Random-field random surfaces

Abstract

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued random-field random surfaces of the $\nabla \phi$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions $d\ge 3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes in dimensions $d\ge 5$. It is further shown that for the integer-valued random-field Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and localizes in dimensions $d\ge 3$. (ii) The surface delocalizes in dimensions $d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at low temperature and weak disorder strength. The behavior in dimensions $d\ge 3$ at high temperature or strong disorder is left open. The proofs rely on several tools: Explicit identities satisfied by the expectation of the random surface, the Efron–Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn (Comm Math Phys 185(1): 1-36, 1997) and the Nash–Aronson estimate.

Appendix A. Nash–Aronson estimates for the Dirichlet problem

In this section, we prove the Nash–Aronson estimate in finite volume stated in Proposition 3.3. The proof builds upon the infinite-volume result of [43, Appendix B] stated below (which itself builds upon the infinite-volume and continuous estimate of Aronson [6]). We first introduce the infinite-volume heat kernel and state the Nash–Aronson estimate for discrete, time-dependent and uniformly elliptic environment of Giacomin–Olla–Spohn [43].

Definition A.1

Let $s_0 \in \mathbb {R}$. For each continuous, time-dependent, uniformly elliptic environment ${\textbf{a}}: [s_0 , \infty ) \times E(\mathbb {Z}^d) \rightarrow [c_- , c_+]$, each initial time $s \in [s_0 , \infty )$, and each point $y \in \Lambda _L$, we introduce the infinite-volume heat kernel $P_{{\textbf{a}}, \infty }$ to be the solution of the parabolic equation

$$\begin{aligned} \left\{ \begin{aligned} \partial _t P_{{\textbf{a}}, \infty } (t , x ; s ,y) - \nabla \cdot {\textbf{a}}\nabla P_{{\textbf{a}}, \infty } (t , x ; s ,y)&= 0{} & {} (t , x) \in (s , \infty ) \times \mathbb {Z}^d, \\ P_{{\textbf{a}}, \infty }(s , x ; s ,y)&= \mathbbm {1}_{\{ x = y \}}{} & {} x \in \mathbb {Z}^d. \end{aligned} \right. \end{aligned}$$

The next proposition establishes lower and upper bounds on the map $P_{{\textbf{a}}, \infty }.$

Proposition A.2

(Nash–Aronson estimates, Propositions B.3 and B.4 of [43]). There exist constants C, c depending on the dimension d and the ellipticity parameters $c_-, c_+$ such that, for any pair of times $s , t \in (s_0 , \infty )$ with $t \ge s$ and any pair of points $x , y \in \mathbb {Z}^d$,

$$\begin{aligned} P_{{\textbf{a}}, \infty } \left( t , x ; s ,y \right) \le \frac{C}{1 \vee (t-s)^{\frac{d}{2}}} \exp \left( - \frac{c|x - y|}{ 1 \vee \sqrt{t- s}} \right) . \end{aligned}$$

(A.1)

Under the additional assumption $|x - y| \le \sqrt{t - s}$, one has the lower bound

$$\begin{aligned} P_{{\textbf{a}}, \infty } \left( t , x ; s ,y \right) \ge \frac{c}{1 \vee (t-s)^{\frac{d}{2}}}. \end{aligned}$$

(A.2)

Remark A.3

The article of Giacomin–Olla–Spohn [43] only establishes the lower bound of the Nash–Aronson estimate in the on-diagonal case (i.e., under the assumption $|x - y| \le \sqrt{t - s}$). While it would be possible to obtain off-diagonal lower bounds (in the case $|x - y| \ge \sqrt{t - s}$) by adapting the techniques of [6, 30] (written in either the continuous setting or the discrete setting with a static environment), they are not necessary in the article [43] or in the proof of Theorem 2.

