Fast flow asymptotics for stochastic incompressible viscous fluids in $\mathbb {R}^2$ and SPDEs on graphs

Abstract

Fast advection asymptotics for a stochastic reaction–diffusion–advection equation are studied in this paper. To describe the asymptotics, we introduce a new class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow. We prove that, as the advection term becomes faster and faster, the solution of the stochastic reaction–diffusion–advection equation on the plane converges in a suitable sense to the solution of such an SPDE defined on the graph. This result is a consequence of averaging, due to a probabilistic representation of solutions, both for the SPDEs on the plane and for the SPDE on the graph.

Appendices

The limiting result for the linear deterministic problem

For every $\epsilon >0$, we consider the linear parabolic Cauchy problem associated with the second order differential operator $\mathcal {L}_\epsilon$

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\partial _t v_\epsilon (t,x)=\mathcal {L}_\epsilon v_\epsilon (t,x),\ \ \ \ \ t>0,\ \ \ x \in \,\mathbb {R}^2,}\\ \\ \displaystyle {v_\epsilon (0,x)=g(x),\ \ \ \ x \in \,\mathbb {R}^2.} \end{array}\right. \end{aligned}$$

(A.1)

The solution of problem (A.1) has a probabilistic representation in terms of the Markov transition semigroup $S_\epsilon (t)$ associated with Eq. (2.12), that, we recall, is defined for any bounded Borel function $\varphi :\mathbb {R}^2\rightarrow \mathbb {R}$ by

$$\begin{aligned} S_\epsilon (t) \varphi (x)=\mathbb {E}_{\,x}\, \varphi (X_\epsilon (t)),\ \ \ \ x \in \,\mathbb {R}^2. \end{aligned}$$

Actually, as a consequence of Itô’s formula, if the initial condition $\varphi$ is taken in $C^2_b(\mathbb {R}^2)$, then

$$\begin{aligned} \frac{d}{dt}[S_\epsilon (\cdot )\varphi (x)](t)=\mathcal {L}_\epsilon \left( S_\epsilon (t)\varphi \right) (x),\ \ \ \ t\ge 0,\ \ \ x \in \,\mathbb {R}^2. \end{aligned}$$

(A.2)

Moreover, as the Hamiltonian H is assumed to be of class $C^4(\mathbb {R})$, with bounded second derivative, we have that the semigroup $S_\epsilon (t)$ has a smoothing effect, namely it maps Borel bounded functions into $C^3_b(\mathbb {R}^2)$, for any $t>0$. Thanks to the semigroup law, this allows to conclude that (A.2) is satisfied on $\mathbb {R}^2$, for any Borel bounded function $\varphi$ and for all $t>0$.

Our purpose here is to study the asymptotic behavior of $S_\epsilon (t)\varphi (x)$, and hence of $v_\epsilon (t,x)$, as $\epsilon \downarrow 0$. As a consequence of (2.15), if the Hamiltonian H satisfies Hypothesis 1, then for any continuous mapping $\psi :\Gamma \rightarrow \mathbb {R}$ we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} S_{\epsilon }(t) \psi ^{\vee }(x)=\bar{S}(t)\psi (\Pi (x)),\ \ \ \ t\ge 0,\ \ \ x \in \,\mathbb {R}^2. \end{aligned}$$

(A.3)

As a first thing, we are going to prove that the limit above is uniform with respect to $t \in \,[0,T]$, for every fixed $T>0$. To this purpose, we introduce some notation.

For any $\eta >0$ we can fix

$$\begin{aligned} z_\eta \ge \max _{i=1,\ldots ,n} H(x_i)+1, \end{aligned}$$

(A.4)

(we recall $x_1,\ldots ,x_n$ are the critical points of the Hamiltonian H), such that, for all $\epsilon >0$ sufficiently small,

$$\begin{aligned} \mathbb {P}_x \left( \rho _{\epsilon ,\eta }\le T\right) \le \eta ,\ \ \ \ \ \ \bar{\mathbb {P}}_{\Pi (x)} \left( \rho _\eta \le T\right) \le \eta , \end{aligned}$$

(A.5)

where

$$\begin{aligned} \rho _{\epsilon ,\eta }:=\inf \,\{t\ge 0\,:\,H(X_\epsilon ^x(t))\ge z_\eta \},\ \ \ \ \ \rho _\eta :=\inf \,\{t\ge 0\,:\,\Pi _1\bar{Y}(t)\ge z_\eta \}, \end{aligned}$$

(A.6)

(here we have denoted $\pi _1(z,k)=z$). Actually, as proved in [5, Lemma 3.2] the family $\{\Pi (X_\epsilon (\cdot ))\}_{\epsilon \in \,(0, \epsilon _0)}$ is tight in $C([0,T];\mathbb {R})$. This means that there exists some $z_\eta >0$ as in (A.4) such that

$$\begin{aligned} \mathbb {P}_x\left( \sup _{t\le T}H(X_\epsilon (t))\ge z_\eta \right) \le \eta , \end{aligned}$$

and (A.5) follows.

In the next proposition, we show that limit (1.5) is in fact uniform with respect to $t \in \,[0,T]$, for every $T>0$.

Proposition A.1

For every $T>0$ and $\psi \in \,C_b(\Gamma )$, we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\sup _{t \in \,[0,T]}\,\left| \mathbb {E}_x\psi ^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)} \psi (\bar{Y}(t))\right| =0. \end{aligned}$$

(A.7)

Proof

For every $\epsilon >0$, let us define

$$\begin{aligned} f_\epsilon (t):=\mathbb {E}_x\psi ^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)} \psi (\bar{Y}(t)). \end{aligned}$$

For every fixed $t\ge 0$, we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} f_\epsilon (t)=0. \end{aligned}$$

(A.8)

If we prove that the family of functions $\{f_\epsilon \}_{\epsilon >0}$ is equibounded and equicontinuous in C([0, T]), by the Ascoli-Arzelà Theorem we have that the limit in (A.8) is uniform.

