Long Time Behavior of the Volume of the Wiener Sausage on Dirichlet Spaces

Abstract

In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces. We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure. We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates. We show that the growth rate of the expectations on a bounded modification of the Euclidian space is identical with the one on the Euclidian space equipped with the Lebesgue measure. We give an example of a metric measure Dirichlet space on which a scaled of the means fluctuates.

Appendices

Appendix: Several Claims used in the Proof of Theorem 1.2

In this section we state several claims used in the proof of Theorem 1.2. Most proofs depend on [26]. We remark that [26] deals with jump processes, however it holds also for diffusions satisfying Vol(V ; α₁,α₂) and HK(ϕ; β₁,β₂).

0-1 Law

Throughout this subsection, we assume that Vol(V ; α₁,α₂) and HK(ϕ; β₁,β₂) hold for certain V and ϕ satisfying Eqs. 1.7 and 1.11 respectively.

Theorem A.1

(The Barlow-Bass zero-one law [3, Theorem 8.4]) Let Abe a tail event, that is,A ∈∩_t> 0σ({X_u,u ≥ t}).Then, eitherP^x(A) = 0 holdsfor everyx ∈ M,or,P^x(A) = 1 holdsfor everyx ∈ M.

By [22, Corollary 5.12 and Theorem 2.12], we have that

Proposition A.2

(Hölder continuity for the heat kernel) There exist two positive constants Cand𝜃suchthat

$$\left|p(t,x,x) - p(t,x,y)\right| \le \frac{C}{V(\phi^{-1}(t))} \left( \frac{d(x,y)}{\phi^{-1}(t)}\right)^{\theta}, \ \ d(x,y) \le \phi^{-1}(t). $$

Since we assume HK(ϕ; β₁,β₂), our framework is a little more general than [3]. However, we can show Theorem A.1 as in [3, Theorem 8.4] once we have Proposition A.2 above, so we do not give the proof. We remark that we need the uniform volume growth condition Eq. 1.8 for the proof, and hence Theorem A.1 is not applicable to the framework of [21, Subsection 4.3]. Informally this occurs due to the lack of homogeneity of the framework of [21, Subsection 4.3]. See also Subsection 2.1.

Chung-Type Liminf Laws of the Iterated Logarithm

Throughout this subsection, we assume that Vol(V ; α₁,α₂) and HK(ϕ; β₁,β₂) hold for certain V and ϕ satisfying Eqs. 1.7 and 1.11 respectively.

Theorem A.3

(cf. [26, Theorem 3.18]) There exists a constant Csuch that for everyx ∈ M,

$$\underset{t \to \infty}{\liminf} \frac{\sup_{s \le t} d(X_{0}, X_{s})}{\phi^{-1}(t /\log\log t)} \le C, \textup{ \(P^{x}\)-a.s.} $$

We remark that we do not need to assume FHK(ϕ; β₁,β₂) above. The proof of Theorem A.3 goes in the same manner as in the proof of [26, Theorem 3.18] if we have the following two assertions.

Lemma A.4

(Estimate for exit time; cf. [26, (3.4)]) There exists a positive constant csuch that for anyx ∈ Mand anyt,r > 0,

$$P^{x} \left( \sup_{s \le t} d(x, X_{s}) \ge r\right) \le c\frac{t}{\phi(r)}. $$

The assumption for the upper bound of the heat kernel Eq. 1.10 is sufficient to show this. The proof goes in the same manner as in the proof of [26, (3.4)].

Proposition A.5

(cf. [26, Proposition 2.12]) Assume that Vol(V ; α₁,α₂) andHK(ϕ; β₁,β₂). Then, there exist apositive constant cand$a_{1}^{*}, a_{2}^{*} \in (0,1)$suchthat

$${(a_{1}^{*})}^{n} \le P^{x} \left( \underset{s \in [0, c \phi(r) n]}{\sup} d(x, X_{s}) \le r \right) \le (a_{2}^{*})^{n}. $$

holds for every n ≥ 1, r > 0 and x ∈ M.

The proof goes in a manner similar to the proof of [26, Proposition 2.12].

Remark A.6

The proof of [26, Proposition 2.12] depend on [26, Proposition 2.11]. The proof of [26, Proposition 2.11] uses a chaining argument of [6, Lemma 2.3]. The chaining argument is done under the assumption that the metric is geodesic. However, [26, Proposition 2.12] does not assume that the metric is geodesic. This does not imply any major problems, because it is easy to show [26, Proposition 2.12] without using the full statement of [26, Proposition 2.11] but using a partial version of [26, Proposition 2.11], which is obtained by replacing B(x,r/2) in [26, Proposition 2.11] with B(x,δ₀r/2) where δ₀ denotes the same constant as in [26, Proposition 2.11].

