Potential Analysis (2005) 22: 61–84

© Springer 2004

The Parabolic Harnack Inequality for the Time Dependent Ginzburg–Landau Type SPDE and its Application HIROSHI KAWABI
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan (e-mail: [email protected]) (Received: 23 October 2002; accepted: 6 November 2003) Abstract. The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg–Landau type stochastic partial differential equation (= SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P (φ)1 -time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry–Emery’s 2 -method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics. Mathematics Subject Classifications (2000): Primary: 60H15, 47D07; secondary: 60J60, 31C25. Key words: SPDE, parabolic Harnack inequality, gradient estimate, Varadhan type small time asymptotics.

1. Introduction In this paper, we consider a dynamics of unbounded continuous spins on R. In quantum field theory, this dynamics is called a P (φ)1 -time evolution which has its origin in Parisi and Wu’s stochastic quantization model. We consider an infinitedimensional diffusion process which is described by the solution of the following time dependent Ginzburg–Landau type SPDE:  1  dXt (x) = {x Xt (x) − ∇U (Xt (x))} dt + dWt (x), x ∈ R, t > 0, (1.1) 2  X0 (x) = w(x), where w is an initial data, U (z) : Rd → R is an interaction potential function, x = d2 /dx 2 , ∇ = (∂/∂zi )di=1 and Wt (x) is a white noise process. The main purpose of this paper is to establish a parabolic Harnack inequality for the transition semigroup {Pt } associated with (1.1). This is an infinite
The author was partially supported by Research Fellowships of the Japan Society for the Pro-

motion of Science for Young Scientists and by Grant-in-Aid for Scientific research 15-03706, Japan Society for the Promotion of Science, Japan.

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HIROSHI KAWABI

dimensional version of the celebrated Li–Yau’s parabolic Harnack inequality. Originally, Wang [22] established this inequality for the transition semigroup of diffusion processes on finite-dimensional non-compact Riemannian manifolds to give a lower bound of the transition probability and the spectral gap of Laplacian. In [1], Aida and the author also proved this inequality in the case of an abstract Wiener space. In these papers, Bakry–Emery’s 2 -method [7] plays a key role in the proof. Recently, Röckner and Wang [21] proved this inequality for generalized Mehler semigroups. Their result is related to our result. However, in this paper, we present a stochastic approach which is different from theirs. Now we describe our framework. First, we give a precise meaning of the solution to the SPDE (1.1). When we discuss the existence and the uniqueness of solution of (1.1), we have to introduce suitable function spaces to control the growth of Xt (x) as |x| → ∞. We introduce Hilbert spaces L2λ (R, Rd ) := L2 (R, e−2λχ(x) dx), λ > 0 where χ ∈ C ∞ (R, R) is a positive symmetric convex function satisfying χ(x) = |x| for |x|
1. L2λ (R, Rd ) has an inner product defined by  (X, Y )λ := (X(x), Y (x))Rd e−2λχ(x)dx, X, Y ∈ L2λ (R, Rd ). R

The corresponding norms are denoted by  · λ . In this paper, we fix λ¯ > 0 and denote E := L2λ¯ (R, Rd ) and H := L2 (R, Rd ). We also define a suitable subspace of C(R, Rd ). For functions of C(R, Rd ), we define |||X|||λ := sup |X(x)|e−λχ(x)

for λ > 0.

x∈R

Let C :=



{X(x) ∈ C(R, Rd ) | |||X|||λ < ∞}.

λ>0

With the system of norms ||| · |||λ , C becomes a Fréchet space. We easily see that the densely inclusion C ⊂ E ∩ C(R, Rd ) holds under the topology of E. We regard these spaces as the state spaces of our dynamics. We denote by Cb (E, R) the set of bounded continuous functions on E. We say a function F : E → R is in class F C ∞ b if there exist a function f := f (α1 , . . . , αn ) ∈ Cb∞ (Rn ), n = 1, 2, . . . and {φk }nk=1 ⊂ C0∞ (R) satisfying F (w) ≡ f ( w, φ1 , . . . , w, φn ).  Here we use the notation u, v by R (u(x), v(x))Rd dx for simplicity. Let ( , F , P , {Ft }t
0 ) be a filtered probability space on which {Ft }t
0 -adapted white noise process W := (Wt )t
0 is defined. Here we call that a family of random linear functionals W on H is a white noise process (H -cylindrical Brownian motion) if the linear functional Wt , φ is a one-dimensional Brownian motion

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multiplied by  φH for every φ ∈ H and W0 , φ = 0 holds. Here we often denote Wt , φ by R (Wt (x), φ(x))Rd dx. Following Funaki [9] and Iwata [15], we call that C-valued {Ft }-adapted continuous stochastic process X := (Xt (x))t
0 is a mild solution of (1.1) with the initial data w ∈ C if X satisfies the stochastic integral equation    1 t Xt (x) = Gt (x, y)w(y) dy − Gt −s (x, y)∇U (Xs (y)) ds dy 2 0 R R  t Gt −s (x, y) dWs (y) dy (1.3) + 0

R

for P -almost surely. Here we denote the heat kernel by   (x − y)2 1 exp − . Gt (x, y) := √ 2t 2π t It is well-known that (1.1) has a unique solution living in C([0, ∞), C) for the initial data w ∈ C under a slightly weaker condition than the conditions (U1) and (U2) below. See Theorem 5.1 and Theorem 5.2 in [15] for the proof. In the sequel, we denote Pw , w ∈ C by the probability measure on C([0, ∞), E) induced by X and M := (X, Pw ). In this paper, we will assume the following conditions for the potential function U . (U1) U ∈ C 2 (Rd , R) and there exists a constant K1 ∈ R such that ∇ 2 U (z)
−K1 holds for any z ∈ Rd . (U2) There exist K2 > 0 and p > 0 such that |∇U (z)|  K2 (1 + |z|p ) holds for any z ∈ Rd . (U3) lim|z|→∞ U (z) = ∞. Typical example of U satisfying above conditions is a double-well potential. That is, U (z) = a(|z|4 − |z|2 ), a > 0. By adding the condition (U3), there exists a Gibbs measure µ ∈ P (C). See Proposition 2.7 in Iwata [14] for details. Here P (C) denotes the class of all probability measures on the space C and µ is said to be a Gibbs measure if its regular conditional probability satisfies the following DLR-equation for every r ∈ N and µ-a.e. ξ ∈ C:  r

−1 ∗ U (w(x)) dx Wr,ξ (dw), (1.4) µ(dw|Br )(ξ ) = Zr,ξ exp − −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. Since the inclusion map of C into E is continuous, we can regard µ ∈ P (E) by identifying it with its image measure under the inclusion map.