Now, the equiboundedness of $\{f_\epsilon \}_{\epsilon >0}$ follows from the fact that $\psi$ is bounded. In order to prove the equicontinuity of $\{f_\epsilon \}_{\epsilon >0}$, first of all we notice that we may assume that both $\psi$ and $\psi ^\vee$ are uniformly continuous. Actually, due to (A.5) for every $\eta >0$ we have

$$\begin{aligned}&\displaystyle \left| \mathbb {E}_x\,\psi ^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\psi (\bar{Y}(t))\right| \le \left| \mathbb {E}_x\,\left( \psi ^\vee (X_\epsilon (t))\,;\,\rho _{\epsilon ,\eta }\le T\right) \right| \\&\qquad +\left| \bar{\mathbb {E}}_{\Pi (x)}\left( \psi (\bar{Y}(t))\,;\,\rho _\eta \le T\right) \right| \\&\qquad \displaystyle {+\left| \mathbb {E}_x\,\left( \psi ^\vee (X_\epsilon (t))\,;\,\rho _{\epsilon ,\eta }> T\right) -\bar{\mathbb {E}}_{\Pi (x)}\left( \psi (\bar{Y}(t))\,;\,\rho _\eta> T\right) \right| }\\&\quad \displaystyle \le 2\,\Vert \psi \Vert _\infty \,\eta +\left| \mathbb {E}_x\,\left( \psi ^\vee (X_\epsilon (t))\,;\,\rho _{\epsilon ,\eta }> T\right) -\bar{\mathbb {E}}_{\Pi (x)}\left( \psi (\bar{Y}(t))\,;\,\rho _\eta > T\right) \right| . \end{aligned}$$

Therefore, as $\psi ^\vee$ is uniformly continuous on $\{H(x)\le z_\eta \}$ and $\psi$ is uniformly continuous on $\{z\le z_\eta \}$, due to the arbitrariness of $\eta$ we can conclude.

Now, if $\psi$ are uniformly continuous, for every $\eta >0$ there exists $\delta _\eta >0$ such that

$$\begin{aligned}&\displaystyle {|\Pi (x_1)-\Pi (x_2)<\delta _\eta ,\ \ \ \text {dist}\left( (z_1,k_1), (z_2,k_2)\right)<\delta _\eta }\nonumber \\&\quad \displaystyle {\Longrightarrow |\psi ^\vee (x_1)-\psi ^\vee (x_2)|+|\psi (z_1,k_1)-\psi (z_2,k_2)|<\frac{\eta }{2}.} \end{aligned}$$

(A.9)

Moreover, as $\{\Pi (X_\epsilon )\}_{\epsilon >0}\subset C([0,T];\Gamma )$ is tight, there exists a compact set $K_\eta \subset C([0,T];\Gamma )$, such that

$$\begin{aligned} \mathbb {P}_x\left( \Pi (X_\epsilon ) \in \,K_\eta ^c\right) +\bar{\mathbb {P}}_{\Pi (x)}\left( \bar{Y} \in \,K_\eta ^c\right) \le \frac{\eta }{4\Vert \psi \Vert _\infty }. \end{aligned}$$

(A.10)

Since functions in $K_\eta$ are equicontinuous, there exists $\theta _\eta >0$ such that on $K_\eta$

$$\begin{aligned} |t-s|<\theta _\eta \Longrightarrow |\Pi (X_\epsilon (t))-\Pi (X_\epsilon (s))|<\delta _\eta ,\ \ \ \ |\bar{Y}(t)-\bar{Y}(s)|<\delta _\eta . \end{aligned}$$

Therefore, thanks to (A.9) and (A.10), for every $t,s \in \,[0,T]$ such that $|t-s|<\theta _\eta$

$$\begin{aligned}&\displaystyle {|f_\epsilon (t)-f_\epsilon (s)|}\\&\quad \displaystyle \le \left| \mathbb {E}_x\left( \psi ^\vee (X_\epsilon (t))-\psi ^\vee (X_\epsilon (s))\,;\,\Pi (X_\epsilon ) \in \,K_\eta \right) \right| \\&\qquad +\left| \bar{\mathbb {E}}_{\Pi (x)}\left( \psi (\bar{Y}(t))-\psi (\bar{Y}(s))\,;\,\bar{Y} \in \,K_\eta \right) \right| \\&\qquad \displaystyle +\,2\,\Vert \psi \Vert _\infty \left( \mathbb {P}_x\left( \Pi (X_\epsilon ) \in \,K_\eta ^c\right) +\bar{\mathbb {P}}_{\Pi (x)}\left( \bar{Y} \in \,K_\eta ^c\right) \right) \\&\quad \le \frac{\eta }{2}+2\,\Vert \psi \Vert _\infty \frac{\eta }{4\Vert \psi \Vert _\infty }=\eta . \end{aligned}$$

$\square$

In what follows, we want to show that, in fact, under suitable conditions, limit (A.7) is also true if $\psi ^\vee$ is replaced by $u:\mathbb {R}^2\rightarrow \mathbb R$ and $\psi$ is replaced by $u^\wedge$. Namely, we want to prove the following result.

Theorem A.2

Assume that the Hamiltonian H satisfies all conditions in Hypotheses 1 and 2. Then, for any $u \in \,C_b(\mathbb {R}^2)$ and $x \in \,\mathbb {R}^2$, and for any $0<\tau \le T$, we have

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}\left| S_\epsilon (t)u(x)-\bar{S}(t)^\vee u(x)\right| \nonumber \\&\quad =\lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x u(X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)} u^\wedge (\bar{Y}(t))\right| =0. \end{aligned}$$

(A.11)

1.1 A preliminary result

Before proceeding with the proof of Theorem A.2 and of all required preliminary results, we introduce some notations. For every $\epsilon , \eta >0$ and $0<\delta ^\prime <\delta$, by using the notations introduced in Sect. 2.2 we define

$$\begin{aligned} \displaystyle \sigma _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }= & {} \min \,\left\{ t\ge \tau _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }\ :\ X_\epsilon (t) \in \,G(\pm \delta )^c\right\} ,\nonumber \\ \displaystyle \tau _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }= & {} \min \,\left\{ t\ge \sigma _{n-1}^{\epsilon ,\eta ,\delta ,\delta ^\prime }\ :\ X_\epsilon (t) \in \,D(\pm \delta ^\prime )\cup C(z_\eta )\right\} , \end{aligned}$$