Remark A.4

Under the assumption $|x - y| \le \sqrt{t - s}$, the ratio $|x - y| / (1 \vee \sqrt{t- s})$ is smaller than 1 and the right-hand sides of (A.1) and (A.2) are of comparable sizes. The estimates are thus sharp up to multiplicative constants.

The proof of Proposition A.2 given below relies on analytic arguments. We mention that a more probabilistic approach, relying on the introduction of the random-walk whose generator is the operator $- \nabla \cdot {\textbf{a}}\nabla$ (following [43, Section 3.2]) and on stopping time arguments, would yield the same result.

Proof of Proposition 3.3

First let us note that by the change of variable $(t - s) \rightarrow (t-s)/c_-$, it is sufficient to prove the result when $c_- = 1$. Let us fix an integer $L\ge 0$, let $c_+ \in [1 , \infty )$ be an ellipticity constant, and let $s_0\in \mathbb {R}$. Let ${\textbf{a}}: [s_0 , \infty ) \times E(\Lambda _L^+) \rightarrow [1 , c_+]$ be a continuous time-dependent (uniformly elliptic) environment. For any $s \in [s_0 , \infty )$ and any vertex $y \in \Lambda _L$, we denote by $P_{\textbf{a}}(\cdot , \cdot ; s, x)$ the solution of the parabolic Eq. (3.3). We extend the environment ${\textbf{a}}$ to the space $[s_0 , \infty ) \times E(\mathbb {Z}^d)$ by setting ${\textbf{a}}(t , e) = c_+$ for any pair $(t , e) \in [s_0 , \infty ) \times \left( E(\mathbb {Z}^d) \setminus E\left( \Lambda _L^+ \right) \right)$, and let $P_{{\textbf{a}}, \infty }$ be the infinite volume heat kernel associated with the extended environment ${\textbf{a}}$ as defined in Definition A.1. We prove the upper and lower bounds of Proposition 3.3 separately. We will make use of the notation

$$\begin{aligned} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) }:= & {} \sum _{x \in \Lambda _L} P_{\textbf{a}}\left( t , x ; s , y \right) ^2 \text{ and } \\ \left\| P_{\textbf{a}}\left( t , x ; s , \cdot \right) \right\| ^2_{L^2 \left( \Lambda _L \right) }:= & {} \sum _{y \in \Lambda _L} P_{\textbf{a}}\left( t , x ; s , y \right) ^2 \end{aligned}$$

as well as, for any directed edge $e = (x , z) \in \textbf{E} \left( \Lambda _L^+ \right)$,

$$\begin{aligned} \nabla P_{\textbf{a}}\left( t , e ; s , y \right) = P_{\textbf{a}}\left( t , x ; s , y \right) - P_{\textbf{a}}\left( t , z ; s , y \right) \end{aligned}$$

and, following the conventions of Section 2.2,

$$\begin{aligned} \left\| \nabla P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L^+ \right) } = \sum _{e \in E \left( \Lambda _L^+ \right) } \left( \nabla P_{\textbf{a}}\left( t , e ; s , y \right) \right) ^2. \end{aligned}$$

Proof of the upper bound. We first note that the estimate (3.5) is equivalent to the two following inequalities: There exist constants c, C depending on $d , c_+$ such that

$$\begin{aligned} \left\{ \begin{aligned} P_{{\textbf{a}}}\left( t , x ; s , y \right)&\le \frac{C}{1 \vee (t-s)^{\frac{d}{2}}} \exp \left( - \frac{c|x - y|}{ 1 \vee \sqrt{t- s}} \right)&~\text{ if }~ (t - s) \le L^2, \\ P_{{\textbf{a}}}\left( t , x ; s , y \right)&\le \frac{C}{1 \vee (t - s)^\frac{d}{2}} \exp \left( - \frac{c(t - s)}{L^2} \right)&~\text{ if }~ (t - s) \ge L^2. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(A.3)