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We note that Gibbs measures are reversible measures of M. Moreover, Gibbs measures coincide with reversible measures by replacing the condition (U3) to the stronger one lim inf|z|→∞ ∇ 2 U (z) > 0. See Theorem 2.21 in [14] for details. Here we recall the C0∞ (R, Rd )-quasi-invariance of the Gibbs measure µ:   F (w + h)µ(dw) = F (w)e (h,w) µ(dw), (1.5) E

E

C0∞ (R, Rd )

and (h, w) is defined by where F ∈ Bb (E, R), h ∈  

(h, w) = U (w(x)) − U (w(x) − h(x)) R

 1 2 − |h (x)| − (w(x), x h(x))Rd dx. 2

(1.6)

For details the reader is referred to Funaki [10] and [14]. This property will be used. Now, we are in a position to state our main theorem. We define the transition semigroup {Pt }t
0 by  F (y)Pw (Xt ∈ dy), F ∈ Cb (E, R), w ∈ C. (1.7) (Pt F )(w) := E

On the other hand, we recall that µ is a stationary measure of our dynamics M. Then Pt F L1 (dµ)  F L1 (dµ) holds for F ∈ Cb (E, R). Hence Riesz–Thorin’s interpolation theorem implies that {Pt } can be extended to a Lp (E, R; dµ), 1  p  ∞-strongly continuous contraction semigroup. Our main theorem is as follows. THEOREM 1.1 (Parabolic Harnack Inequality). Let F ∈ Cb (E, R). Then for any h ∈ H ∩ C and α > 1, the following dimension free parabolic Harnack inequality holds for all w ∈ C.

K1 αh2H α α · . (1.8) |Pt F (w)|  Pt |F | (w + h) · exp 2(α − 1) 1 − e−K1 t 1 := 1t if Here K1 is the constant denoted in the condition (U1) and we set 1−eK−K 1t K1 = 0. In the case of F ∈ L∞ (dµ) and h ∈ C0∞ (R, Rd ), this inequality also holds for µ-a.e. w ∈ E.

The organization of this paper is as follows: In Section 2, we prove the gradient estimate for the transition semigroup {Pt }, which plays an important role in this paper. In Section 3, we give a proof of Theorem 1.1. We also discuss a certain smoothing property of {Pt }. In Section 4, we present an application of Theorem 1.1. We devote ourselves to give a lower estimate for the transition probability of our dynamics. Finally, we show the celebrated Varadhan type short time asymptotics as a corollary of this estimate.

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2. Gradient Estimate for the Semigroup {Pt } In this section, we prove the gradient estimate for the transition semigroup {Pt } which plays a key role in the proof of Theorem 1.1. We note that this type estimate is studied in Proposition 2.3 in Bakry [6] by using 2 -method. Here we note that the existence of a suitable core A ⊂ L2 (dµ) which is stable under the operation Pt is assumed in [6]. In finite-dimensional cases, we can easily check this assumption. But in infinite-dimensional situations, it is not trivial to find such a core. In this paper, we adopt another approach to prove this estimate. Here we represent Pt F as the expectation of the functional associated with our dynamics. We begin to prepare the following lemmas. LEMMA 2.1. Let X w be a solution of the SPDE (1.1) with the initial condition X0w = w ∈ C. Then for every h ∈ H ∩C, the following estimate holds for P -almost surely. Xtw+h − Xtw H  e

K1 t 2

hH .

(2.1)

Proof. We realize X w and X w+h on the same probability space as solutions of (1.1) with the same cylindrical Brownian motion. Here we set Y := X w+h − X w . By (1.3), Y satisfies the following integral equation:   t Gt (x, y)h(y) dy + Gt −s (x, y)(s, y) ds dy, Yt (x) = R

0

R

where (s, y) := − 12 (∇U (Xsw+h (y)) − ∇U (Xsw (y))). For  and r > 0, we set 2 r (s, y) := e−r|y| (s, y). We define its smooth approximate functions {m,r }∞ m=1 by m,r (s, y)  s+ 1 m :=



1 (s− m )∨0

R

m2 ρ(m(s − s ))ρ(m(y − y ))r (s , y ) dy ds ,

(2.2)

∞ where  ρ ∈ C0 (R, R) is a symmetric nonnegative function with supp ρ ⊂ (−1, 1) and R ρ(y) dy = 1. We set   t Gt (x, y)h(y) dy + Gt −s (x, y)m,r (s, y) ds dy. Yt(m,r) (x) := R

0

R

Then (2.2) derives that Y (m,r) ∈ C 1,2 ([0, ∞) × R, R), x Yt(m,r) ∈ L2 (R; dx) and Y (m,r) satisfies the following semi-linear heat equation:   ∂ Y (m,r)(x) = 1  Y (m,r)(x) +  (t, x), x ∈ R, t > 0, x t m,r t ∂t 2  (m,r) Y0 (x) = h(x).

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HIROSHI KAWABI

Here we take λ ∈ (0, λ¯ ], multiply both sides by 2Yt(m,r)(x)e−2λχ(x) and integrate over (0, t) × R. We remark that the convexity of χ implies ∇χL∞  1. Then by applying integration by parts, we obtain  |Yt(m,r)(x)|2 e−2λχ(x) dx R  t  2 −2λχ(x) 2 |h(x)| e dx + 2λ |Ys(m,r) (x)|2 e−2λχ(x) ds dx  0 R R  t (m,r (s, x), Ys(m,r) (x))Rd e−2λχ(x) ds dx. (2.4) + 0

R

Take a sufficient large number T such that T > t. By X·w+h , X·w ∈ C([0, ∞), C) and the condition (U2), we have m,r L∞ ((0,T )×R)  r L∞ ((0,T )×R) < ∞ and limm→∞ m,r = r in Lp ((0, T ) × R), 1  p < ∞. Then by letting m → ∞ and r → 0 on both sides of (2.4) and using Lebesgue’s dominated convergence theorem, we have  |Yt (x)|2 e−2λχ(x) dx R  t  |h(x)|2 e−2λχ(x) dx + 2λ2 |Ys (x)|2 e−2λχ(x) ds dx  R R 0  t ((s, x), Ys (x))Rd e−2λχ(x) ds dx. (2.5) + 0

R

The condition (U1) and (2.5) lead us to the following estimate:  |Yt (x)|2 e−2λχ(x) dx R  t  2 −2λχ(x) 2 |h(x)| e dx + (K1 + 2λ ) |Ys (x)|2 e−2λχ(x) ds dx.  R

0

R

Hence by using Gronwall’s lemma,   2 −2λχ(x) (K1 +2λ2 )t |Yt (x)| e dx  e |h(x)|2 e−2λχ(x) dx. R

(2.6)

R

Since h ∈ H , we complete the proof by letting λ ↓ 0.