(A.12)

with $\tau _0^{\epsilon ,\eta ,\delta ,\delta ^\prime }=0$ and $z_\eta$ defined as in (A.4). Clearly, after the process $X_\epsilon (t)$ reaches $C(z_\eta )$, all $\tau _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }$ and $\sigma _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }$ coincide with the stopping time $\rho _{\epsilon ,\eta }$ introduced in (A.6), and for any $n \in \,\mathbb N$

$$\begin{aligned} X_\epsilon \left( \sigma _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }\right) \in \,D(\pm \delta )\cup C(z_\eta ),\ \ \ \ \ X_\epsilon \left( \tau _n^{\epsilon ,\eta ,\delta ,\delta ^\prime }\right) \in \,D(\pm \delta ^\prime )\cup C(z_\eta ). \end{aligned}$$

(A.13)

Moreover, if $X_\epsilon (0) \in \,G(\pm \delta )^c$, we have that $\sigma _0^{\epsilon ,\eta ,\delta ,\delta ^\prime }=0$ and $\tau _1^{\epsilon ,\eta ,\delta ,\delta ^\prime }$ is the first time the process $X_\epsilon$ touches $D(\pm \delta ^\prime )\cup C(z_\eta )$. In particular, if $X_\epsilon (0)\ge z_\eta$, then $\tau _1^{\epsilon ,\eta ,\delta ,\delta ^\prime }$ is the first time the process $X_\epsilon$ touches $C(z_\eta )$ and all successive stopping times coincide with $\rho _{\epsilon ,\eta }$.

Lemma A.3

Assume that the same assumptions of Theorem A.2 are verified. Then, for every $u \in \,C_b(\mathbb {R}^2)$ and $x \in \,\mathbb {R}^2$, and for every $0<\tau <T$

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\,\sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x\,(u^\wedge )^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,u^\wedge (\bar{Y}(t))\right| =0. \end{aligned}$$

(A.14)

Proof

Thanks to (A.7), if $u^\wedge$ were a continuous function on $\Gamma$, then (A.14) would follow immediately. Unfortunately, because of the presence of the interior vertices, even if u is continuous on $\mathbb {R}^2$ we cannot conclude that $u^\wedge$ is continuous on $\Gamma$, in general. This means that we have to treat separately the internal vertices and the rest of the points of the graph $\Gamma$.

First of all, we fix $\delta >0$ and $f_\delta \in \,C_b(\Gamma )$ such that

$$\begin{aligned} \Vert f_\delta \Vert _\infty \le \Vert u^\wedge \Vert _\infty \le \Vert u\Vert _\infty ,\ \ \ \ f_\delta =u^\wedge ,\ \text {on}\ \Pi (G(\pm \delta /2)^c). \end{aligned}$$

Thus, for every $\delta \in \,(0,1)$ and $t\ge 0$, we can write

$$\begin{aligned}&\displaystyle {\mathbb {E}_x\,(u^\wedge )^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,u^\wedge (\bar{Y}(t))=\mathbb {E}_x\,\left[ (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\right] }\\&\qquad \displaystyle +\left[ \mathbb {E}_x\,f_\delta ^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,f_\delta (\bar{Y}(t))\right] +\bar{\mathbb {E}}_{\Pi (x)}\left[ f_\delta (\bar{Y}(t))-u^\wedge (\bar{Y}(t))\right] \\&\quad =:I_1^{\epsilon ,\delta }(t)+I_2^{\epsilon ,\delta }(t)+I^\delta (t). \end{aligned}$$

If we prove that for any $\eta >0$ there exist $\delta _\eta , \epsilon _\eta >0$ such that

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}\left( |I^{\epsilon ,\delta _\eta }_1(t)|+|I^{\delta _\eta }(t)|\right)<\eta ,\ \ \ \ \ \epsilon <\epsilon _\eta , \end{aligned}$$

(A.15)

then

$$\begin{aligned}&\displaystyle {\sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x\,(u^\wedge )^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,u^\wedge (\bar{Y}(t))\right| }\\&\quad \displaystyle {\le \eta +\sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x\,f_{\delta _\eta }^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,f_{\delta _\eta } (\bar{Y}(t))\right| ,\ \ \ \ \epsilon <\epsilon _\eta .} \end{aligned}$$

Since $f_{\delta _\eta } \in \,C_b(\Gamma )$, due to (A.7) and the arbitrariness of $\eta >0$, this implies (A.14).

Thus, let us prove (A.15). If we take $\eta ^\prime =\eta /8\Vert u \Vert _\infty$, due to (A.5) we have

$$\begin{aligned}&\displaystyle {I_1^{\epsilon ,\delta }(t)}\nonumber \\&\quad \displaystyle =\mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t<\rho _{\epsilon ,\eta ^\prime }\right) \nonumber \\&\qquad +\,\mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t\ge \rho _{\epsilon ,\eta ^\prime }\right) \nonumber \\&\quad \displaystyle {\le \mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t<\rho _{\epsilon ,\eta ^\prime }\right) +2\,\Vert u \Vert _{\infty }\,\mathbb {P}_x( t\ge \rho _{\epsilon ,\eta ^\prime })}\nonumber \\&\quad \displaystyle {\le \mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t<\rho _{\epsilon ,\eta ^\prime }\right) +\frac{\eta }{4}=:J^{\epsilon ,\eta ^\prime ,\delta }_1(t)+\frac{\eta }{4}.}\nonumber \\ \end{aligned}$$

(A.16)

Recalling that $f_\delta =u^\wedge$ on $\Pi (G(\pm \delta /2))^c$, we have that $f_\delta ^\vee =(u^\wedge )^\vee$ on $G(\pm \delta /2)^c$, so that

$$\begin{aligned} \displaystyle J_1^{\epsilon ,\eta ^\prime ,\delta }(t)= & {} \mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t \in \,\bigcup _{n=0}^\infty \left[ \tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right) \nonumber \\= & {} \displaystyle \sum _{n=0}^\infty \, \mathbb {E}_x\,\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t \in \,\left[ \tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right) \nonumber \\=: & {} \sum _{n=0}^\infty J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t). \end{aligned}$$

(A.17)