We first treat the case $(t - s) \le L^2$. By the maximum principle for the parabolic operator $\partial _t - \nabla \cdot {\textbf{a}}\nabla$, one has the estimate, for any $t, s \in [ s_0 , \infty )$ with $t \ge s$ and any $x , y \in \Lambda _L$,

$$\begin{aligned} P_{\textbf{a}}\left( t , x ; s , y \right) \le P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) . \end{aligned}$$

(A.4)

Combining the inequality (A.4) with Proposition A.2 yields the upper bound, for any $s , t \in (s_0 , \infty )$ with $t \ge s$ and any $x , y \in \mathbb {Z}^d$,

$$\begin{aligned} P_{\textbf{a}}\left( t , x ; s , y \right) \le \frac{C}{1 \vee (t-s)^{\frac{d}{2}}} \exp \left( - \frac{c|x - y|}{ 1 \vee \sqrt{t- s}} \right) . \end{aligned}$$

(A.5)

This is (A.3) in the case $t \le L^2$. We now focus on the case $t \ge L^2$. To this end, we denote by $C_{\textrm{Poinc}}$ the constant which appears in the Poincaré inequality, that is, the smallest constant which satisfies $\left\| u \right\| _{L^2 \left( \Lambda _\ell \right) }^2 \le C_\textrm{Poinc} \ell ^2 \left\| \nabla u \right\| _{L^2 \left( \Lambda _\ell ^+\right) }^2$ for any side length $\ell \in \mathbb {N}$ and any function $u : \Lambda _\ell ^+ \rightarrow \mathbb {R}$ normalized to be 0 on the boundary $\partial \Lambda _\ell$. We then let $c_1 := 1/C_\textrm{Poinc}$, and note that this constant depends only on the parameter d.

Using that the heat kernel $P_{\textbf{a}}$ solves the parabolic Eq. (3.3) and the discrete integration by parts (2.6), we have

$$\begin{aligned} \partial _t \left( e^{\frac{c_1(t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \right)&= \frac{c_1 e^{\frac{c_1(t-s)}{L^2}}}{L^2} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \nonumber \\&\quad - 2 e^{\frac{c_1(t-s)}{ L^2}}\sum _{e \in E \left( \Lambda _L^+ \right) } {\textbf{a}}(t , e) \left( \nabla P_{\textbf{a}}\left( t , e ; s , y \right) \right) ^2. \end{aligned}$$

(A.6)

Using the lower bound ${\textbf{a}}\ge 1$ on the environment, and the Poincaré inequality in the box $\Lambda _L$ (which can be applied since the mapping $x \mapsto P_{\textbf{a}}(t , x ; s , y )$ is equal to 0 on $\partial \Lambda _L$), we obtain

$$\begin{aligned} \sum _{e \in E \left( \Lambda _L^+ \right) } {\textbf{a}}(t , e) \left( \nabla P_{\textbf{a}}\left( t , e ; s , y \right) \right) ^2 \ge \frac{1}{C_{\textrm{Poinc}} L^2} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| _{L^2 \left( \Lambda _L \right) }^2. \end{aligned}$$

A combination of (A.6) and (A.7) with the definition of the constant $c_1$ above yields

$$\begin{aligned} \partial _t \left( e^{ \frac{c_1 (t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \right) \le - c_- e^{\frac{c_1 (t-s)}{L^2}} \left\| \nabla P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L^+ \right) }.\qquad \end{aligned}$$

(A.7)

By the Nash inequality (see [57]) and the non-negativity of the map $P_{\textbf{a}}$, there exists a constant $C_{\textrm{Nash}}$ depending only on the dimension d such that

$$\begin{aligned} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| _{L^2 \left( \Lambda _L \right) }^{1 + \frac{2}{d}} \le C_{\textrm{Nash}} \left( \sum _{x \in \Lambda _L} P_{\textbf{a}}\left( t , x ; s , y \right) \right) ^\frac{2}{d} \left\| \nabla P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| _{L^2 \left( \Lambda _L^+ \right) }.\nonumber \\ \end{aligned}$$

