk · kfkLp(µ), f ∈ L (µ). (3.1) Here we give a remark in the case of p = 2. In this case, this operator is represented by Z

νˆ(α − L) := νˆ(α + λ)dEλ, [0,∞) 2 where {Eλ}λ≥0 is the spectral decomposition of −L in L (µ). By Lemma 2.3 in [1], there exist finite signed measures ν and ν such that the Laplace √ √ 1 2 transform are given byν ˆ (λ) = 1+√λ andν ˆ (λ) = √1+ λ , respectively. For ε > 0, we denote 1 1+ λ 2 1+λ (ε) by νi , i = 1, 2, the image measure of νi under the mapping λ 7→ λ/ε. Then we have √ (ε) ε + λ (ε) νˆ (λ) = √ √ , kν k ≤ kν1k, (3.2) 1 ε + λ 1 √ √ (ε) ε + λ (ε) νˆ (λ) = √ , kν k ≤ kν2k. (3.3) 2 ε + λ 2 √ √ √ ε+(α−L) ε+ α−L (3.1), (3.2) and (3.3) imply the resulting operators √ √ and √ on Lp(µ) have ε+ α−L ε+(α−L) the operator norms not more than kν1k and kν2k, respectively. We also have p √ √ √ √ p ³ ε + (α − L)´³ ε + α − L ´ ³ ε + α − L ´³ ε + (α − L)´ √ √ p = p √ √ = I. ε + α − L ε + (α − L) ε + (α − L) ε + α − L

16 Then we obtain the following relation for q > 1: ³p ´ ¡p ¢−q √ √ ε + (α − L) −q ε + (α − L) = ( ε + α − L)−q √ √ ε + α − L √ √ √ √ ³ ε + α − L ´q = ( ε + α − L)−q p . (3.4) ε + (α − L) Now we are in a position to give the proof of Theorem 1.3. Proof of Theorem 1.3. First, we set β ∈ R and ε > 0 such that α = β + ε and β > R. 2 p Note 0 < ε < αR. Let f ∈ L ∩ L (µ) and we consider √ √ ³ ε + β − L ´q g := p f. ε + (β − L) By (3.4), we have ¡ √ ¢ ¡ √ p ¢ Γ ( α − L)−qf = Γ ( ε + β − L)−qg µ Z ¶ ∞ √ 2 1 q−1 − εt (β) 1/2 ≤ t e Γ(Qt g) dt . Γ(q) 0 Here we use Theorem 1.2. By recalling q > 1, we have the following estimate: Z ° ¡ √ ¢ ° ° ∞ √ ° ° −q 1/2° 1 ° q−1 − εt (β) 1/2 ° Γ ( α − L) f p ≤ ° t e Γ(Qt g) dt° L (µ) Γ(q) Lp(µ) Z0 Z ° ∞ √ ³ ∞ ´1/2° 1 °¡ ¢1/2 (β) ° ≤ ° t2q−3e−2 εtdt tΓ(Q g)dt ° t p Γ(q) 0 0 L (µ) ³ ´1/2 1 Γ(2q − 2) ↑ = · kG k p Γ(q) (4ε)q−1 g L (µ) 1/2 −(q−1)/2 Γ(2q − 2) . (4ε) · kgk p . (3.5) Γ(q) L (µ)

However the left hand side of (3.5) does not depend on ε. Hence we can let ε % αR on the right hand side, and it leads us to ° √ ° ¡ −q ¢1/2 −(q−1)/2 ° ° p Γ ( α − L) f Lp(µ) ≤ CK,p,q αR · kgkL (µ). (3.6) On the other hand, we have

q kgkLp(µ) ≤ kν1k · kfkLp(µ). (3.7) Then by combining (3.6) with (3.7), we complete the proof of the item (1). For the proof of the item (2), we use the same argument as used in Kawabi [11]. Since p {Pt}t≥0 is an analytic semigroup on L (µ) (see Chapter III of Stein [23] for details), there exists a positive constant Cp such that ° ° ° ° ° ° −1° ° p LPtf Lp(µ) ≤ Cpt f Lp(µ), f ∈ L (µ), (3.8)