Due to the strong Markov property, we have

$$\begin{aligned}&\displaystyle \left| J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)\right| =\left| \mathbb {E}_x\,\left( \left[ (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^{\vee }(X_\epsilon (t))\right] \, \mathbb {I}_{\{t\ge \tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\}}\,\mathbb {I} _{\{\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}> t\}}\right) \right| \\&\quad \le \displaystyle \mathbb {E}_x\,\left( \mathbb {I}_{\left\{ \tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n\le t\right\} }\left| \mathbb {E}_{X_\epsilon \left( \tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) } \left( (u^\wedge )^\vee (X_\epsilon (t)) -f_\delta ^\vee (X_\epsilon (t))\,;\,t <\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| \right) . \end{aligned}$$

Thanks to (A.13), this implies

$$\begin{aligned}&\displaystyle {|J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)|}\nonumber \\&\quad \displaystyle \le \mathbb {P}_x\left( \tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n\le t\right) \sup _{y \in \,D(\pm \delta /2)\cup C(z_\eta ^\prime )} \left| \mathbb {E}_y\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t<\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| \nonumber \\&\quad \displaystyle \le e^t\,\mathbb {E}_x\,e^{-\tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n} \sup _{y \in \,D(\pm \delta /2)\cup C(z_{\eta ^\prime })} \left| \mathbb {E}_y\left( (u^\wedge )^\vee (X_\epsilon (t))-f_\delta ^\vee (X_\epsilon (t))\,;\,t <\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| .\nonumber \\ \end{aligned}$$

(A.18)

According to what proved in [5, Section 8.3, see (8.3.14)], there exist a constant $c>0$ and $\delta _1>0$ such that for all $x \in \,\mathbb {R}^2$ and for all $\delta \le \delta _1$ and $\epsilon >0$ sufficiently small it holds

$$\begin{aligned} \begin{array}{l} \displaystyle {\sum _{n=0}^\infty \mathbb {E}_x\,e^{-\tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n}=\sum _{n=0}^\infty \mathbb {E}_x\,\left( e^{-\tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n}\,;\,\tau ^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}_n\le \rho _{\epsilon ,\eta ^\prime }\right) \le \frac{c}{\delta }.} \end{array} \end{aligned}$$

(A.19)

Therefore, from (A.18) we get

$$\begin{aligned} \sum _{n=0}^\infty |J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)|\le \frac{c}{\delta }\,\frac{2\Vert u \Vert _\infty e^t}{t}\,\sup _{y \in \,D(\pm \delta /2)}\mathbb {E}_y\,\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}. \end{aligned}$$

(A.20)

Because of our definition, $\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}$ is the first exit time of the process $X_\epsilon (t)$ from $G(\pm \delta )$. In [5, Section 8.5, (8.5.17)] it is proved that there exists $\delta _2>0$ such that for all $\delta <\delta _2$ and $\epsilon >0$ sufficiently small

$$\begin{aligned} \mathbb {E}_y\,\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\le c\,\delta ^2|\log \delta |,\ \ \ \ y \in \,G(\pm \delta ). \end{aligned}$$

(A.21)

In particular, due to (A.16), (A.17) and (A.20), this implies that for all $\delta <\delta _0:=\delta _1\wedge \delta _2$ and all $\epsilon$ small enough

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}|I_1^{\epsilon ,\delta }(t)|\le \frac{c}{\delta }\,\frac{2\Vert u \Vert _\infty e^T}{\tau }\delta ^2|\log \delta |+\frac{\eta }{4}. \end{aligned}$$

This means that for any $\eta >0$ fixed, there exist $\epsilon _{1,\eta }>0$ and $\delta _{1,\eta }>0$ such that

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}|I_1^{\epsilon ,\delta }(t)|\le \frac{\eta }{2},\ \ \ \epsilon< \epsilon _{1,\eta },\ \ \delta < \delta _{1,\eta }. \end{aligned}$$

(A.22)

Concerning $I^\delta (t)$, recalling how $f_\delta$ was defined, for every $\delta >0$ we have

$$\begin{aligned} \displaystyle |I^\delta (t)|\le & {} 2\,\Vert u\Vert _\infty \, \bar{\mathbb {P}}_{\Pi (x)}\left( \bar{Y}(t) \in \,\Pi (G(\pm \delta /2))\right) \le 2\,\Vert u\Vert _\infty \, \bar{\mathbb {E}}_{\Pi (x)}\,\psi _\delta (\bar{Y}(t))\nonumber \\\le & {} \displaystyle 2\,\Vert u\Vert _\infty \,\left| \bar{\mathbb {E}}_{\Pi (x)}\,\psi _\delta (\bar{Y}(t))-\mathbb {E}_x\psi _\delta ^\vee (X_\epsilon (t))\right| \nonumber \\&\quad +\,2\,\Vert u\Vert _\infty {\mathbb {E}}_{x}\,\psi _\delta ^\vee (X_\epsilon (t)), \end{aligned}$$

(A.23)

where $\psi _\delta$ is a function in $C_b(\Gamma )$ such that

$$\begin{aligned} \mathbb {I}_{\Pi (G(\pm \delta /2))}\le \psi _\delta \le 1,\ \ \ \ \psi _\delta \equiv 0,\ \ \text {on}\ \Pi (G(\pm (\delta ))^c. \end{aligned}$$

Now, by proceeding as in the proof of (A.22), we can find $\epsilon _{2,\eta }>0$ and $\delta _{2,\eta }>0$ such that

$$\begin{aligned} 2\,\Vert u\Vert _\infty \sup _{t \in \,[\tau ,T]}\,{\mathbb {E}}_{x}\,\psi _\delta ^\vee (X_\epsilon (t))\le \frac{\eta }{4},\ \ \ \epsilon \le \epsilon _{2,\eta },\ \ \delta \le \delta _{2,\eta }. \end{aligned}$$

(A.24)

Therefore, if we set $\delta _\eta :=\delta _{1,\eta }\wedge \delta _{2,\eta }$, from (A.22) and (A.23) we get

$$\begin{aligned}&\sup _{t \in \,[\tau ,T]} \left( |I^{\epsilon ,\delta _\eta }_1(t)|+|I^{\delta _\eta }(t)|\right) \\&\quad \le \frac{\eta }{2}+2\,\Vert u\Vert _\infty \sup _{t \in \,[\tau ,T]}\left| \bar{\mathbb {E}}_{\Pi (x)}\,\psi _{\delta _\eta }(\bar{Y}(t))-\mathbb {E}_x \psi _{\delta _\eta }^\vee (X_\epsilon (t))\right| , \end{aligned}$$

for every $\epsilon \le \epsilon _{1,\eta }\wedge \epsilon _{2,\eta }.$ As $\psi _{\delta _\eta } \in \,C_b(\Gamma )$, due to (A.7), this implies that there exists $\epsilon _\eta \le \epsilon _{1,\eta }\wedge \epsilon _{2,\eta }$ such that (A.15) holds and hence (A.14) follows. $\square$

