Liouville distorted Brownian motion

TL;DR

This paper extends the concept of Liouville Brownian motion to a distorted Brownian motion setting, constructing the associated positive continuous additive functional with respect to the Liouville distorted measure.

Contribution

It introduces the construction of the positive continuous additive functional for Liouville distorted Brownian motion starting from all points in 2, generalizing previous work on Liouville Brownian motion.

Findings

Construction of the additive functional $(F_t)_{t \u2265 0}$ for Liouville distorted Brownian motion.

Extension of Liouville Brownian motion framework to distorted Brownian motions.

Theoretical foundation for further analysis of Liouville distorted measures.

Abstract

The Liouville Brownian motion was introduced in \cite{GRV} as a time changed process $B_{A_t^{-1}}$ of a planar Brownian motion $(B_t)_{t \ge 0}$, where $(A_t)_{t \ge 0}$ is the positive continuous additive functional of $(B_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville measure. We first consider a distorted Brownian motion $(X_t)_{t\ge0}$ starting from all points in $\R^2$ associated to a Dirichlet form $(\E, D(\E))$ (see \cite{ShTr14}). We show that the positive continuous additive functional $(F_t)_{t \ge 0}$ of $(X_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville distorted measure can be constructed.

Equations

$${\mathcal{E}}^{\prime }(f,g):=\frac{1}{2}{\int }_{{\mathbb{R}}^2}⟨\nabla f,\nabla g⟩dx,$$

$$\mathcal{E}(f,g):=\frac{1}{2}{\int }_{{\mathbb{R}}^2}⟨\nabla f(x),\nabla g(x)⟩\rho (x)dx,f,g\in C_0^{\infty }({\mathbb{R}}^2),$$

$$(m^2-\Delta )G^{(m)}(x,\cdot )=2\pi {\delta }_x,x\in {\mathbb{R}}^2,$$

$$G^{(m)}(x,y)={\int }_0^{\infty }{e}^{-\frac{m^2}{2}s-\frac{\Vert x-y{\Vert }^2}{2s}}\frac{ds}{2s}={\int }_1^{\infty }\frac{k_m(s(x-y))}{s}ds,x,y\in {\mathbb{R}}^2,$$

$$k_m(z)=\frac{1}{2}{\int }_0^{\infty }{e}^{-\frac{m^2}{2s}\Vert z{\Vert }^2-\frac{s}{2}}ds.$$

$$E[Y_n(x)Y_n(y)]={\int }_{c_{n-1}}^{c_n}\frac{k_m(s(x-y))}{s}ds.$$

$$X(x)=\sum _{k\ge 1}{Y}_k(x).$$

$$X_n(x)=\sum _{k=1}^n{Y}_k(x),n\ge 1$$

$$M_{n,\gamma}^{\rho}(dz)=\exp\Big{(}\gamma X_{n}(z)-\frac{\gamma^{2}}{2}E[X_{n}(z)^{2}]\Big{)}\ \rho(z)dz,\quad\gamma\in(0,2),$$

$$M_{\gamma}^{\rho}(dz)=\exp\Big{(}\gamma X(z)-\frac{\gamma^{2}}{2}E[X(z)^{2}]\Big{)}\ \rho(z)dz.$$

$$\mathcal{E}(f,g):=\frac{1}{2}{\int }_{{\mathbb{R}}^2}⟨\nabla f(x),\nabla g(x)⟩m(dx),f,g\in C_0^{\infty }({\mathbb{R}}^2),$$

$$\mathbb{M}:=(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},\zeta ,(X_t{)}_{t\ge 0},({\mathbb{P}}_x{)}_{x\in {\mathbb{R}}^2})$$

$$P_{t}f(x):={\int }_{{\mathbb{R}}^2}{p}_t(x,y)f(y)m(dy),t>0,x,y\in {\mathbb{R}}^2,f\in {\mathcal{B}}_b({\mathbb{R}}^2)$$