2

LEMMA 2.2. The transition semigroup {Pt } is Feller continuous, i.e., for any F ∈ Cb (E, R) and t
0, Pt F : C ⊂ E → R is a bounded continuous function. Proof. For w, w ∈ C ⊂ E, we have the following estimate by recalling (2.6).

¯2

E[Xtw − Xtw 2E ]  e(K1 +2λ )t w − w 2E .

(2.7)

Then we have the Feller continuity by combining (2.7) and the same argument in the proof of Lemma 8.1.4 in Øksendal [19]. 2

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PARABOLIC HARNACK INEQUALITY

Before giving the gradient estimate, we discuss the relation between the solution of the SPDE (1.1) and a certain Dirichlet form. For F ∈ F C ∞ b , we also define the Fréchet derivative DF : E → H by DF (w)(x) :=

n  ∂f ( w, φ1 , . . . , w, φn )φk (x), ∂α k k=1

x ∈ R.

(2.8)

Now, we consider a symmetric bilinear form E which is given by  1 DF (w)2H µ(dw), F ∈ F C ∞ E(F ) = b . 2 E We also define E1 (F ) := E(F ) + F 2L2 (dµ) and D(E) by the completion of F C ∞ b 1/2

with respect to E1 -norm. For F ∈ D(E), we also denote DF by the closed extension of (2.8). By virtue of the C0∞ (R, Rd )-quasi-invariance and the strictly positive property of the Gibbs measure µ, Theorem 1 and Proposition 3.6 in Kusuoka [16] derive that (E, D(E)) is a Dirichlet form on L2 (E, dµ), i.e., (E, D(E)) is a closed Markovian symmetric bilinear form. Now we can summarize the relation between this Dirichlet form and our dynamics as the following proposition. This is essentially due to Theorem 2.1 in [10]. So we omit the proof. ˜ := (X˜ t , P˜w ) on E PROPOSITION 2.3. (1) There exists a diffusion process M associated with the Dirichlet form (E, D(E)). (2) If the initial distribution of X˜ 0 is the Gibbs measure µ, the distribution on C([0, ∞), E) of the process X˜ t coincides with that of Xt . Here Xt is the solution of the SPDE (1.1). Here we give a remark. Let {P˜t } be a L2 (dµ)-strongly contraction semigroup associated with the Dirichlet form (E, D(E)). Then the assertion (2) implies that   (Pt F )(w)G(w)µ(dw) = (P˜t F )(w)G(w)µ(dw) E

E

holds for any F, G ∈ L (dµ). So by Riesz’s representation theorem, we have Pt F = P˜t F holds for any F ∈ L2 (dµ). This means that {P˜t } coincides with {Pt } as a L2 (dµ)-strongly continuous contraction semigroup. We state the gradient estimate as follows. 2

PROPOSITION 2.4 (Gradient Estimate for {Pt }). For F ∈ D(E), the following gradient estimate holds for any t ∈ [0, ∞) and µ-a.e. w ∈ E. D(Pt F )(w)H  e

K1 t 2

Pt (DF H )(w).

(2.10)

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HIROSHI KAWABI

Proof. We first assume that F ∈ F C ∞ b , i.e., F (w) = f ( w, φ1 , . . . , w, φn ). ∞ ∞ d Here {φi }i=1 ⊂ C0 (R, R ) denotes a C.O.N.S of H for simplicity. By noting (2.6) and Schwarz’s inequality, we have the following estimate for any w, w ∈ C: |(Pt F )(w) − (Pt F )(w )|

= |E[F (Xtw )] − E[F (Xtw )]| n 

  1 ∂f 

 w w w w

E

∂α (1 − θ) Xt , φ + θ Xt , φ · Xt − Xt , φi

dθ i

0

i=1



 ∇f L∞ · E

n 

w

Xtw − Xt 2E ·

i=1 K +2λ¯ 2 ( 1 2 t)

 K(λ¯ )e

 R

¯

1/2 

|φi (x)|2 e2λχ(x) dx

· w − w E ,

(2.11)

where K(λ¯ ) is a positive constant defined by  n 

1/2  ¯ ∇f L∞ (Rn ) · |φi (x)|2 e2λχ(x) dx i=1

R

and Xtw , φ is denoted by ( Xtw , φ1 , . . . , Xtw , φn ) for simplicity. This estimate means that Pt F is uniformly continuous on C. Hence Pt F can be extended to a function of Cb (E, R). In the sequel, we also denote it by Pt F . ∞ For w ∈ E, h ∈ H , we take approximate sequences {wn }∞ n=1 ⊂ C, {hn }n=1 ⊂ H ∩ C such that limn→∞ wn = w in E and limn→∞ hn = h in H . Then by Lemma 2.1, we have |(Pt F )(w + h) − (Pt F )(w)|  lim inf E[|F (Xtwn +hn ) − F (Xtwn )|] n→∞

 ∇f L∞ · lim inf E[Xtwn +hn − Xtwn H ] n→∞

e

K1 t 2

=e

K1 t 2

∇f L∞ · lim hn H n→∞

∇f L∞ · hH .

(2.12)

Then by Lemma 1.3 in [16], there exists 0 ∈ B(E) such that 0 ⊂ C, µ( 0 ) = 1 and the following identity holds: 1 lim {(Pt F )(w + εh) − (Pt F )(w)} = (D(Pt F )(w), h)H ε→0 ε for any w ∈ 0 , h ∈ H. For w ∈ 0 , ε > 0 and h ∈ H ∩ C, we define Ztw,ε,h := 1ε (Xtw+εh − Xtw ). By Lemma 2.1, we can easily see that Ztw,ε,h H  e

K1 t 2

hH holds for P -almost

PARABOLIC HARNACK INEQUALITY

69

surely. Then for any t > 0, w ∈ 0 and h ∈ H ∩ C, we have 1 {(Pt F )(w + εh) − (Pt F )(w)} ε   1 = E (F (Xtw+εh ) − F (Xtw )) ε  n   1 ∂f   (1 − θ) Xtw , φ + θ Xtw+εh , φ · Ztw,ε,h , φi dθ =E ∂α i 0 i=1   n 

2 1/2  1 ∂f

w w+εh

, φ )

dθ · Ztw,ε,h H E

∂α ((1 − θ) Xt , φ + θ Xt i 0 i=1  n 

2 1/2   1 ∂f

K1 t w w+εh

dθ

((1 − θ) X , φ

+ θ X , φ ) .  e 2 hH · E t t

∂α i 0 i=1 (2.13) Then by combining (2.13) and Lebesgue’s dominated convergence theorem, we have the following estimate for any w ∈ 0 : (D(Pt F )(w), h)H



 n 

2 1/2   1 ∂f

 e hH · E lim ((1 − θ) Xtw , φ + θ Xtw+εh , φ )

dθ

ε→0 ∂αi i=1 0  n

2 1/2  ∂f

K1 t w

 e 2 hH · E

∂α ( Xt , φ ) i i=1 K1 t 2

=e

K1 t 2

hH · Pt (DF H )(w).