1.2 Proof of Theorem A.2

In Lemma A.3 we have proved that for any $u \in \,C_b(\mathbb {R}^2)$ and $x \in \,\mathbb {R}^2$ and for any $0<\tau <T$

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\,\sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x\,(u^\wedge )^\vee (X_\epsilon (t))-\bar{\mathbb {E}}_{\Pi (x)}\,u^\wedge (\bar{Y}(t))\right| =0. \end{aligned}$$

Thus, in order to prove (A.11), it is sufficient to prove that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x \left[ u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\right] \right| =0. \end{aligned}$$

(A.25)

In what follows, we shall assume that $u \in \,C^1_b(\mathbb {R}^2)$. Actually, if this is not the case, we can fix a sequence $\{u_n\}_{n \in \,\mathbb N} \subset C^1_b(\mathbb {R}^2)$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert u-u_n\Vert _\infty =0,\ \ \ \ \Vert u_n\Vert _\infty \le \Vert u\Vert _\infty . \end{aligned}$$

Since this also implies that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert (u^\wedge )^\vee -(u_n^\wedge )^\vee \Vert _\infty =0, \end{aligned}$$

we get

$$\begin{aligned}&\displaystyle {\lim _{n\rightarrow \infty }\ \sup _{\epsilon>0}\,\sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x \left[ u(X_\epsilon (t))-u_n(X_\epsilon (t))(X_\epsilon (t))\right] \right| }\\&\quad \displaystyle {=\lim _{n\rightarrow \infty }\ \sup _{\epsilon >0}\,\sup _{t \in \,[\tau ,T]} \left| \mathbb {E}_x \left[ (u_n^\wedge )^\vee (X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\right] \right| =0.} \end{aligned}$$

Therefore, in order to prove (A.25), we have to prove that for any fixed $n \in \,\mathbb {N}$

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x \left[ u_n(X_\epsilon (t))-(u_n^\wedge )^\vee (X_\epsilon (t))\right] \right| =0. \end{aligned}$$

To this purpose, let us fix $\alpha >0$ and take $\epsilon >0$ small enough so that $\tau -\epsilon ^\alpha >0$. If we fix $\eta >0$ and take $\eta ^\prime =\eta /4\Vert u\Vert _\infty$, we have

$$\begin{aligned} \sup _{t\ge 0}\left| \mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha \ge \rho _{\epsilon ,\eta ^\prime }\right) \right| \le \frac{\eta }{2},\ \ \ \ \epsilon >0, \end{aligned}$$

where $\rho _{\epsilon ,\eta ^\prime }$ is the stopping time defined in (A.6) and satisfying (A.5). This implies

$$\begin{aligned}&\displaystyle \left| \mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\right) \right| \nonumber \\&\quad \le \left| \mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha <\rho _{\epsilon ,\eta ^\prime }\right) \right| +\frac{\eta }{2}. \end{aligned}$$

(A.26)

Now, as in the proof of Lemma A.3, we have

$$\begin{aligned}&\displaystyle {\mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\, t-\epsilon ^\alpha <\rho _{\epsilon ,\eta ^\prime }\right) }\nonumber \\&\quad \displaystyle {=\sum _{n \in \,\mathbb {N}}\,\mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha \in \,\left[ \tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right) }\nonumber \\&\qquad \displaystyle {+\sum _{n \in \,\mathbb {N}}\,\mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha \in \, \left[ \sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\tau _{n+1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right) }\nonumber \\&\quad \displaystyle {=:\sum _{n \in \,\mathbb {N}}\,J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)+\sum _{n \in \,\mathbb {N}}\,J_{2,n}^{\epsilon ,\eta ^\prime ,\delta }(t).} \end{aligned}$$

(A.27)

As in the proof of Lemma A.3, due to (A.20), we have that there exist $\delta _1>0$ and a constant $c>0$ such that for all $\epsilon$ sufficiently small and $\delta < \delta _1$

$$\begin{aligned} \sum _{n \in \,\mathbb {N}}\,J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)\le \frac{c}{\delta }\,\frac{2\Vert u\Vert _\infty e^{t-\epsilon ^\alpha }}{t-\epsilon ^\alpha }\,\sup _{y \in \,D(\pm \delta /2)}\mathbb {E}_y\,\sigma _0^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}, \end{aligned}$$

so that, thanks to (A.21), we get

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}\,\sum _{n \in \,\mathbb {N}}\,J_{1,n}^{\epsilon ,\eta ^\prime ,\delta }(t)\le c\,\frac{\Vert u\Vert _\infty e^T}{\tau -\epsilon ^\alpha }\,\delta |\log \delta |. \end{aligned}$$

(A.28)

On the other hand, by using once more the strong Markov property, we have

$$\begin{aligned}&\displaystyle {|J_{2,n}^{\epsilon ,\eta ^\prime ,\delta }(t)|}\\&\quad \displaystyle \le \mathbb {E}_x \left( \mathbb {I}_{\{\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\le t-\epsilon ^\alpha \}}\left| \mathbb {E}_{X_\epsilon (\sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2})}\right. \right. \\&\quad \left. \left. \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| \right) \\&\quad \displaystyle \le \mathbb {P}_x\left( \sigma _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\le t-\epsilon ^\alpha \right) \\&\quad \sup _{y \in \,D(\pm \delta )}\,\left| \mathbb {E}_{y} \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| \\&\quad \displaystyle \le e^{t-\epsilon ^\alpha }\, \mathbb {E}_x\left( e^{-\tau _n^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}}\right) \\&\quad \sup _{y \in \,D(\pm \delta )}\left| \mathbb {E}_{y}\left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| , \end{aligned}$$

and thanks to (A.19), this implies that there exists $\delta _2>0$, such that for any $\delta \le \delta _2$

$$\begin{aligned}&\sum _{n=1}^\infty |J_{2,n}^{\epsilon ,\eta ^\prime ,\delta }(t)|\le \frac{c\, e^{t-\epsilon ^\alpha }}{\delta }\\&\quad \sup _{y \in \,D(\pm \delta )}\left| \mathbb {E}_{y}\left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t -\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) \right| . \end{aligned}$$