$$R_{\beta }f(x):={\int }_{{\mathbb{R}}^2}{r}_{\beta }(x,y)f(y)m(dy),\beta >0,x\in {\mathbb{R}}^2,f\in {\mathcal{B}}_b({\mathbb{R}}^2),$$

$$\text{Cap(A)}=\inf_{\begin{subarray}{\ }\text{open}\ B\subset{\mathbb{R}}^{2}\\ A\subset B\end{subarray}}\inf_{\begin{subarray}{\ }f\in D({\mathcal{E}})\\ 1_{B}\cdot f\geq 1\ \text{m-a.e.}\end{subarray}}{\mathcal{E}}(f,f)+(f,f)_{L^{2}({\mathbb{R}}^{2},m)}.$$

$$P_t^{B}f(x)={\int }_B{p}_t^B(x,y)f(y)m(dy),t>0,x\in B$$

$$R_{\beta }^{B}f(x)={\int }_B{r}_{\beta }^B(x,y)f(y)m(dy),\beta >0,x\in B$$

$$R_{\beta }^B\mu (x)={\int }_B{r}_{\beta }^B(x,y)\mu (dy),\beta >0,x\in B$$

$$X_t^B(\omega )=\{\begin{array}{l}{X}_t(\omega ),0\le t<D_{B^c}(\omega )\\ \Delta ,t\ge D_{B^c}(\omega )\end{array}$$

$${\int }_B|f(x)|\mu (dx)\le c\sqrt{{\mathcal{E}}_1^B(f,f)},f\in D({\mathcal{E}}^B)\cap C_0(B),$$

$${\mathcal{E}}_1^B(U_1^B\mu ,f)={\int }_{B}f(x)\mu (dx),$$

$$E_k:=\{x\in {\mathbb{R}}^2\mid 1/k<|x|<k\},\text{and}E:=\bigcup _{k\ge 1}{E}_k={\mathbb{R}}^2\setminus \{0\}.$$

$${\mathbb{P}}_{x}\Big{(}\lim_{k\rightarrow\infty}D_{E_{k}^{c}}=\infty\Big{)}={\mathbb{P}}_{x}\Big{(}\lim_{k\rightarrow\infty}\sigma_{E_{k}^{c}}=\infty\Big{)}=1.$$

$$\rho (x)=|x{|}^{\alpha },\alpha \in [2,\infty ).$$

$$M_{\gamma }^{\rho }(G)\le \underset{x\in G}{\sup }\rho (x)M_{\gamma }(G).$$

$$M_{n,\gamma}^{\rho}(G)\leq\sup_{x\in G}\rho(x)\int_{G}\exp\Big{(}\gamma X_{n}(z)-\frac{\gamma^{2}}{2}E[X_{n}(z)^{2}]\Big{)}\ dz.$$

$$r_1^G(x,y)\le c_1\frac{1}{|x-y{|}^{\delta }},$$

$$R_1^G(1_G\cdot M_{\gamma }^{\rho })(x)$$

$$M_{\gamma }(B_r(x))\le c_2{r}^{\alpha }.$$

Full text

Liouville distorted Brownian motion

Jiyong Shin

**Abstract. The Liouville Brownian motion was introduced in [3] as a time changed process $B_{A_{t}^{-1}}$ of a planar Brownian motion $(B_{t})_{t\geq 0}$, where $(A_{t})_{t\geq 0}$ is the positive continuous additive functional of $(B_{t})_{t\geq 0}$ in the strict sense w.r.t. the Liouville measure. We first consider a distorted Brownian motion $(X_{t})_{t\geq 0}$ starting from all points in ${\mathbb{R}}^{2}$ associated to a Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$ (see [7]). We show that the positive continuous additive functional $(F_{t})_{t\geq 0}$ of $(X_{t})_{t\geq 0}$ in the strict sense w.r.t. the Liouville distorted measure can be constructed.

**

2010 Mathematics Subject Classification: Primary 31C25, 60J60, 60J45; Secondary 31C15, 60G15, 60H20.

Key words: Dirichlet forms, Distorted Brownian motion, Gaussian free field, Gaussian multiplicative chaos, Revuz correspondence.