Therefore we have the following for any w ∈ 0 :   D(Pt F )(w)H = sup (D(Pt F )(w), h)H | h ∈ C ∩ H, hH = 1 e

K1 t 2

Pt (DF H )(w).

Next we consider the case of F ∈ D(E). By the symmetry of (E, D(E)), we can regard {Pt } as the strongly continuous contraction semigroup on L2 (dµ). For ∞ F ∈ D(E), we can take a sequence {Fj }∞ j =1 ⊂ F C b such that Fj → F in D(E) as j → ∞. Then we easily have Pt {DFj H } → Pt {DF H } in L2 (dµ) as j → ∞. Hence we have the convergence of the right-hand side of (2.10). On the other hand, we obtain the following estimate by using (2.10):  eK1 t sup Pt (DFj H )(w)2 µ(dw) < ∞. sup E(Pt Fj )  2 j ∈N E j ∈N Hence by recalling Lemma 2.12 in Ma and Röckner [18], there exists a subse1 j ∞ quence {Pt Fjk }∞ k=1 of {Pt Fj }j =1 such that its Cesaro mean fj := j k=1 Pt Fjk →

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HIROSHI KAWABI

Pt F in D(E) as j → ∞. Therefore we also have the convergence of the left-hand side of (2.10). This completes the proof. 2

3. Proof of Theorem 1.1 In this section, we devote ourselves to give a proof of Theorem 1.1. In this paper, we employ a stochastic approach positively. So it is different from the original functional analytic proof as Wang [22] and [1]. Especially, we use Itô’s formula for martingales when we need to expand the term (Pt F )α . This method is due to Albeverio and Kusuoka [4]. First, we prepare a new probability measure which is important to show the differentiability property of functions in D(E). We fix h ∈ C0∞ (R, Rd ) and t > 0 in this section. We assume supp h ⊂ (−T , T ). We define a cut-off function φ ∈ C0∞ (R, R) by φ(x) ≡ 1 for |x|  T and φ(x) ≡ 0 for |x|
T + 1. For w ∈ C, we define V (w(x)) := K3 (1 + |w(x)|p+1 )φ(x),

x ∈ R.

(3.1)

Here K3 := K3 (p, K2 ) is a sufficient large constant which will be determined in the proof of Lemma 3.1. p and K2 are positive constants in the condition (U2). By using this function, we define a weighted Gibbs measure µV by 

−1 V µ (dw) := ZV exp − V (w(x)) dx µ(dw), R

where ZV is the normalization constant. Clearly, this measure is equivalent to the original Gibbs measure µ. Then we can state the following: LEMMA 3.1. (1) For F ∈ Bb (E, R) and k ∈ C0∞ (R, Rd ), the following quasiinvariance of µV holds:  F (w + k)µV (dw) E

  F (w) exp (k, w) + (V (w(x)) − V (w(x) − k(x))) dx µV (dw). = E

R

(3.3)

(2) Let F ∈ L2 (dµ) and v(·) ∈ C([0, t], R). Then there exists a positive constant K4 := K4 (hL∞ , K2 , K3 , p, T , vL∞ ) such that

1/2   −1 K4 V 2 |F (w + v(s)h)|µ (dw)  ZV e |F (w)| µ(dw) (3.4) E

for any 0  s  t.

E

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PARABOLIC HARNACK INEQUALITY

(3) Let F ∈ D(E) and v(·) ∈ C 1 ([0, t], R) such that v(0) = 0 and v(t) = 1. Then F (· + v(s)h) : s ∈ [0, t] → L1 (dµV ) is a C 1 -function. Moreover the following identity holds for 0 < s < t: d F (· + v(s)h) = (DF (· + v(s)h), v (s)h)H . ds

(3.5)

Proof. By recalling (1.5) and (1.6), we can easily show (3.3). So we aim to prove the assertion (2). By (3.3) and Schwarz’s inequality, we have  |F (w + v(s)h)|µV (dw) E

 ZV−1

 |F (w)|2 µ(dw) E

E

 −

1/2   · exp 2 (w, v(s)h)

R



1/2 V (w(x) − v(s)h(x)) dx µ(dw) .

(3.6)

We can get the following estimate by recalling (3.1) and using Young’s inequality: 

(w, v(s)h) − V (w(x) − v(s)h(x)) dx R  T   U (w(x)) − U (w(x) − v(s)h(x)) dx  −T



− K3 

T

−T

(1 + |w(x) − v(s)h(x)|p+1 )φ(x) dx

T

v(s)(w(x), x h(x))Rd dx − −T    T p+2 p+2 p p+1 (1 + |w(x)| ) dx  2 K2 (1 + hL∞ )(1 + vL∞ ) −T

 K3 T p+1 p+1 p+1 |w(x)| dx + 2K3 T hL∞ vL∞ − p 2 −T

 T 2pT 1 (p+1)/p (p+1)/p p+1 x hL∞ |w(x)| dx + vL∞ + p + 1 −T p+1  p+2 p+2 p+1 p+1 = 2p+1 K2 T (1 + hL∞ )(1 + vL∞ ) + 2K3 T hL∞ vL∞

 p+2 p+2 + 2p K2 (1 + hL∞ )(1 + vL∞ ) +

2pT (p+1)/p (p+1)/p x hL∞ + vL∞ p+1

K3 1 − p p+1 2



T

−T

|w(x)|p+1 dx.



(3.7)

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HIROSHI KAWABI p+2

p+2

1 Now we choose K3 such that 2p K2 (1 + hL∞ )(1 + vL∞ ) + p+1 − K2p3 < 0 and set  p+2  p+2  p+1 p+1 K4 := 2p+1 K2 T 1 + hL∞ 1 + vL∞ + 2K3 T hL∞ vL∞ 2pT (p+1)/p (p+1)/p x hL∞ + vL∞ . p+1

Hence by combining (3.6) and (3.7), we have (3.4). Finally, we prove the assertion (3). For F ∈ D(E), we can choose {Fn }∞ n=1 ⊂ 1/2 ∞ F C b such that Fn → F in E1 -norm. Note that for any 0  s  1 and w ∈ E,  s (DFn (w + v(s)), v (τ )h)H dτ. (3.8) Fn (w + v(s)h) = Fn (w) + 0

By (3.4), we have the following:  |Fn (w + v(s)h) − F (w + v(s)h)|µV (dw) E



ZV−1 eK4

1/2

 |Fn (w) − F (w)| µ(dw) 2

−→ 0

as n → ∞,

(3.9)

E

  s

 

DFn (w + v(τ )h) − DF (w + v(τ )h), v (τ )h H dτ

µV (dw)

E 0  s 

dτ DFn (w + v(τ )h) −  sup v (τ )hH · 0τ t

0

E

− DF (w + v(τ )h)H µ (dw) V



ZV−1



sup v (τ )hH ·

0τ t

−→ 0

DFn (w) −

K4

dτ e 0

1/2



t

DF (w)2H µ(dw)

E

as n → ∞.