For every $y \in \,\mathbb {R}^2$, we have

$$\begin{aligned}&\displaystyle {\mathbb {E}_{y}\left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) }\\&\quad \displaystyle {=\mathbb {E}_{y}\left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t-\epsilon ^\alpha ))\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) }\\&\qquad \displaystyle +\,{\mathbb {E}_{y}\left( (u^\wedge )^\vee (X_\epsilon (t-\epsilon ^\alpha ))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\,t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) }\\&\qquad \displaystyle +\,\mathbb {E}_{y}\left( (u^\wedge )^\vee (X_\epsilon (t-\epsilon ^\alpha ))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t -\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\,t \ge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \\&\quad =:\sum _{i=1}^3 L_i^{\epsilon ,\delta }(t,y). \end{aligned}$$

Let us start considering $L_2^{\epsilon ,\delta }(t)$. If $y \in \,D(\pm \delta )$, then $\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}$ is the first time the process $X_\epsilon$ touches $D(\pm \delta /4)\cup C(z_{\eta ^\prime })$. This means that

$$\begin{aligned} t <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\Longrightarrow X_\epsilon (s) \in \,G(\pm \delta /4)^c\cap \{H\le z_{\eta ^\prime }\},\ \ \ \ s\le t. \end{aligned}$$

In particular, $\Pi (X_\epsilon (s))$ remains in the interior of the same edge of the graph $\Gamma$ where y is, for all $s\le t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}$. As we are assuming that $u \in \,C^1_b(\mathbb {R}^2)$, due to Lemma 3.2 we have that $u^\wedge$ is continuously differentiable on $\Pi (G(\pm \delta /4))^c$, with uniformly bounded derivative.

As a consequence of Itô’s formula, for every $s<t$ we have

$$\begin{aligned} \begin{array}{l} \displaystyle {H(X_\epsilon (t))-H(X_\epsilon (s))=\frac{1}{2} \int _s^t \Delta H(X_\epsilon (r))\,dr+\int _s^t \langle \nabla H(X_\epsilon (r)), dw(r)\rangle .} \end{array} \end{aligned}$$

Hence, since for $y \in \,D(\pm \delta )$ and $s<t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}$, the process $\Pi (X_\epsilon (s))$ remains in the same edge of the graph $\Gamma$, we get

$$\begin{aligned}&\displaystyle \mathbb {E}_y\left( \left| \Pi (X_\epsilon (t))-\Pi (X_\epsilon (s))\right| ^2\,;\,t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \nonumber \\&\quad =\mathbb {E}_y\left( \left| H(X_\epsilon (t))-H(X_\epsilon (s))\right| ^2\,;\,t <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \nonumber \\&\quad \displaystyle {\le c\,\Vert \Delta H\Vert ^2_\infty (t-s)^2+\sup _{y^\prime \in \{H\le z_{\eta ^\prime }\}}|\nabla H(y^\prime )|(t-s)\le c_{T,\eta ^\prime } (t-s).} \end{aligned}$$

(A.29)

In particular, for every $y \in \,D(\pm \delta )$

$$\begin{aligned}&\displaystyle {|L_2^{\epsilon ,\delta }(t,y)|}\nonumber \\&\quad \displaystyle \le \mathbb {E}_{y}\left( \Vert u^\wedge \Vert _{C^1(\Pi (G(\pm \delta /4))^c)}|\Pi (X_\epsilon (t-\epsilon ^\alpha )))\right. \nonumber \\&\qquad \left. -\,\Pi (X_\epsilon (t))|\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\,t <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \nonumber \\&\quad \displaystyle {\le \Vert u^\wedge \Vert _{C^1(\Pi (G(\pm \delta /4))^c)}\,c_{T,\eta ^\prime }^{1/2}\,\epsilon ^{\alpha /2}.} \end{aligned}$$

(A.30)

Next, concerning $L_3^{\epsilon ,\delta }(t,y)$, we have

$$\begin{aligned}&\displaystyle {\mathbb {P}_y\left( t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\,t \ge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) }\nonumber \\&\quad \displaystyle \le \mathbb {P}_y\left( |H\left( X_\epsilon \left( \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \right) -H(X_\epsilon (t-\epsilon ^\alpha ))|\right. \nonumber \\&\quad \left. \ge \delta /4\,,\,t -\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2},\,t \ge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\right) \nonumber \\&\quad \displaystyle {\le \mathbb {P}_y\left( |H\left( X_\epsilon \left( t\wedge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4} \right) \right) -H\left( X_\epsilon \left( (t-\epsilon ^\alpha )\wedge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4} \right) \right) |\ge \delta /4\right) }\nonumber \\&\quad \displaystyle {\le \frac{16}{\delta ^2}\,\mathbb {E}_y\left( |H\left( X_\epsilon \left( t\wedge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4} \right) \right) -H\left( X_\epsilon \left( (t-\epsilon ^\alpha )\wedge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4} \right) \right) |^2\right) ,}\nonumber \\ \end{aligned}$$

(A.31)

and, thanks to (A.29), this yields

$$\begin{aligned} \sup _{y \in \,D(\pm \delta )}\,|L_3^{\epsilon ,\delta }(t,y)|\le \frac{16}{\delta ^2}\,c_{T,\eta ^\prime }\,\epsilon ^\alpha . \end{aligned}$$

(A.32)

Finally, we consider $L^{\epsilon ,\delta }_1(t,y)$. As a consequence of the Markov property,

$$\begin{aligned}&\displaystyle {\mathbb {E}_{y}\left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t-\epsilon ^\alpha ))\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) }\\&\quad \displaystyle {= \mathbb {E}_{y}\left( \psi _\epsilon (\epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha ))\,;\,t -\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\right) ,} \end{aligned}$$

where

$$\begin{aligned} \psi _\epsilon (s,x)=\mathbb {E}_x u(X_\epsilon (s))-(u^\wedge )^\vee (x). \end{aligned}$$