1 Introduction

The Liouville Brownian motion, introduced by C. Garban, R. Rhodes, V. Vargas in [3], is a Markov process defined as a time changed process on ${\mathbb{R}}^{2}$. By classical theory of Gaussian multiplicative chaos (cf. [5]), the Liouville measure $M_{\gamma}$, $\gamma\in(0,2)$ is well defined (see Section 2). In [3] the positive continuous additive functional $(A_{t})_{t\geq 0}$ of a planar Brownian motion $(B_{t})_{t\geq 0}$ in the strict sense w.r.t. the Liouville measure $M_{\gamma}$ is constructed and then the Liouville Brownian motion is defined as $B_{A_{t}^{-1}}$.

In this paper we are concerned with the extension of the Liouville Brownian motion $B_{A_{t}^{-1}}$ to more general Markov processes. Note that the planar Brownian motion $(B_{t})_{t\geq 0}$ is associated with the Dirichlet form

[TABLE]

where $f,g\in\mathcal{F}=\{f\in L^{2}({\mathbb{R}}^{2},dx)\mid\nabla f\in L^{2}({\mathbb{R}}^{2},dx)\}$. As an extension of the Dirichlet form $({\mathcal{E}}^{\prime},\mathcal{F})$, we first consider a more general Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$ which is defined as the closure of the symmetric bilinear form

[TABLE]

on $L^{2}({\mathbb{R}}^{2},\rho dx)$ where $\rho(x):=|x|^{\alpha}$, $\alpha\in(-2,\infty)$ (see [7]). It is known from [6] and [7, Section 3] that there exists the distorted Brownian motion $((X_{t})_{t\geq 0},({\mathbb{P}}_{x})_{x\in{\mathbb{R}}^{2}})$ starting from all points in ${\mathbb{R}}^{2}$ associated with the Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$. Then using the estimates of the resolvent kernel and part Dirichlet form method, we can construct the positive continuous additive functional $(F_{t})_{t\geq 0}$ of $(X_{t})_{t\geq 0}$ in the strict sense w.r.t. the Liouville distorted measure $M_{\gamma}^{\rho}$ (see Section 2). Similarly to the Liouville Brownian motion, the Liouville distorted Brownian motion is defined as $X_{F_{t}^{-1}}$.

Notations:

For any open set $U$ in ${\mathbb{R}}^{2}$, we denote the set of all Borel measurable functions and the set of all bounded Borel measurable functions on $U$ by $\mathcal{B}(U)$ and $\mathcal{B}_{b}(U)$, respectively. The usual $L^{q}$-spaces $L^{q}(U,m)$, $q\in[1,\infty]$ are equipped with $L^{q}$-norm $\|\cdot\|_{L^{q}(U,m)}$ with respect to the measure $m$ on $U$ and $\mathcal{A}_{b}$ : = $\mathcal{A}\cap\mathcal{B}_{b}(U)$ for $\mathcal{A}\subset L^{q}(U,m)$. The inner product on $L^{2}(U,m)$ is denoted by $(\cdot,\cdot)_{L^{2}(U,m)}$. The indicator function of a set $A$ is denoted by $1_{A}$. Let $\nabla f:=(\partial_{1}f,\dots,\partial_{d}f)$ and $\Delta f:=\sum_{j=1}^{d}\partial_{jj}f$ where $\partial_{j}f$ is the $j$-th weak partial derivative of $f$ and $\partial_{jj}f:=\partial_{j}(\partial_{j}f)$, $j=1,\dots,d$. As usual $dx$ denotes the Lebesgue measure on ${\mathbb{R}}^{2}$. Here $C_{0}^{\infty}({\mathbb{R}}^{2})$ denotes the set of all infinitely differentiable functions with compact support in ${\mathbb{R}}^{2}$. We equip ${\mathbb{R}}^{2}$ with the Euclidean norm $|\cdot|$ and the corresponding inner product $\langle\cdot,\cdot\rangle$.