(3.10)

By virtue of (3.9) and (3.10), we can take the limit n → ∞ on both sides of (3.8). Then we have  s (DF (· + v(τ )h), v (τ )h)H dτ, 0  s  t. F (· + v(s)h) = F + 0

Therefore we can easily see (3.5). By the continuity of v (·) and (3.4), we have the continuity of dsd F (· + v(s)h) : s ∈ [0, t] → L1 (dµV ). This completes the proof. 2 Now we are in a position to give the proof of our main theorem.

73

PARABOLIC HARNACK INEQUALITY

Proof of Theorem 1.1. We may assume that F ∈ F C ∞ b , F (w) > δ > 0 since |Pt F (w)|  Pt |F |(w) holds generally. For fixed t > 0, we define v(·) ∈ C ∞ ([0, t], R) by  s −K r e 1 dr . v(s) := 0t −K1 r dr 0 e For α > 1 and h ∈ C0∞ (R, Rd ), we will consider a function G : [0, t] → D(E) ⊂ L1 (dµV ) by G(s) := Ps (Pt −s F )α (· + v(s)h). First we study the differentiability of G with respect to s. This is the most important property in this proof. We claim the following lemma: LEMMA 3.2. The following identity holds in L1 (dµV ): G (s) =

α(α − 1) Ps {(Pt −s F )α−2 D(Pt −s F )2H }(· + v(s)h) 2 + (D{Ps (Pt −s F )α }(· + v(s)h), v (s)h)H , 0 < s < t.

(3.13)

Proof. We consider a function H (r1 , r2 , r3 ) : (0, t) × (0, t) × (0, t) → L1 (dµV ) which is defined by H (r1 , r2 , r3 ) := Pr1 (Pt −r2 F )α (· + v(r3 )h). To show that H (r1 , r2 , r3 ) is a C 1 -function, we expand this function. Let L be the infinitesimal generator of the transition semigroup {Pt } on L2 (dµ). We note that Pt −r2 F ∈ D(L) ⊂ L2 (dµ) for F ∈ F C ∞ b . Hence we can consider a continuous [Pt−r2 F ] }0r1 t defined by {Fr1 }-martingale {Mr1  r1 [Pt−r2 F ] := (Pt −r2 F )(Xr1 ) − (Pt −r2 F )(X0 ) − L(Pt −r2 F )(Xτ ) dτ. Mr1 0

Here we mention that the quadratic variation of M [Pt−r2 F ] is given by  r1 [Pt−r2 F ]

r1 = D(Pt −r2 F )(Xτ )2H dτ. M

(3.14)

(3.15)

0

See Proposition 4.5 in Albeverio and Röckner [5] for the proof. We apply Itô’s formula for (3.14). Then we can expand (Pt −r2 F )α as  r1 [Pt−r F ] α α (Pt −r2 F )α−1 (Xτ ) dMτ 2 (Pt −r2 F ) (Xr1 ) = (Pt −r2 F ) (X0 ) + α 0  r1 +α (Pt −r2 F )α−1 (Xτ )L(Pt −r2 F )(Xτ ) dτ 0  α(α − 1) r1 (Pt −r2 F )α−2 (Xτ ) d M [Pt−r2 F ] τ . (3.16) + 2 0

74

HIROSHI KAWABI

By combining (3.15) and (3.16), we obtain the following expansion for any r1 , r2 , r3 ∈ [0, t]: Pr1 (Pt −r2 F )α (· + v(r3 )h) = E[(Pt −r2 F )α (X0·+v(r3)h )]  r1  α−1 ·+v(r3 )h ·+v(r3 )h + αE (Pt −r2 F ) (Xτ )Pt −r2 (LF )(Xτ ) dτ 0  r1  α(α − 1) α−2 ·+v(r3 )h ·+v(r3 )h 2 E (Pt −r2 F ) (Xτ )D(Pt −r2 F )(Xτ )H dτ + 2 0   r1 [Pt−r F ] (Pt −r2 F )α−1 (Xτ·+v(r3)h ) dMτ 2 + αE 0

= (Pt −r2 F )α (· + v(r3 )h)  r1 Pτ {(Pt −r2 F )α−1 Pt −r2 (LF )}(· + v(r3 )h) dτ +α 0  α(α − 1) r1 Pτ {(Pt −r2 F )α−2 D(Pt −r2 F )2H }(· + v(r3 )h) dτ. + 2 0

(3.17)

Hence for any r1 , r2 , r3 ∈ (0, t), we have ∂H (r1 , r2 , r3 ) = αPr1 {(Pt −r2 F )α−1 Pt −r2 (LF )}(· + v(r3 )h) ∂r1 α(α − 1) Pr1 {(Pt −r2 F )α−2 D(Pt −r2 F )2H }(· + v(r3 )h) + 2 (3.18) =: H1 (r1 , r2 , r3 ) + H2 (r1 , r2 , r3 ). (r , r , r ). By recalling the assertion (2) in Now we discuss the continuity of ∂H ∂r1 1 2 3 Lemma 3.1 and the contraction property of {Pt } in L2 (dµ), we have the following estimate for sufficient small numbers ε1 , ε2 , ε3 :     ∂H ∂H    ∂r (r1 + ε1 , r2 + ε2 , r3 + ε3 ) − ∂r (r1 , r2 , r3 ) 1 V 1

1

L (dµ )

2    Hi (r1 , r2 , r3 + ε3 ) − Hi (r1 , r2 , r3 )L1 (dµV ) i=1

+ ZV−1 eK4 Hi (r1 + ε1 , r2 + ε2 , 0) − Hi (r1 , r2 , 0)L2 (dµ) 



2   Hi (r1 , r2 , r3 + ε3 ) − Hi (r1 , r2 , r3 )L1 (dµV ) i=1

+ ZV−1 eK4 Hi (r1 + ε1 , r2 + ε2 , 0) − Hi (r1 + ε1 , r2 , 0)L2 (dµ)  + ZV−1 eK4 Hi (r1 + ε1 , r2 , 0) − Hi (r1 , r2 , 0)L2 (dµ) 

2   Hi (r1 , r2 , r3 + ε3 ) − Hi (r1 , r2 , r3 )L1 (dµV ) + i=1

75

PARABOLIC HARNACK INEQUALITY

+ ZV−1 eK4 Hi (0, r2 + ε2 , 0) − Hi (0, r2 , 0)L2 (dµ)

 + ZV−1 eK4 Hi (r1 + ε1 , r2 , 0) − Hi (r1 , r2 , 0)L2 (dµ) .