Since the family $\{\Pi (X_\epsilon )\}_{\epsilon >0}$ is weakly convergent in $C([0,+\infty );\Gamma )$ and $H(x)\uparrow \infty$, as $|x|\uparrow \infty$, we have that for any $\eta >0$ there exists $M_\eta >0$ such that

$$\begin{aligned}&\displaystyle {\left| \mathbb {E}_{y}\left( \psi _\epsilon (\epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha ))\,;\,t -\epsilon ^\alpha <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\,,\,|X_\epsilon (t-\epsilon ^\alpha )|> M_\eta \right) \right| }\\&\quad \displaystyle {\le 2\,\Vert u\Vert _\infty \mathbb {P}_y\left( |X_\epsilon (t-\epsilon ^\alpha )|> M_\eta \right) \le \eta .} \end{aligned}$$

Therefore,

$$\begin{aligned}&\displaystyle |L_1^{\epsilon ,\delta }(t,y)|\le \eta +\mathbb {E}_{y}\left( |\psi _\epsilon (\epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha ))|\,;\,t -\epsilon ^\alpha \right. \\&\quad \left. <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\, ,\,|X_\epsilon (t-\epsilon ^\alpha )|\le M_\eta \right) . \end{aligned}$$

As above, we write

$$\begin{aligned}&\displaystyle {\mathbb {E}_{y}\left( |\psi _\epsilon \left( \epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha )\right) |\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\, ,\,|X_\epsilon (t-\epsilon ^\alpha )|\le M_\eta \right) }\\&\quad \displaystyle =\mathbb {E}_{y}\left( |\psi _\epsilon \left( \epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha )\right) |\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\, ,\,t\right. \\&\quad \left. \ge \tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\,\,|X_\epsilon (t-\epsilon ^\alpha )|\le M_\eta \right) \\&\qquad \displaystyle +\,\mathbb {E}_{y}\left( |\psi _\epsilon \left( \epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha )\right) |\,;\,t -\epsilon ^\alpha \right. \\&\quad \left.<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\, ,\,t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}\,\,|X_\epsilon (t-\epsilon ^\alpha )|\le M_\eta \right) \\&\quad \displaystyle {=:I_{\epsilon ,1}(t)+I_{\epsilon ,2}(t).} \end{aligned}$$

Due to (A.29) and (A.31), we have

$$\begin{aligned} I_{\epsilon ,1}(t)\le 2\,\Vert u\Vert _\infty \,c_T\frac{\epsilon ^\alpha }{\delta ^2}. \end{aligned}$$

Now, in [4, Lemma 4.3], it has been proved that, as a consequence of the averaging principle, under the crucial assumption (2.3) given in Hypothesis 2, if $\alpha \in \,(4/7,2/3)$, then for every fixed $\delta >0$

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{x \in \,K}\,|\psi _\epsilon (\epsilon ^\alpha ,x)|=\lim _{\epsilon \rightarrow 0} \sup _{x \in \,K}\, \left| \mathbb {E}_x\, u(X_\epsilon (\epsilon ^\alpha ))-(u^\wedge )^\vee (x)\right| =0, \end{aligned}$$

(A.33)

for any compact subset K in $G(\pm \delta /2)^c$ and any function u whose support is contained in $G(\pm \delta /4)^c$. Since $X_\epsilon (t) \in \,G(\pm \delta /4)^c$, if $t<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4}$, (A.33) implies that

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0}\sup _{y \in \,D(\pm \delta )}\mathbb {E}_{y}\left( |\psi _\epsilon \left( \epsilon ^\alpha ,X_\epsilon (t-\epsilon ^\alpha )\right) |\,;\,t -\epsilon ^\alpha<\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /2}\, ,\,t\right. \\&\quad \left. <\tau _{1}^{\epsilon ,\eta ^\prime ,\delta ,\delta /4},\,|X_\epsilon (t-\epsilon ^\alpha )|\le M_\eta \right) =0, \end{aligned}$$

and, because of the arbitrariness of $\eta >0$, we conclude that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\sup _{t \in \,[\tau ,T]}\,\sup _{y \in \,D(\pm \delta )}\,|L_1^{\epsilon ,\delta }(t,y)|=0. \end{aligned}$$

This, together with (A.30) and (A.32), implies that for every $\delta \le \delta _2$ fixed

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}\,\sum _{n=1}^\infty |J_{2,n}^{\epsilon ,\eta ^\prime ,\delta }(t)|=0. \end{aligned}$$

Therefore, if in (A.28) we pick $\bar{\delta } \in \,(0,\delta _2]$ such that

$$\begin{aligned} \sup _{t \in \,[\tau , T]}\sum _{n \in \,\mathbb {N}}\,J_{1,n}^{\epsilon ,\eta ^\prime ,\bar{\delta }}(t)<\frac{\eta }{4}, \end{aligned}$$

and then we pick $\epsilon _\eta >0$ such that

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}\,\sum _{n=1}^\infty |J_{2,n}^{\epsilon ,\eta ^\prime ,\bar{\delta }}(t)|<\frac{\eta }{4},\ \ \ \ \epsilon \le \epsilon _\eta , \end{aligned}$$

because of (A.26) and (A.27), we can conclude that

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}\left| \mathbb {E}_x \left( u(X_\epsilon (t))-(u^\wedge )^\vee (X_\epsilon (t))\,;\,t-\epsilon ^\alpha<\rho _{\epsilon ,\eta ^\prime }\right) \right|<\frac{\eta }{2},\ \ \ \ \ \epsilon <\epsilon _\eta , \end{aligned}$$

and (A.25) follows.

Some consequences of Theorem A.2

The first immediate consequence of Theorem A.2 is that the semigroup $S_\epsilon (t)$ converges to the semigroup $\bar{S}(t)$ in $H_\gamma$, as $\epsilon \downarrow 0$.