2 Massive Gaussian free field and Gaussian multiplicative chaos

We first state the definition of the massive Gaussian free field as stated in [3]. The massive Gaussian free field on ${\mathbb{R}}^{2}$ is a centered Gaussian random distribution (in the sense of Schwartz) on a probability space $(\Omega,\mathcal{A},P)$ with covariance function given by the Green function $G^{(m)}$ of the operator $m^{2}-\Delta$, $m>0$, i.e.

[TABLE]

where $\delta_{x}$ stands for the Dirac mass at $x$. The massive Green function with the operator $(m^{2}-\Delta)$ can be written as

[TABLE]

where

[TABLE]

Let $(c_{n})_{n\geq 1}$ be an unbounded strictly increasing sequence such that $c_{1}=1$ and $(Y_{n})_{n\geq 1}$ be a family of independent centered continuous Gaussian fields on ${\mathbb{R}}^{2}$ on the probability space $(\Omega,\mathcal{A},P)$ with covariance kernel given by

[TABLE]

The massive Gaussian free field is the Gaussian distribution defined by

[TABLE]

We define $n$-regularized field by

[TABLE]

and the associated $n$-regularized measure by

[TABLE]

where $\rho dz$ is a positive Radon measure on ${\mathbb{R}}^{2}$. By the classical theory of Gaussian multiplicative chaos (see [5]), $P$-a.s. the family $(M_{n,\gamma})_{n\geq 1}$ weakly converges to the Liouville distorted measure

[TABLE]

It is known from [5] that $M_{\gamma}^{\rho}$ is a Radon measure on ${\mathbb{R}}^{2}$. If $\rho=1$, we denote the n-regularized Liouville measure $M_{n,\gamma}^{\rho}$ and the Liouville measure $M_{\gamma}^{\rho}$ by $M_{n,\gamma}$ and $M_{\gamma}$, respectively.

3 Liouville distorted Brownian motion

We consider $\rho(x):=|x|^{\alpha}$, $\alpha\in(-2,\infty)$ and the symmetric bilinear form

[TABLE]

where $m:=\rho dx$. It is known that $({\mathcal{E}},C_{0}^{\infty}({\mathbb{R}}^{2}))$ is closable and its closure is a strongly local, regular Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$ on $L^{2}({\mathbb{R}}^{d},m)$ (cf. e.g. [7]). Let $(T_{t})_{t>0}$ and $(G_{\beta})_{\beta>0}$ be the $L^{2}({\mathbb{R}}^{2},m)$-semigroup and resolvent associated to $({\mathcal{E}},D({\mathcal{E}}))$ (see [2]). By [6] and [7, Section 3] there exists the distorted Brownian motion associated with the Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$

[TABLE]

with transition function $(P_{t})_{t\geq 0}$ where $\zeta$ is the lifetime. Moreover, it is known from [7, Section 3] that there exists a jointly continuous transition kernel density $p_{t}(x,y)$ such that

[TABLE]

is an $m$-version of $T_{t}f$ if $f\in L^{2}({\mathbb{R}}^{2},m)_{b}$. We set $P_{0}:=id$. Taking the Laplace transform of $p_{\cdot}(x,y)$, we obtain a $\mathcal{B}({\mathbb{R}}^{2})\times\mathcal{B}({\mathbb{R}}^{2})$ measurable non-negative resolvent kernel density $r_{\beta}(x,y)$ such that

[TABLE]

is an $m$-version of $G_{\beta}f$ if $f\in L^{2}({\mathbb{R}}^{2},m)_{b}$.