(A.8)

Applying the inequality (3.4) obtained in Section 3.3, we may simplify the inequality (A.8) and write

$$\begin{aligned} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| _{L^2 \left( \Lambda _L \right) }^{1 + \frac{2}{d}} \le C_{\textrm{Nash}} \left\| \nabla P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| _{L^2 \left( \Lambda _L^+ \right) }. \end{aligned}$$

(A.9)

Combining (A.7) and (A.9) yields

$$\begin{aligned} \partial _t \left( e^{\frac{c_1(t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \right)&\le - \frac{1}{C_{\textrm{Nash}}} e^{\frac{c_1(t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^{\frac{2d}{d+2}}_{L^2 \left( \Lambda _L \right) } \nonumber \\&\le - \frac{1}{C_{\textrm{Nash}}} \left( e^{\frac{c_1(t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^{2}_{L^2 \left( \Lambda _L \right) } \right) ^{\frac{d}{d+2}}. \nonumber \\ \end{aligned}$$

(A.10)

Integrating the differential inequality (A.10) and using the identity $\left\| P_{\textbf{a}}\left( s , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } = 1$, we obtain that there exists a constant C depending on d and $c_-$ such that, for any $t \ge s$,

$$\begin{aligned} e^{\frac{c_1(t-s)}{L^2}} \left\| P_{\textbf{a}}\left( t , \cdot ; s , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \le \frac{C}{1 \vee (t - s)^\frac{d}{2}}. \end{aligned}$$

(A.11)

We now show that the estimate (A.11) implies the inequality (A.3) in the case $(t -s) \ge L^2$. Using the convolution property for the heat kernel $P_{\textbf{a}}$, we have the identity, for any $t, s\in [s + L^2 , \infty )$ and any $x, y \in \Lambda _L$,

$$\begin{aligned} P_{\textbf{a}}(t , x ; s , y) = \sum _{z \in \Lambda _L} P_{\textbf{a}}\left( t , x ; \frac{t + s}{2} , z\right) P_{\textbf{a}}\left( \frac{t + s}{2} , z ; s , y \right) . \end{aligned}$$

The Cauchy-Schwarz inequality then yields

$$\begin{aligned} P_{\textbf{a}}(t , x ; s , y) \le \left\| P_{\textbf{a}}\left( t , \cdot ; \frac{t + s}{2} , y \right) \right\| _{L^2 \left( \Lambda _L \right) } \left\| P_{\textbf{a}}\left( \frac{t + s}{2} , x ; s , \cdot \right) \right\| _{L^2 \left( \Lambda _L \right) }.\nonumber \\ \end{aligned}$$

(A.12)

The first term of the right-hand side of (A.12) can be estimated by the upper bound (A.11) applied with the initial time $\frac{s + t}{2}$ instead of s. We obtain

$$\begin{aligned} \left\| P_{\textbf{a}}\left( t , \cdot ; \frac{s + t}{2} , y \right) \right\| ^2_{L^2 \left( \Lambda _L \right) } \le \frac{C e^{-\frac{c_1(t-s)}{2 L^2}} }{1 \vee (t - s)^\frac{d}{2}}. \end{aligned}$$

(A.13)

To estimate the second term in the right-hand side, we use the estimate (A.5), the assumption $t - s \ge L^2$, the observations that, for any pair of points $x , z \in \Lambda _L$, $|x - z| \le C L$, and that the cardinality of the box $\Lambda _L$ is equal to $(2L+1)^d$. We obtain

$$\begin{aligned} \left\| P_{\textbf{a}}\left( \frac{t + s}{2} , x ; s , \cdot \right) \right\| _{L^2 \left( \Lambda _L \right) }^2&\le \sum _{z \in \Lambda _L} \left( \frac{C}{1 \vee (t-s)^{\frac{d}{2}}} \exp \left( - \frac{c|x - y|}{ 1 \vee \sqrt{t- s}} \right) \right) ^2 \nonumber \\&\le \sum _{z \in \Lambda _L} \frac{C}{1 \vee (t-s)^{d}} \nonumber \\&\le \frac{CL^d}{1 \vee (t-s)^{d}} \nonumber \\&\le \frac{C}{1 \vee (t-s)^{\frac{d}{2}}}. \end{aligned}$$