(3.19)

On the other hand, we note that the strongly continuity of {Pt } in L2 (dµ) implies lim E(Pε F − F )= − lim (Pε F − F, Pε (LF ) − LF )L2 (dµ)

ε→0

ε→0

= 0.

(3.20)

Hence we have lim H2 (0, r2 + ε2 , 0) − H2 (0, r2 , 0)L2 (dµ) = 0.

ε2 →0

(3.21)

By combining the assertion (3) in Lemma 3.1, (3.21) and the strongly continuity of {Pt }, (3.19) implies    ∂H  ∂H (r1 + ε1 , r2 + ε2 , r3 + ε3 ) − (r1 , r2 , r3 ) lim    1 V ,h = 0. ε1 ,ε2 ,ε3 →0 ∂r1 ∂r1 L (dµ ) Next, we discuss the continuity of for r1 , r2 , r3 ∈ (0, t):

∂H (r , r , r ) ∂r2 1 2 3

which is given by the following

  ∂H (r1 , r2 , r3 ) = −αPr1 (Pt −r2 F )α−1 Pt −r2 (LF ) (· + v(r3 )h). ∂r2

(3.23)

By using the same argument in (3.19) and the strongly continuity of {Pt }, we can easily have     ∂H ∂H (r1 + ε1 , r2 + ε2 , r3 + ε3 ) − (r1 , r2 , r3 ) lim   1 V = 0.  ε1 ,ε2 ,ε3 →0 ∂r2 ∂r2 L (dµ ) Next, we consider

∂H (r , r , r ). ∂r3 1 2 3

By virtue of (3.5), we have

∂H (r1 , r2 , r3 ) = (D{Pr1 (Pt −r2 F )α }(· + v(r3 )h), v (r3 )h)H . ∂r3

(3.25)

Here we denote H3 (r1 , r2 , r3 , r4 ) := (D{Pr1 (Pt −r2 F )α }(· + v(r3 )h), v (r4 )h)H . By using the similar argument in (3.19), we have the following estimate for sufficient small numbers ε1 , ε2 , ε3 : H3 (r1 + ε1 , r2 + ε2 , r3 + ε3 , r3 + ε3 ) − H3 (r1 , r2 , r3 , r3 )L1 (dµV )  H3 (r1 , r2 , r3 , r3 + ε3 ) − H3 (r1 , r2 , r3 , r3 )L1 (dµV )  + ZV−1 eK4 H3 (r1 + ε1 , r2 + ε2 , 0, r3 + ε3 ) − H3 (r1 + ε1 , r2 + ε2 , 0, r3 )L2 (dµ) + H3 (r1 + ε1 , r2 + ε2 , 0, r3 ) − H3 (r1 + ε1 , r2 , 0, r3 )L2 (dµ)  + H3 (r1 + ε1 , r2 , 0, r3 ) − H3 (r1 , r2 , 0, r3 )L2 (dµ) . (3.26)

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HIROSHI KAWABI

We treat the third term of the right-hand side in (3.26). We have H3 (r1 + ε1 , r2 + ε2 , 0, r3 ) − H3 (r1 + ε1 , r2 , 0, r3 )2L2 (dµ)    E Pr1 +ε1 {(Pt −r2 −ε2 F )α − (Pt −r2 F )α } v (r3 )h2H  = Pr1 +ε1 {(Pt −r2 −ε2 F )α − (Pt −r2 F )α },  LPr1 +ε1 {(Pt −r2 −ε2 F )α − (Pt −r2 F )α } L2 (dµ) · v (r3 )h2H .

(3.27)

By recalling (3.17), we have Pr1 +ε1 (Pt −r2 −ε2 F )α ∈ D(L) and LPr1 +ε1 (Pt −r2 −ε2 F )α = αPr1 +ε1 (Pt −r2 −ε2 F )α−1 · Pt −r2 −ε2 (LF ) α(α − 1) Pr1 +ε1 (Pt −r2 −ε2 F )α−2 · D(Pt −r2 −ε2 F )2H . + 2

(3.28)

Then by noting (3.20), (3.27) and (3.28), we have lim H3 (r1 + ε1 , r2 + ε2 , 0, r3 ) − H3 (r1 + ε1 , r2 , 0, r3 )L2 (dµ) = 0.

ε2 →0

(3.29)

Hence we can obtain lim

ε1 ,ε2 ,ε3 →0

H3 (r1 + ε1 , r2 + ε2 , r3 + ε3 , r3 + ε3 ) − H3 (r1 , r2 , r3 , r3 )L1 (dµV ) = 0

by using (3.26), (3.29) and the continuity of v (·). Therefore we can conclude that H (r1 , r2 , r3 ) is a C 1 -function. Hence we have the following calculation by combining (3.18), (3.23) and (3.25):

3 

∂H (r1 , r2 , r3 )

G (s) = ∂ri r1 =r2 =r3 =s i=1 =

α(α − 1) Ps {(Pt −s F )α−2 D(Pt −s F )2H }(· + v(s)h) 2 + (D{Ps (Pt −s F )α }(· + v(s)h), v (s)h)H .

This completes the proof.

2

Continuation of the proof of Theorem 1.1. By virtue of Lemma 3.2, we have the following estimate for 0 < s < t: G (s)


α(α − 1) Ps {(Pt −s F )α−2 D(Pt −s F )2H }(· + v(s)) 2 − D{Ps (Pt −s F )α }(· + v(s))H · v (s)hH .

Here we use Proposition 2.4. Then we can continue as G (s)


α(α − 1) Ps {(Pt −s F )α−2 D(Pt −s F )2H }(· + v(s)h) − 2

77

PARABOLIC HARNACK INEQUALITY K1 s

− e 2 Ps {D(Pt −s F )α H }(· + v(s)h) · v (s)hH α  = Ps (α − 1)(Pt −s F )α−2 D(Pt −s F )2H 2 K1 s  − 2e 2 v (s)hH · (Pt −s F )α−1 D(Pt −s F )H (· + v(s)h)   α Ps eK1 s v (s)h2H · (Pt −s F )α (· + v(s)h).
− 2(α − 1)   K12 e−2K1 s αeK1 s 2 α · h =− H · Ps (Pt −s F ) (· + v(s)h). (3.31) 2(α − 1) (1 − e−K1 t )2 By (3.31), we can get the following estimate for 0 < s < t: G (s) d log G(s) = ds G(s) K12 e−2K1 s αeK1 s · h2H .
− 2(α − 1) (1 − e−K1 t )2

(3.32)