Corollary B.1

Under Hypotheses 1, 3 and 2, for every $0<\tau <T$ and $u \in \,H_\gamma$ we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}|S_\epsilon (t) u-\bar{S}(t)^{\vee } u|_{H_\gamma }= \lim _{\epsilon \rightarrow 0} \sup _{t \in \,[\tau ,T]}|(S_\epsilon (t) u)^{\wedge } -\bar{S}(t) u^{\wedge }|_{\bar{H}_{\gamma }}=0. \end{aligned}$$

(B.1)

Proof

First of all, we notice that in view of (3.2), the first limit in (B.1) implies the second one. So, we will only prove the first limit. We have

$$\begin{aligned} |S_\epsilon (t) u-\bar{S}(t)^{\vee } u|_{H_\gamma }^2=\int _{\mathbb {R}^2}|S_\epsilon (t) u(x)-\bar{S}(t)^{\vee } u(x)|^2\gamma ^\vee (x)\,dx. \end{aligned}$$

If $u \in \,C_b(\mathbb {R}^2)$, we have

$$\begin{aligned} \sup _{t\ge 0}|S_\epsilon (t) u(x)-\bar{S}(t)^{\vee } u(x)|\le 2\,|u|_\infty . \end{aligned}$$

Hence, since $\gamma ^\vee \in \,L^1(\mathbb {R}^2)$, in view of (A.11) and the dominated convergence theorem, we have that the first limit in (B.1) is true for every $u \in \,C_b(\mathbb {R}^2)$. Moreover, in view of Hypothesis 3, we have that

$$\begin{aligned} |S_\epsilon (t) u|_{H_\gamma }\le c_T\,|u|_{H_\gamma },\ \ \ \ \ t \in \,[0,T],\ \ \ \epsilon >0, \end{aligned}$$

(B.2)

so that $\bar{S}(t)^\vee u \in \,H_\gamma$ and

$$\begin{aligned} |\bar{S}(t) ^\vee u|_{H_\gamma }\le c_T\,|u|_{H_\gamma },\ \ \ \ \ t \in \,[0,T]. \end{aligned}$$

(B.3)

Since $C_c(\mathbb {R}^2)$ is dense in $L^2(\mathbb {R}^2)$ and we assume the weight $\gamma ^\vee$ to be continuous and strictly positive, we have that $C_c(\mathbb {R}^2)$ is dense in $H_\gamma$. Actually, if $u \in \,H_\gamma$, then $u\sqrt{\gamma ^\vee } \in \,L^2(\mathbb {R}^2)$. Then, if $\{u_n\}_{n \in \,\mathbb N}$ is a sequence in $C_c(\mathbb {R}^2)$, converging to $u\sqrt{\gamma ^\vee } \in \,H_\gamma$ in $L^2(\mathbb {R}^2)$, we have that the sequence $\{u_n/\sqrt{\gamma ^\vee }\}_{n \in \,\mathbb N}$ converges to u in $H_\gamma$. Moreover, as $u_n$ has compact support and $1/\sqrt{\gamma ^\vee }$ is continuous and positive, it follows that $\{u_n/\sqrt{\gamma ^\vee }\}_{n \in \,\mathbb N}\subset C_c(\mathbb {R}^2)$.

Thus, for any $u \in \,H_\gamma$, we can fix a sequence $\{u_n\}_{n \in \,\mathbb N}\subset C_c(\mathbb {R}^2)$ converging to u in $H_\gamma$. Thanks to (B.3), we get that the sequence $\{\bar{S}(t)^\vee u_n\}_{n \in \,\mathbb N}$ is Cauchy in $H_\gamma$, so that we conclude that

$$\begin{aligned} \exists \lim _{n\rightarrow \infty }\bar{S}(t)^\vee u_n=:\bar{S}(t)^\vee u \in \,H_\gamma , \end{aligned}$$

the limit does not depend on the sequence $\{u_n\}_{n \in \,\mathbb N}$ and and (B.3) holds for every $u \in \,H_\gamma$.

Finally, since we have

$$\begin{aligned}&\displaystyle |S_\epsilon (t)u-\bar{S}(t)^\vee u|_{H_\gamma }\le |S_\epsilon (t)(u-u_n)|_{H_\gamma }\\&\quad +|\bar{S}(t)^\vee (u-u_n)|_{H_\gamma }+|S_\epsilon (t)u_n-\bar{S}(t)^\vee u_n|_{H_\gamma }, \end{aligned}$$

according to (B.2) and (B.3), for every $\eta >0$ we can find $\eta _\eta \in \,\mathbb N$ such that

$$\begin{aligned} \sup _{t \in \,[\tau ,T]}|S_\epsilon (t)u-\bar{S}(t)^\vee u|_{H_\gamma }\le \eta +\sup _{t \in \,[\tau ,T]}|S_\epsilon (t)u_{n_\eta }-\bar{S}(t)^\vee u_{n_\eta }|_{H_\gamma }. \end{aligned}$$

This allows to conclude, as $u_{u_\eta } \in \,C_c(\mathbb {R}^2)$. $\square$

Remark B.2

From the proof of the corollary above, it is clear that from the pointwise convergence of $S_\epsilon (t)u$ to $\bar{S}(t)^\vee u$, as stated in Theorem A.2, we cannot conclude that limit (B.1) is also true in $L^2(\mathbb {R}^2)$, as the Lebesgue measure in $\mathbb {R}^2$ is not finite. It is only after introducing a weight that we can prove the convergence in ${H}_\gamma$.

Corollary B.3

Under Hypotheses 1, 3 and 2, we have that the semigroup $\bar{S}(t)$ is well defined in $\bar{H}_\gamma$ and for any $T>0$ there exists $c_T>0$ such that

$$\begin{aligned} \Vert \bar{S}(t)\Vert _{\mathcal {L}(\bar{H}_\gamma )}\le c_T,\ \ \ \ t \in \,[0,T]. \end{aligned}$$

(B.4)

Proof

In (B.3) we have seen that $\bar{S}(t)^\vee$ is well defined in $H_\gamma$ and for any $T>0$ there exists $c_T>0$ such that for any $u \in \,H_\gamma$

$$\begin{aligned} |\bar{S}(t) ^\vee u|_{H_\gamma }\le c_T\,|u|_{H_\gamma },\ \ \ \ \ t \in \,[0,T]. \end{aligned}$$

Therefore, thanks to (3.3) and (3.2), if $f \in \,\bar{H}$

$$\begin{aligned} |\bar{S}(t)f|_{\bar{H}_\gamma }=|(\bar{S}(t)f)^\vee |_{{H}_\gamma }=|\bar{S}(t)^\vee (f^\vee )|_{{H}_\gamma } \le c_T\,|f^\vee |_{{H}_\gamma }=c_T\,|f|_{\bar{H}_\gamma },\ \ \ \ t \in \,[0,T], \end{aligned}$$

and this allows to conclude. $\square$