Laws of Iterated Logarithms for Local Time

Throughout this subsection, we assume that Vol(V ; α₁α₂) and FHK(ϕ; β₁,β₂) hold for certain V and ϕ satisfying Eqs. 1.7 and 1.11 respectively, and furthermore α₂ < β₁. Under these assumptions we have Eq. 2.9, and in the same manner as in the proof of [26, Proposition 4.3], we can show that a version of the local time of X exists, specifically, there exists a random field ℓ(t,x)(ω) satisfying the following conditions: (1) ℓ(t,x)(ω) is jointly measurable with respect to (t,x,ω), and

$${{\int}_{0}^{t}} h(X_{s}) ds = {\int}_{M} h(x) \ell (t,x) \mu(dx) $$

holds for every t > 0 and every Borel measurable function h on M.

(2) We have that for every λ > 0 and every x,y ∈ M,

$$E^{x} \left[ {\int}_{0}^{+\infty} \exp(-\lambda t) d\ell(t,y)\right] = {\int}_{0}^{+\infty} \exp(-\lambda t) p(t,x,y) dt, $$

where for every fixed yinM, dℓ(t,y) denotes the measure with respect to t. These conditions correspond to [26, property (1) and (2) of Proposition 4.3] respectively.

Theorem A.7

(cf. [26, Upper bounds for Theorems 4.11 and 4.15]) There exist twoconstants$c_{\sup }$and$c_{\inf }$suchthat for everyx ∈ M,

• (i)
  $$\underset{t \to \infty}{\limsup} \frac{\sup_{z \in M} \ell(t,z)}{t/V(\phi^{-1}(t/\log\log t))} \le c_{\sup}, \textup{ \(P^{x}\)-a.s.} $$
• (ii)
  $$\underset{t \to \infty}{\liminf} \frac{\sup_{z \in M} \ell(t,z)}{(t/\log\log t)/V(\phi^{-1}(t/\log\log t))} \le c_{\inf}, \textup{ \(P^{x}\)-a.s.} $$

The most important step of the proof is establishing the following assertion.

Proposition A.8)

(Modified statement of [26, Proposition 4.8] There exist two positive constantsc₁,c₂suchthat for everyA,L,u > 0,

$$\begin{array}{@{}rcl@{}} &&P^{z} \left( \sup\limits_{d(x,y) \le L} \sup\limits_{t \in [0, u]} |\ell(t,x) - \ell(t,y)| > A \right)\\ &\le& c_{1} \frac{V(\max\{L, \phi^{-1}(u)\})^{2}}{V(L)^{2}} \exp\left( -c_{2} A \left( \frac{V(L) V(\max\{L, \phi^{-1}(u)\})}{\phi(L) \phi(\max\{L, \phi^{-1}(u)\})}\right)^{1/2}\right). \end{array} $$

Given this assertion, Theorem A.7 can be proved similarly to [26, Proof of Theorems 4.11 and 4.15]. The statement is changed from [26, Proposition 4.8], accompanying a modification Eq. 1 in the proof below.

Outline of Proof of Proposition A.8

The proof goes in the same manner as in the proof of [26, Proposition 4.8] with minor modifications.

For δ > 0, we define the following: d_(δ)(x,y) := d(x,y)/δ. μ_(δ)(A) := μ(A)/V (δ). Let $X^{(\delta )}_{t} := X_{\phi (\delta ) t}$. Then,

$$p_{(\delta)}(t,x,y) = V(\delta) p(\phi(\delta)t, x, y).$$

Let

$$V_{(\delta)}(r) := V(\delta r) / V(\delta), \ \phi_{(\delta)}(r) := \phi(\delta r) / \phi(\delta). $$

Then,

$$\frac{V(\delta)}{V(\phi^{-1}(\phi(\delta)t))} = \frac{V(\delta)}{V\left( \delta \frac{\phi^{-1}(\phi(\delta)t)}{\delta}\right)} = \frac{1}{V_{(\delta)}\left( \frac{\phi^{-1}(\phi(\delta)t)}{\delta}\right)} = \frac{1}{V_{(\delta)}(\phi_{(\delta)}^{-1}(t))}. $$

By Eqs. 1.7 and 1.11, we also have that

$$\left( \frac{R}{r}\right)^{\alpha_{1}} \le \frac{V_{(\delta)} (R)}{V_{(\delta)} (r)} \le \left( \frac{R}{r}\right)^{\alpha_{2}}, \ 0 < r < R, $$

and,

$$\left( \frac{R}{r}\right)^{\beta_{1}} \le \frac{\phi_{(\delta)} (R)}{\phi_{(\delta)} (r)} \le \left( \frac{R}{r}\right)^{\beta_{2}}, \ 0 < r < R. $$

Let

$${\Psi}_{(\delta)} (r, t) = \sup_{s > 0} \frac{r}{s} - \frac{t}{\phi_{\delta}(s)}. $$