We present some definitions and properties concerning $({\mathcal{E}},D({\mathcal{E}}))$. We will refer to [2] till the end, hence some of its standard notations may be adopted below without definition. For any set $A\subset{\mathbb{R}}^{2}$ the capacity of $A$ is defined as

[TABLE]

Definition 3.1**.**

Let $B$ be an open set in ${\mathbb{R}}^{2}$. For $x\in B,t\geq 0,\beta>0$ and $p\in[1,\infty)$ let

• •

$\sigma_{B^{c}}:=\inf\{t>0\,|\;X_{t}\in B^{c}\},\quad D_{B^{c}}:=\inf\{t\geq 0\,|\;X_{t}\in B^{c}\}$,

• •

$P^{B}_{t}f(x):={\mathbb{E}}_{x}[f(X_{t});t<\sigma_{B^{c}}],\quad f\in\mathcal{B}_{b}(B)$,

• •

R^{B}_{\beta}f(x):={\mathbb{E}}_{x}\big{[}\int_{0}^{\sigma_{B^{c}}}e^{-\beta s}f(X_{s})\,ds\big{]},\quad f\in\mathcal{B}_{b}(B)* ,*

• •

$D({\mathcal{E}}^{B}):=\{u\in D({\mathcal{E}})\,|\;u=0\,\,\mathcal{E}\text{-q.e}\;on\;B^{c}\}$.

• •

${\mathcal{E}}^{B}:=\mathcal{E}\,|_{D({\mathcal{E}}^{B})\times D({\mathcal{E}}^{B})}$.

• •

${\mathcal{E}}^{B}_{1}(f,g):={\mathcal{E}}^{B}(f,g)+\int_{B}fg\ dm,\quad f,g\in D({\mathcal{E}}^{B})$.

• •

$\|\,f\,\|_{D(\mathcal{E}^{B})}:={\mathcal{E}}^{B}_{1}(f,f)^{1/2},\quad f\in D({\mathcal{E}}^{B})$.

It is known that $({\mathcal{E}}^{B},D({\mathcal{E}}^{B}))$ is a regular Dirichlet form on $L^{2}(B,m)$, which is called the part Dirichlet form of $({\mathcal{E}},D({\mathcal{E}}))$ on $B$ (cf. [2, Section 4.4]). Let $(T^{B}_{t})_{t>0}$ and $(G^{B}_{\beta})_{\beta>0}$ be the $L^{2}(B,m)$-semigroup and resolvent associated to $({\mathcal{E}}^{B},D({\mathcal{E}}^{B}))$. Then $P^{B}_{t}f,\;R^{B}_{\beta}f$ is an $m$-version of $T^{B}_{t}f,G^{B}_{\beta}f$, respectively for any $f\in L^{2}(B,m)_{b}$. Since $P_{t}^{B}1_{A}(x)\leq P_{t}1_{A}(x)$ for any $A\in\mathcal{B}(B)$, $x\in B$ and $m$ has full support on $E$, $A\mapsto P_{t}^{B}1_{A}(x),\;A\in\mathcal{B}(B)$ is absolutely continuous with respect to $1_{B}\cdot m$. Hence there exists a (measurable) transition kernel density $p^{B}_{t}(x,y)$, $x,y\in B$, such that

[TABLE]

for $f\in\mathcal{B}_{b}(B)$. Correspondingly, there exists a (measurable) resolvent kernel density $r_{\beta}^{B}(x,y)$, such that

[TABLE]

for $f\in\mathcal{B}_{b}(B)$. For a signed Radon measure $\mu$ on $B$, let us define

[TABLE]

whenever this makes sense. The process defined by

[TABLE]

is called the part process associated to ${\mathcal{E}}^{B}$ and is denoted by ${\mathbb{M}}|_{B}$. The part process ${\mathbb{M}}|_{B}$ is a Hunt process on $B$ (see [2, p.174 and Theorem A.2.10]). In particular, by (3.1) ${\mathbb{M}}|_{B}$ satisfies the absolute continuity condition on $B$.

A positive Radon measure $\mu$ on $B$ is said to be of finite energy integral if

[TABLE]

where $c$ is some constant independent of $f$ and $C_{0}(B)$ is the set of all compactly supported continuous functions on $B$. A positive Radon measure $\mu$ on $B$ is of finite energy integral (on $B$) if and only if there exists a unique function $U_{1}^{B}\,\mu\in D({\mathcal{E}}^{B})$ such that

[TABLE]

for all $f\in D({\mathcal{E}}^{B})\cap C_{0}(B)$. $U_{1}^{B}\,\mu$ is called $1$-potential of $\mu$. In particular, $R_{1}^{B}\mu$ is a version of $U_{1}^{B}\mu$ (see e.g. [2, Exercise 4.2.2]). The measures of finite energy integral are denoted by $S_{0}^{B}$. We further define $S_{00}^{B}:=\{\mu\in S_{0}^{B}\mid\mu(B)<\infty,\|U_{1}^{B}\mu\|_{L^{\infty}(B,m)}<\infty\}$.