17 (α) −αt and hence Pt := e Pt also satisfies ° ° ¡ ° ° ° (α) ° −αt −1 ° ° p (α − L)Pt f Lp(µ) ≤ e Cpt + α) f Lp(µ), f ∈ L (µ). (3.9) Then by noting 1 < q < 2 and (3.9), the left hand side of (1.7) is dominated as ° ° ° ¡ ¢ ° ° 1/2° αt° (α) 1/2° Γ(Ptf) Lp(µ) = e Γ Pt f Lp(µ) ° √ ° αt (q) ° q (α) ° ≤ e kRα (L)kp,p ( α − L) Pt f Lp(µ) ° √ ° αt (q) ° q−2 (α) ° = e kRα (L)kp,p ( α − L) (α − L)Pt f Lp(µ) Z αt (q) ∞ ° ° e kRα (L)kp,p −q/2° (α) ° ≤ s (α − L)Ps+tf Lp(µ)ds Γ(1 − q/2) 0 Z αt (q) ∞ n ¡ ¢° ° o e kRα (L)kp,p −q/2 −α(s+t) Cp ° ° ≤ s e + α f Lp(µ) ds, (3.10) Γ(1 − q/2) 0 s + t where we used (1.6) for the second line. Moreover, we have Z eαt ∞ ³ C ´ s−q/2e−α(s+t) p + α ds Γ(1 − q/2) s + t 0 Z Z C ∞ α ∞ ≤ p s−q/2(s + t)−1ds + s−q/2e−αsds Γ(1 − q/2) Γ(1 − q/2) 0 Z 0 C ³ ∞ ´ = p t−q/2 τ −q/2(1 + τ)−1dτ + αq/2 Γ(1 − q/2) 0 ¡ −q/2 q/2¢ ≤ Cp,q t + α , (3.11) where we changed the variable s to tτ in the third line. Hence by combining (3.10) with (3.11), we obtain our desired estimate (1.7). This completes the proof.

4 Examples

4.1 Diffusion processes on a path space with Gibbs measures In this subsection, we present an example on an infinite dimensional setting. This is studied in Kawabi [10], [12]. We consider diffusion processes on an infinite volume path space C(R, Rd) with Gibbs measures associated with the (formal) Hamiltonian Z Z 1 0 2 H(w) := |w (x)|Rd dx + U(w(x))dx, 2 R R where U : Rd → R is an interaction potential. Our diffusion processes are defined through the time dependent Ginzburg-Landau type SPDE © ª √ dXt(x) = ∆xXt(x) − ∇U(Xt(x)) dt + 2dWt(x), x ∈ R, t > 0, (4.1)

18 2 2 d where ∆x = d /dx , ∇ = (∂/∂zi)i=1 and (Wt)t≥0 is a white noise process. This dynamics is called the P (φ)1-time evolution which has its origin in Parisi and Wu’s stochastic quantization model. In what follows we describe the framework. We introduce some spaces of functions to

control the growth of Xt(x) as |x| → ∞. For fixed λ > 0, we consider a Hilbert spaces E := L2(R, Rd; e−2λχ(x)dx), λ > 0 where χ ∈ C∞(R, R) is a positive symmetric convex function satisfying χ(x) = |x| for |x| ≥ 1. We also consider n ¯ o d ¯ −λχ(x) C := X(·) ∈ C(R, R ) sup |X(x)|Rd e < ∞ for every λ > 0 . x∈R We regard these spaces as state spaces of our dynamics. Let µ be a (U-)Gibbs measure. This means that the regular conditional probability satisfies the following DLR-equation for every r ∈ N and µ-a.e. ξ ∈ C: Z r ∗ −1 ¡ ¢ µ(dw|Br )(ξ) = Zr,ξ exp − U(w(x))dx Wr,ξ(dw), −r ∗ where Br is the σ-field generated by C|[−r,r]c , Wr,ξ¡ is the path measure of the¢ Brownian bridge on [−r, r] with a boundary condition Wr,ξ w(r) = ξ(r), w(−r) = ξ(−r) = 1 and Zr,ξ is the normalization constant. We impose the following conditions for the potential function U. 1 d (U1) U ∈ C (R , R) and there exists a constant K1 ∈ R such that ¡ ¢ 2 d ∇U(z1) − ∇U(z2), z1 − z2 Rd ≥ −K1|z1 − z2|Rd , z1, z2 ∈ R .

(U2) There exist K2 > 0 and p > 0 such that p d |∇U(z)|Rd ≤ K2(1 + |z|Rd ), z ∈ R . (U3) lim U(z) = ∞. |z|Rd →∞ As examples of U satisfying above conditions, we are interested in a square potential and 2 4 2 a double-well potential. Those are, U(z) = a|z|Rd and U(z) = a(|z|Rd − |z|Rd ), a > 0, respectively. We remark that conditions (U1) and (U2) imply that SPDE (4.1) has a unique (mild) solution living in C([0, ∞), C) for initial datum w ∈ C. See Theorems 5.1 and 5.2 in Iwata [9] for the proof. We also note that condition (U3) is sufficient for the existence of a Gibbs measure. Moreover it is known that Gibbs measures are reversible under the solution X := {Xt(x)}t≥0 of SPDE (4.1). See Proposition 2.7 and Lemma 2.9 in Iwata [8] for details. We denote by {Pt}t≥0 the transition semigroup related to the diffusion process X. Now we introduce the relationship between our dynamics and a certain Dirichlet form. We define H := L2(R, Rd; dx) and n ¯ ∞ ˜ ¯ n ∞ d FCb := f(w) = f(hw, φ1i, ··· , hw, φni) ¯ n ∈ N, {φk}k=1 ⊂ C0 (R, R ), Z o ˜ ˜ ∞ n f = f(α1, ··· , αn) ∈ Cb (R ), hw, φki := (w(x), φk(x))Rd dx . R