Then

$${\Psi} (d(x,y), \phi(\delta)t) = {\Psi}_{(\delta)}(d_{(\delta)(x,y)}, t). $$

By these results and Eqs. 1.9 and 1.10, we have that for every δ > 0,

$$\frac{c_{5}}{ \mu_{(\delta)}(B_{(\delta)}(x,\phi^{-1}(t)))} \le p_{(\delta)}(t,x,y), \ d_{(\delta)}(x,y) \le c_{6}\phi_{(\delta)}^{-1}(t), $$

and,

$$p_{(\delta)}(t,x,y) \le \frac{c_{7} \exp\left( -c_{8} {\Psi}_{(\delta)}(d_{(\delta)}(x,y), t) \right)}{ \mu_{(\delta)}(B_{(\delta)}(x,\phi_{(\delta)}^{-1}(t)))}, \ x,y \in M, t > 0, $$

where c₅,c₆,c₇ and c₈ are the same constants as in Eqs. 1.9 and 1.10.

Let ℓ^(δ)(t,x) be a local time for the scaled process X^(δ) with respect to μ_(δ). We assume that it satisfies [26, property (1) and (2) of Proposition 4.3] for the scaled setting and furthermore it is jointly continuous almost surely. Let $P_{(\delta )}^{x}$ be the probability law of X^(δ) starting at x ∈ M.

Henceforth, we let $\delta ^{\prime } := 1/\delta$. Then for $\delta = \min \{1/\phi ^{-1}(u), 1/L\}$,

$$\begin{array}{@{}rcl@{}} &&P^{z} \left( \sup\limits_{d(x,y) \le L} \sup\limits_{t \in [0, u]} |\ell(t,x) - \ell(t,y)| > A \right)\\ &\le& P_{(\delta^{\prime})}^{z} \left( \sup\limits_{d_{(\delta^{\prime})} (x,y) \le \delta L} \sup\limits_{t \in [0, 1]} \left|\ell^{(\delta^{\prime})}(t,x) - \ell^{(\delta^{\prime})}(t,y)\right| > A\frac{V(\delta^{\prime})}{\phi(\delta^{\prime})} \right). \end{array} $$

See [26, (4.20)] for details.

Let

$$ U_{(\delta^{\prime})}(r) := \sqrt{\frac{\phi_{(\delta^{\prime})}(r)}{V_{(\delta^{\prime})}(r)}}. $$

(20)

(This part is different from the proof of [26, Proposition 4.8].) Let

$$F_{\delta^{\prime}} \!:=\! \int{\int}_{d_{(\delta^{\prime})}(x,y) \le 1} \left( \!\exp\!\left( c_{*} \frac{\sup_{t \in [0, 1]} \left|\ell^{(\delta^{\prime})}(t,x) - \ell^{(\delta^{\prime})}(t,y)\right| }{U_{(\delta^{\prime})}(d_{(\delta^{\prime})}(x,y))} \right) - 1\!\right) \mu_{(\delta^{\prime})}(dx)\mu_{(\delta^{\prime})}(dy) $$

for a sufficiently small constant c_∗ > 0.

We assume that

$$ \sup_{\delta > 0} E_{(\delta^{\prime})}\left[F_{\delta^{\prime}}\right]< +\infty. $$

(21)

By Garsia’s lemma stated in [26, Lemma A.1], there exists two positive constants c₃ and c₄ independent from δ such that

$$\left|\ell^{(\delta^{\prime})}(t,x) - \ell^{(\delta^{\prime})}(t,y)\right| \le c_{3} U_{(\delta^{\prime})}(\delta L) \log\left( 1 + c_{4} \frac{F_{\delta^{\prime}}}{V_{(\delta^{\prime})}(\delta L)^{2}}\right) $$

holds for $\mu _{(\delta ^{\prime })}$-a.e. x,y ∈ M satisfying that $d_{(\delta ^{\prime })}(x,y) \le L$ and t ∈ [0, 1].

By using this and the joint continuity of local time, we can derive that

$$\begin{array}{@{}rcl@{}} &&P_{(\delta^{\prime})}^{z} \left( \sup\limits_{d_{(\delta^{\prime})} (x,y) \le \delta L} \sup\limits_{t \in [0, 1]} \left|\ell^{(\delta^{\prime})}(t,x) - \ell^{(\delta^{\prime})}(t,y)\right| > A\frac{V(\delta^{\prime})}{\phi(\delta^{\prime})} \right)\\ &\le& \frac{c_{5}}{V_{(\delta^{\prime})}(\delta L)^{2}} \exp\left( -\frac{A V(\delta^{\prime})}{c_{3} U_{(\delta^{\prime})}(\delta L) \phi(\delta^{\prime})}\right) \left( 1 + c_{4} \frac{E_{(\delta^{\prime})} [F_{\delta^{\prime}}]}{V_{(\delta^{\prime})}(\delta L)^{2}}\right) \end{array} $$

Hence the assertion follows once we show Eq. 21.

We can show Eq. 21 in the same manner as in the proof of [26, Proposition 4.8]. In the proof, [26, Proposition 2.11] is used. However, the issue raised in Remark A.6 does not affect any estimates in our proof. □