By integrating both sides of (3.32) over s from 0 to t and letting δ ↓ 0, we obtain the inequality (1.8). We remark that the Feller continuity implies our assertion in the case of F ∈ Cb (E, R). If F ∈ L∞ (dµ), the C0∞ (R, Rd )-quasi-invariance of µ also implies our assertion. 2 Before closing this section, we present an application of Theorem 1.1. The following corollary is an H -smoothing property of the transition semigroup {Pt }: COROLLARY 3.3 (H -Smoothing Property). Let F ∈ L∞ (dµ). Then for every t > 0, the function (Pt F )(w + ·) : C0∞ (R, Rd ) ⊂ H → R is continuous for µ-a.e. w ∈ E. Proof. For F ∈ L∞ (dµ), we can take an approximate sequence {Fn }∞ n=1 ⊂ such that lim F − F  4 = 0. Since {P } is a contraction semigroup F C∞ n→∞ n t L (dµ) b on L2 (dµ), we have Pt |Fn − F |2 L2 (dµ)  Fn − F 2L4 (dµ) . Then for every t > 0, limn→∞ Pt |Fn − F |2 L2 (dµ) = 0. This implies that limn→∞ Pt (|Fn − F |2 )(w) = 0 for µ-a.e. w ∈ E. By substituting Fn − F to both sides of (1.8) and α = 2, we have sup |(Pt Fn )(w + h) − (Pt F )(w + h)|

hH R

 K1 R 2 . · exp 2(1 − e−K1 t ) 

 (Pt |Fn − F | )(w) 2

1/2

(3.33)

Then by Pt Fn ∈ Cb (E, R) and (3.33), the sequence {(Pt Fn )(w + ·)}∞ n=1 converges to (Pt F )(w + ·) uniformly in the wide sense. This and the C0∞ (R, Rd )-quasiinvariance of µ lead to our assertion. 2

78

HIROSHI KAWABI

4. Certain Lower Estimate and the Varadhan Type Asymptotics of the Transition Probability In this section, we present an application of Theorem 1.1. Here we give a certain lower estimate of pt (A, B) in terms of the geometric H -distance, where pt (A, B) is defined for Borel measurable sets A, B ⊂ E by   Pt 1B (w)µ(dw) = Pt 1A (w)µ(dw). (4.1) pt (A, B) := A

B

Here 1A is the indicator function on A. Briefly speaking, this is the probability of our dynamics M starting from A and reaching B at time t. We note that the symmetry of {Pt } leads to the second equality of (4.1). Following to [1] and [3], we define the H -distance between two Borel measurable sets in E. For u, v ∈ E, we define dH (u, v) by  u − vH if u − v ∈ H , (4.2) dH (u, v) := +∞ otherwise. For a Borel measurable set A ⊂ E, we define the distance function dH (·, A) : E → R by dH (u, A) := infv∈A dH (u, v). We also define the distance dH (A, B) between two Borel measurable sets A, B ⊂ E with µ(A), µ(B) > 0 as follows:  ˜ ess inf dH (v, A) ˜ | dH (A, B) := sup ess inf dH (u, B), u∈A

v∈B

˜ B˜ ⊂ E are σ -compact sets with A,

 ˜ ∪ (A˜ \ A)) = µ((B \ B) ˜ ∪ (B˜ \ B)) = 0 . µ((A \ A)

(4.3)

Before giving our lower estimate, we recall the notion of H -open set from [1]. We call that a Borel measurable set A ⊂ E is a H -open set if for any u ∈ A, there exists ε > 0 such that {u + h | h ∈ H, hH < ε} ⊂ A holds. This is a weaker notion than a open set. The following direct lower estimate of pt (A, B) is our main theorem in this section: THEOREM 4.1. Let A, B ⊂ E be Borel measurable sets with µ(A), µ(B) > 0 and assume that dH (A, B) < ∞ and A or B is H -open. Then for any α > 2 and ε > 0, there exists C(α, ε, A, B) > 0 such that   K1 α(dH (A, B) + ε)2 · (4.4) pt (A, B)
C(α, ε, A, B) · exp − 2(α − 1) 1 − e−K1 t holds. Next we study the small time behavior of pt (A, B) as a corollary of Theorem 4.1. Especially, we establish the Varadhan type asymptotics as follows:

PARABOLIC HARNACK INEQUALITY

79

THEOREM 4.2 (Varadhan Type Asymptotics). Let A, B ⊂ E be Borel measurable sets with µ(A), µ(B) > 0 and assume that A or B is H -open. Then the following asymptotics holds: lim 2t log pt (A, B) = −dH (A, B)2 .

t →0

(4.5)

Recently, some authors have studied this asymptotics for infinite-dimensional diffusion processes. Fang [8] proved this asymptotics for the standard Ornstein– Uhlenbeck process on the Wiener space. Zhang [23] generalized Fang’s result by using a standard large deviation theory. After [1], Hino [12], Ramirez [20] and Hino and Ramirez [13] also studied this asymptotics for symmetric diffusion processes in general state spaces. They established this asymptotics in terms of the intrinsic distance. Here we emphasize that they studied only this small time asymptotics. On the other hand, our result Theorem 4.1 gives a explicit lower estimate of pt (A, B). Hence it seems that Theorem 4.1 is a stronger result in this viewpoint. Now we will give the proof of Theorem 4.1 and Theorem 4.2. As a preparation, we study some fundamental properties of the distance function dH (·, A) : E → [0, ∞]. We can summarize as the following proposition: PROPOSITION 4.3. Let A be a σ -compact subset of E with µ(A) > 0, then the following assertions hold. (1) dH (·, A) : E → [0, ∞] is Borel measurable. (2) Let F := dH (·, A) ∧ n for a fixed n ∈ N, then |F (w + h) − F (w)|  hH

(4.6)

holds for every w ∈ E and h ∈ H . (3) F ∈ D(E) and DF (w)H  1 for µ-a.e. w ∈ E. (4) Suppose that µ satisfies the Poincaré inequality relative to E, i.e., there exists CP > 0 such that f − Eµ [f ]2L2 (dµ)  CP E(f )

(4.7)

holds for any f ∈ D(E). Then there exist C1 , C2 > 0 such that Eµ [exp(C1 dH (·, A))]  C2 . Consequently, µ({w ∈ E | dH (w, A) < ∞}) = 1 holds. REMARK 4.4. If the potential function U is strictly convex, i.e., infz∈Rd ∇ 2 U (z) > 0, the logarithmic Sobolev inequality holds. See Theorem 4.1 in Funaki [11] for details. Hence it can be a sufficient condition such that the assertion (4) in Proposition 4.3 and dH (A, B) < ∞ hold. Proof. By a slight modification of the proof of Theorem 4.1 in Kusuoka [17], we have the assertion (1). We can easily show the assertion (2) by using the same

80

HIROSHI KAWABI

argument in [1] and [8]. Now we prove the assertion (3). As the same argument of (2.11) and (2.12), we can get |(Pt F )(w + h) − (Pt F )(w)|  e

K1 t 2

hH .