(A.14)

A combination of (A.12), (A.13) and (A.14) completes the proof of (A.3) in the case $t \ge L^2$.

Proof of the lower bound. We first claim that there exists a constant $c_0 \in (0 , 1)$ depending on the parameters $d , c_+$ such that, for any $t , s \in [ s_0 , \infty )$ satisfying $\sqrt{t - s} \le c_0 L$ and any $y \in \Lambda _{L/2}$,

$$\begin{aligned} \sup _{(t' , x) \in [s , t] \times \partial \Lambda _{L}} P_{{\textbf{a}}, \infty } \left( t' , x ; s , y \right) \le \frac{1}{2} \inf _{x \in y + \Lambda _{\sqrt{t-s}}} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) . \end{aligned}$$

(A.15)

The proof of this inequality relies on Proposition A.2. First by the lower bound (A.2), we have the estimate, for any $t, s \in [s_0 , \infty )$ such that $t - s \le L^2$,

$$\begin{aligned} \inf _{x \in y + \Lambda _{\sqrt{t-s}}} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) \ge \frac{c}{1 \vee (t - s)^\frac{d}{2}} \ge \frac{c}{L^d}. \end{aligned}$$

(A.16)

Let us fix a constant $c_1 \in (0 , 1)$. Using that for any point $x \in \partial \Lambda _L$ and any point $y \in \Lambda _{L/2}$, we have $|x - y| \ge L/2$ together with the upper bound (A.1), we obtain the estimate, for any $L \ge c_1^{-1}$ and any $t, s \in [s_0 , \infty )$ satisfying $\sqrt{t - s} \le c_1 L$,

$$\begin{aligned} \sup _{(t' , x) \in [s , t] \times \partial \Lambda _{L}} P_{{\textbf{a}}, \infty } \left( t' , x ; s , y \right)&\le \sup _{(t' , x) \in [s , t] \times \partial \Lambda _L} \frac{C}{1 \vee (t'-s)^{\frac{d}{2}}} \exp \left( - \frac{c|x - y|}{ 1 \vee \sqrt{t'- s}} \right) \nonumber \\&\le \sup _{t' \in [s , t]} \frac{C}{1 \vee (t'-s)^{\frac{d}{2}}} \exp \left( - \frac{cL}{2 \left( 1 \vee \sqrt{t'- s} \right) } \right) \nonumber \\&\le \sup _{t' \in [0, c_1^2 L^2]} \frac{C}{ t'^{d/2}} \exp \left( - \frac{cL}{2\sqrt{t'}} \right) \nonumber \\&\le \sup _{t' \in [0, c_1^2]} \frac{C}{L^d t'^{d/2}} \exp \left( - \frac{c}{2\sqrt{t'}} \right) . \end{aligned}$$

(A.17)

Using that the mapping $t' \mapsto t'^{-d/2} \exp \left( - c/ (2 \sqrt{t'}) \right)$ tends to 0 as $t'$ tends to 0, we may select a constant $c_0 \in (0 , 1]$ such that

$$\begin{aligned} \sup _{t' \in [0, c_0^2]} \frac{C}{ t'^{d/2}} \exp \left( - \frac{c}{2 \sqrt{t'}} \right) \le \frac{c}{2}, \end{aligned}$$