19 ∞ For f ∈ FCb , we define the Fr´echet derivative Df : E −→ H by

Xn ∂f˜ Df(w) := (hw, φ1i, ··· , hw, φni)φk. (4.2) ∂αk k=1 We consider a symmetric bilinear form E which is given by Z 2 ∞ E(f) = |Df(w)|H µ(dw), f ∈ FCb . E

2 ∞ We set E1(f) := E(f) + kfkL2(µ) and denote by D(E) the completion of FCb with respect 1/2 to E1 -norm. For f ∈ D(E), we also denote by Df the closed extension of (4.2). ∞ d By virtue of the C0 (R, R )-quasi-invariance and the strictly positive property of the Gibbs measure µ,(E, D(E)) is a Dirichlet form on L2(µ), i.e., (E, D(E)) is a closed Marko- ∞ vian symmetric bilinear form. Hence condition (A) holds by putting A = FCb . Moreover our diffusion process X is associated with the Dirichlet form (E, D(E)). See Proposition 2 2.3 in [10] for the detail. We note that Γ(f) = |Df|H in this case. Then the following gradient estimate of the transition semigroup {Pt}t≥0 holds for any f ∈ D(E): ¡ ¢ K1t |D(Ptf)(w)|H ≤ e Pt |Df|H (w) for µ-a.e. w ∈ E. See Proposition 2.4 in [10] and Proposition 2.1 in [12] for details. Therefore Theorems 1.2 and 1.3 hold for α > K1 ∨ 0. These results play important roles when we study analytic properties for SPDEs containing rotation. See Theorem 4.4 in Kawabi [11] for details.

4.2 Superprocesses with immigration In this subsection, we give a simple example which comes from superprocesses (or Dawson- Watanabe processes) with immigration. Recently, Stannat [21], [22] studied these measure- valued processes from analytic view points. Following [21] and [22], we consider the one of the most elementary superprocesses. In what follows, we introduce the framework precisely. We assume that the type space S is a finite set {1, ··· , d} and the mutation

A = 0. Let E := M+(S) be the set of finite positive Borel measures on S. Note that we ∼ d d can identify E = R+ := {x ∈ R : xi ≥ 0, 1 ≤ i ≤ d} with the usual topology. For immi- gration ν ∈ E, we use the notation νi := ν({i}), 1 ≤ i ≤ d. The branching mechanism is given by 2 Ψ(i, λ) := −aiλ − biλ, λ ≥ 0, where ai, bi > 0 for every i ∈ S. We consider a (0, Ψ)-superprocess M on S with immigration ν ∈ E. It is a diffusion process on E whose generator is given by

Xd ∂2f Xd ∂f Lf(x) = a x (x) + (ν − b x ) (x), f ∈ C2(E), x = (x )d ∈ E. i i ∂x2 i i i ∂x 0 i i=1 i=1 i i=1 i

20 We may think of the diffusion process M as a continuous time limit of rescaled Galton- Watson processes modelling the random evolution of a given population where each in- dividual i ∈ S, independently of the others, produces a random number of children distributed according to a given offspring distribution and an additional immigration rate ν. The immigration ν induces an additional state-independent drift. Ψ We define a Gamma measure mν on E by

d Y ³ ´ν /a bi i i mΨ(dx) := Γ(ν /a )−1xνi/ai−1e−bixi/ai dx , ν a i i i i i=1 i and consider a symmetric bilinear form Z Xd ¡ ∂f ¢ E Ψ(f) = a x (x) 2mΨ(dx), f ∈ C2(E). ν i i ∂x ν 0 E i=1 i

Ψ 2 2 Ψ Then by Theorem 3.1 in [22], the closure of (Eν ,C0 (E)) in L (mν ) is a Dirichlet form Ψ ν,Ψ and it corresponds to the mν -symmetric diffusion process M. We denote by (Pt )t≥0 its 2 transition semigroup. We note that condition (A) holds by putting A = C0 (E) and

Xd ¡ ∂f ¢ Γ(f)(x) = a x (x) 2, x = (x )d ∈ E. i i ∂x i i=1 i=1 i Here we assume ν 1 min i ≥ , (4.3) 1≤i≤d ai 2 and set a0 := min1≤i≤d ai, ad+1 := max1≤i≤d ai and b0 := min1≤i≤d bi. Then by Theorem 2.9 in [21], we can see condition (G) ¡a ¢ © ª ν,Ψ d+1 −b0t ν,Ψ 1 Γ(Pt f) ≤ · e Pt Γ(f) , f ∈ Cb (E), a0 holds under the condition (4.3). Therefore Theorems 1.2 and 1.3 hold for all α > 0. Acknowledgment. We would like to express our sincere gratitude to Professor Shigeo Kusuoka and the anonymous referee for their valuable suggestions and pointing out some errors during the preparation of this paper. We also thank Professors Thierry Coulhon, Beniamin Goldys, Kazuhiro Kuwae and Wilhelm Stannat for useful comments. The first author was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and is supported by 21st century COE program “Development of Dynamic Mathematics with High Functionality” at Faculty of Mathematics, Kyushu University.