(4.8)

Here we choose T > 0 such that the Lipschitz estimate |(Pt F )(w +h)−(Pt F )(w)|  2hH holds for any w ∈ E, h ∈ H and 0 < t < T . Then this estimate implies D(Pt F )(w)H  2 for µ-a.e. w ∈ E by Lemma 1.3 in [16]. Now we show F ∈ D(E). We note that E1 (Pt F )  2 + n2 and Pt F ∈ D(E) hold for any 0 < t < T . Hence there exist a sequence {tm }∞ m=1 with limm→∞ tm = 0 and F∗ ∈ D(E) such that Ptm F → F∗ weakly in D(E) as m → ∞. Therefore we have Ptm F → F∗ weakly in L2 (dµ) as m → ∞. On the other hand, we have Pt F → F strongly in L2 (dµ) as t → 0. Hence F = F∗ ∈ D(E) holds. By recalling (4.6) and F ∈ D(E), we also have DF (w)H  1 for µ-a.e. w ∈ E. This completes the proof of the assertion (3). Next we begin to show the assertion (4). For F which is defined in the assertion (2), we have µ({w ∈ E | F (·, A)  1})
µ(A). Therefore (4.7) and Lemma 2.2 in Aida and Stroock [2] imply that |Eµ [F ]|  1 + µ(A)−1/2 · F − Eµ [F ]L2 (dµ)  1 + µ(A)−1/2 CP E(F )1/2 . 1/2

(4.9)

Now we use the following inequality due to Theorem 2.5 in [2]: 



F Eµ [F ] µ  K exp √ , E exp √ 2CP |||F |||∞ 2CP |||F |||∞  −m −2m ) and |||F |||∞ is defined by where K = ∞ m=1 (1 − 4   1 DF (u)2H G(u)µ(du) | |||F |||2∞ := sup 2 E G ∈ D(E) ∩ Bb (E), GL1 (dµ)

(4.10)

 =1 .

√ In our case, |||F |||∞ = 1/ 2 holds by remembering the assertion (3). Then by (4.9) and (4.10), we have 

 µ

dH (·, A) ∧ n E [F ] µ √  K exp √ E exp CP CP  1/2  1 + 2−1/2 µ(A)−1/2 CP √ . (4.12)  K exp CP Finally, by letting n → ∞ in (4.12), we obtain the inequality in the assertion (4). Hence we complete the proof. 2

PARABOLIC HARNACK INEQUALITY

81

By noticing the assertion (3) in Proposition 4.3, we can use Lyons–Zheng’s decomposition formula. Hence we can present the following proposition by following a parallel argument to [8, 1] and [3]: PROPOSITION 4.5. Let A, B ⊂ E be Borel measurable sets with µ(A), µ(B) > 0. Then lim sup 2t log pt (A, B)  −dH (A, B)2 . t →0

(4.13)

Before starting the proof of Theorem 4.1, we have to prepare a diagonal lower estimate. We note that the symmetry of {Pt } and Proposition 4.5 lead to the following proposition. The reader is referred to Theorem 2.11 in [1] for the proof. PROPOSITION 4.6. Let A ⊂ E be a Borel measurable set with µ(A) > 0. Then the following estimate holds for any 1 < p < ∞ and t > 0:

1 2 µ(A)2 · 1− , (4.14) pt (A, A)
µ(AK∗ √t ) p  2p ). where Ar := {w ∈ E | dH (w, A)  r} and K∗ := 2 log( µ(A) Now we are in a position to give the proof. This proof is based on [3]. Proof of Theorem 4.1. Without loss of generality, we can assume that B is H open. First, we recall Definition 3.4 and Lemma 3.5 in [1]. Since B is H -open and dH (A, B) < ∞, for any ε > 0, there exist a Borel set D ⊂ A with µ(D) > 0 and h ∈ C0∞ (R, Rd ) such that D + h ⊂ B and hH  dH (A, B) + ε hold. Then by using C0∞ (R, Rd )-quasi-invariance of µ, we easily get  1D+h (w)Pt 1D (w)µ(dw) pt (A, B)
E 1D+h (w + h)Pt 1D (w + h) exp( (−h, w))µ(dw) = E Pt (1D )α (w + h) exp ( (−h, w))µ(dw). (4.15) = D

Now we use Theorem 1.1. We have

 K1 −αh2H · |Pt 1D (w)|α exp( (−h, w))µ(dw) · pt (A, B)
exp 2(α − 1) 1 − e−K1 t D

−1

 K1 −αh2H · exp(− (−h, w))µ(dw)
exp 2(α − 1) 1 − e−K1 t E 

2 × |Pt 1D (w)|α/2 µ(dw) , (4.16) D

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HIROSHI KAWABI

where, in the passage from the first to the second line, we have used Schwarz’s inequality. On the other hand, we use Hölder’s inequality by noting α > 2. Then we have 

α 

2 Pt 1D (w)µ(dw)  |Pt 1D (w)|α/2 µ(dw) · µ(D)α−2 . (4.17) D

D

Next we use Proposition 4.6 in the case of p = 1/2. Then we can obtain the following by combining (4.16) and (4.17).

α  2−α Pt 1D (w)µ(dw) pt (A, B)
µ(D)  ×

D

exp(− (−h, w))µ(dw) E

2α

−1



K1 −αh2H · · exp 2(α − 1) 1 − e−K1 t



1 µ(DK∗ √t )−α µ(D)2α µ(D)2−α 2



−1 K1 −αh2H · × exp(− (−h, w))µ(dw) · exp 2(α − 1) 1 − e−K1 t E 2α

−1  1
µ(D)α+2 exp(− (−h, w))µ(dw) 2 E   K1 −α(dH (A, B) + ε)2 · × exp , (4.18) 2(α − 1) 1 − e−K1 t  where K∗ = 2 − log µ(D). Finally, by setting

−1  2α 1 α+2 µ(D) exp(− (−h, w))µ(dw) C(α, ε, A, B) := 2 E


in (4.18), we complete the proof.

2

Proof of Theorem 4.2. In the case of dH (A, B) = ∞, Proposition 4.5 implies (4.5). So we only show in the case of dH (A, B) < ∞. By letting α → ∞ and ε ↓ 0 in (4.4), we have lim 2t log pt (A, B)
−dH (A, B)2 .

t →0

Hence we can complete the proof by combining this estimate and Proposition 4.5. 2

Acknowledgment The author would like to express his sincere gratitude to Professors Shigeki Aida, Shigeo Kusuoka and anonymous referee for their valuable suggestions and pointing

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out some errors during the preparation of this paper. He also thanks Professors Tadahisa Funaki, Kazuhiro Kuwae and Michael Röckner for hearty comments.

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22. 23.