(A.18)

where the constants c, C are the ones which appear in the right-hand sides of (A.16) and (A.17). Let us note that, since the constants c, C depend only on the dimension d and the ellipticity constant $c_+$, the constant $c_0$ may be chosen so that it depends only on $d , c_+$. Multiplying both sides of the inequality (A.18) by $L^{-d}$ yields, for any $t, s \in [s_0 , \infty )$ such that $\sqrt{t - s} \le c_0 L$,

$$\begin{aligned} \sup _{(t' , x) \in [s , t] \times \partial \Lambda _{L}} P_{{\textbf{a}}, \infty } \left( t' , x ; s , y \right)&\le \sup _{t' \in [0, c_0]} \frac{C}{2 L^d t'^{d/2}} \exp \left( - \frac{1}{2C \sqrt{t'}} \right) \\&\le \frac{c}{2L^d} \\&\le \inf _{x \in y + \Lambda _{\sqrt{t-s}}} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) . \end{aligned}$$

The proof of (A.15) is complete.

We now deduce the lower bound (3.6) from the inequality (A.15). To this end, let us fix a time $t \in (s , s + c_0 L^2)$, set $\varepsilon := \frac{1}{2} \inf _{x \in y + \Lambda _{\sqrt{t-s}}} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right)$, and define the map $P_{{\textbf{a}}, \infty }^\varepsilon := P_{{\textbf{a}}, \infty } - \varepsilon$. Let us note that the mapping $P_{{\textbf{a}}, \infty }^\varepsilon$ solves the parabolic equation

$$\begin{aligned} \partial _t P_{{\textbf{a}}, \infty }^\varepsilon (\cdot , \cdot ; s ,y) - \nabla \cdot {\textbf{a}}\nabla P_{{\textbf{a}}, \infty }^\varepsilon (\cdot , \cdot ; s ,y) = 0 \text{ in }~ (s , t) \times \Lambda _L . \end{aligned}$$

By the definition of the parameter $\varepsilon$ and the inequality (A.15), the map $P_{{\textbf{a}}, \infty }^\varepsilon$ satisfies the boundary estimates

$$\begin{aligned} \left\{ \begin{aligned} P_{{\textbf{a}}, \infty }^\varepsilon (t' , x ; s ,y)&\le 0 \le P_{\textbf{a}}(t' , x ; s ,y){} & {} (t' , x) \in [s , t] \times \partial \Lambda _L, \\ P_{{\textbf{a}}, \infty }^\varepsilon (s , x ; s ,y)&= \mathbbm {1}_{\{ x = y \}} - \varepsilon \le \mathbbm {1}_{\{ x = y \}} = P_{\textbf{a}}(s , x ; s ,y){} & {} x \in \Lambda _L^+. \end{aligned} \right. \end{aligned}$$

Applying the maximum principle for the parabolic operator $\partial _t - \nabla \cdot {\textbf{a}}\nabla$, we obtain the inequality, for any $(t', x) \in [s , t] \times \Lambda _L$,

$$\begin{aligned} P_{{\textbf{a}}, \infty }^\varepsilon (t' , x ; s , y) \le P_{\textbf{a}}(t' , x ; s , y). \end{aligned}$$

(A.19)

Applying the estimate (A.19) at time $t' = t$ and using the definition of the parameter $\varepsilon$ yields, for any vertex x satisfying $|x - y| \le \sqrt{t - s}$,

$$\begin{aligned} \frac{1}{2} P_{{\textbf{a}}, \infty } \left( t , x ; s , y \right) \le P_{\textbf{a}}^\varepsilon \left( t , x ; y , s \right) \le P_{\textbf{a}}\left( t , x ; y , s \right) . \end{aligned}$$

(A.20)

Combining the estimate (A.20) with the lower bound of Proposition A.2 implies

$$\begin{aligned} P_{\textbf{a}}\left( t , x ; y , s \right) \ge \frac{c}{1 \vee (t-s)^\frac{d}{2}}. \end{aligned}$$

The proof of the lower bound (3.6) is complete. $\square$