21 References

[1] D. Bakry: Etude des transformations de Riesz dans les vari´et´esriemaniennes ´acour- bure de Ricci minor´ee, S´eminairede Prob. XXI, Lecture Notes in Math., 1247, Springer-Velrag, Berlin-Heidelberg-New York, (1987), pp. 137–172.

[2] D. Bakry: On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in “New Trends in Stochastic Analysis” ( K. D. Elworthy, S. Kusuoka and I. Shigekawa eds.), World Sci. Publishing, River Edge, NJ (1997), pp. 43–75.

[3] A.N. Borodin and P. Salminen: Handbook of Brownian Motion–Facts and Formulae, Second edition. Probability and its Applications. Birkh¨auserVerlag, Basel, 2002.

[4] N. Bouleau and F. Hirsch: Dirichlet Forms and Analysis on Wiener space, Walter de Gruyter, 1991.

[5] T. Coulhon and X.T. Duong: Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56, (2003), pp. 1728–1751.

[6] P. J. Fitzsimmons: On the quasi-regularity of semi-Dirichlet forms, Potential Anal- ysis 15, (2001), pp. 151–185.

[7] M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994.

[8] K. Iwata: Reversible measures of a P (φ)1-time evolution, in Prob. Meth. in Math. Phys. : Proceedings of Taniguchi symposium, eds. K. Itˆoand N. Ikeda (Kinokuniya, 1985), pp. 195–209.

[9] K. Iwata: An infinite dimensional stochastic differential equation with state space C(R), Probab. Theory Relat. Fields 74 (1987), pp. 141–159.

[10] H. Kawabi: The parabolic Harnack inequality for the time dependent Ginzburg- Landau type SPDE and its application, Potential Analysis 22 (2005), pp. 61–84.

[11] H. Kawabi: Functional inequalities and an application for parabolic stochastic partial differential equations containing rotation, Bull. Sci. Math. 128 (2004), pp. 687–725.

[12] H. Kawabi: A Simple proof of log-Sobolev inequalities on a path space with Gibbs measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, pp. 321–329.

[13] E. Lenglart, D. L´epingleand M. Pratelli: Pr´esentationunifi´eede certaines in´egalit´es de th´eoriedes martingales, S´eminairede Prob. XIV, Lecture Notes in Math., 784, Springer-Verlag, Berlin-Heidelberg-New York, (1980), pp. 26–48.

22 [14] X.-D. Li: Riesz transforms for symmetric diffusion operators on complete Rieman- nian manifolds, To appear in Rev. Mat. Iberoamericana, (2005).

[15] Z.-M. Ma and M. R¨ockner: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin-Heidelberg-New York, 1992.

[16] P.A. Meyer: D´emonstration probabiliste de certaines in´egalit´esde Littlewood-Paley, S´eminaire de Prob. X, Lecture Notes in Math., 511, Springer-Velrag, Berlin- Heidelberg-New York, (1976), pp. 125–183.

[17] H. Okura: A new approach to the skew product of symmetric Markov processes, Mem. Fac. Eng. and Design Kyoto Inst. Tech., 46 (1998), pp. 1–12.

[18] I. Shigekawa: Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary, Osaka J. Math. 39 (2002), pp. 897–930.

[19] I. Shigekawa: Stochastic Analysis, Translations of Mathematical Monographs, 224. Iwanami Series in Modern Mathematics. American Mathematical Society, Provi- dence, RI, 2004.

[20] I. Shigekawa and N. Yoshida: Littlewood-Paley-Stein inequality for a symmetric dif- fusion, J. Math. Soc. Japan 44 (1992), pp. 251–280.

[21] W. Stannat: On transition semigroups of (A, Ψ)-superprocesses with immigration, Ann. Probab. 31 (2003), pp. 1377–1412.

[22] W. Stannat: Spectral properties for a class of continuous state branching processes with immigration, J. Funct. Anal. 201 (2003), pp. 185–227.

[23] E.M. Stein: Topics in Harmonic Analysis Related to Littlewood-Paley Theory, An- nals of Math. Studies, 63, Princeton Univ. Press, 1970.

[24] N. Yoshida: Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ. 32 (1992), pp. 621–654.

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