If $\mu\in S_{00}^{B}$, then there exists a unique $A\in A_{c,1}^{+,B}$ with $\mu=\mu_{A}$, i.e. $\mu$ is the Revuz measure of $A$ (see [2, Theorem 5.1.6]). Here, $A_{c,1}^{+,B}$ denotes the positive continuous additive functionals on $B$ in the strict sense.

We define

[TABLE]

The following proposition recalls some properties of the Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$ as stated in [7, Theorem 2.10, Lemma 3.13]:

Proposition 3.2**.**

• (i)

For any $\alpha\in(-2,\infty)$, the Dirichlet form $({\mathcal{E}},D({\mathcal{E}}))$ is conservative.

• (ii)

For any $\alpha\in[0,\infty)$, $\emph{Cap}(\{0\})=0$.

• (iii)

For any $\alpha\in[0,\infty)$ and any $x\in E$

[TABLE]

From now on till the end of this paper, we consider

[TABLE]

Lemma 3.3**.**

Almost surely in $X$, for any relatively compact open set $G\subset E$,

[TABLE]

Proof.

The statement follows from

[TABLE]

$\Box$

Lemma 3.4**.**

Let $G$ be any relatively compact open set in $E$ with $G\subset\overline{G}\subset E$. For any $x,y\in G$ and any $\delta>0$

[TABLE]

where $c_{1}>0$ is some constant.

Proof.

By [7, Lemma 3.10] for $m$-a.e. $x,y\in G$ and any $\delta>0$

[TABLE]

where $c_{1}>0$ is some constant. Since $p_{t}(x,y)$ is jointly continuous, the statement holds for all $x,y\in G$. $\Box$

Theorem 3.5**.**

Almost surely in $X$, for any relatively compact open set $G\subset\overline{G}\subset E$, $1_{G}\cdot M^{\rho}_{\gamma}\in S^{G}_{00}$.

Proof.

Clearly, $M^{\rho}_{\gamma}(G)<\infty$. By Lemma 3.3 and Lemma 3.4, for any $x\in G$ and $\delta>0$,

[TABLE]

Since $G$ is a relatively compact open set, we can find a constant $R>0$ such that $G\subset B_{R}(0)$. By [3, Theorem 2.2], there exist a constant $c_{2}>0$ and $\alpha>0$ (depending on $R>0$ and $\gamma$) such that for all $r\in(0,R)$ and $x\in B_{R}(0)$

[TABLE]

By taking $0<\delta<\alpha$, we obtain

[TABLE]

Hence,

[TABLE]

and

[TABLE]

By [2, Exercise 4.2.2], (3.2) implies that $1_{G}\cdot M^{\rho}_{\gamma}\in S^{G}_{0}$. Therefore, $1_{G}\cdot M^{\rho}_{\gamma}\in S^{G}_{00}$. $\Box$

Corollary 3.6**.**

Almost surely in $X$, $M^{\rho}_{\gamma}$ does not charge capacity zero sets.

Proof.

Let $N\subset{\mathbb{R}}^{2}$ be an open set such that Cap$(N)=0$. Note that by [2, Lemma 2.2.3], $M^{\rho}_{\gamma}(1_{E_{k}}\cap N)=0$ (see Theorem 3.5). Then the statement follows from

[TABLE]

$\Box$

The proof of the following theorem is a slight modification of [6, Lemma 5.11] in our setting.

Theorem 3.7**.**

Let $F_{t}^{k}$ be the positive continuous additive functional of $(X_{t}^{E_{k}})_{t\geq 0}$ in the strict sense associated to $1_{E_{k}}\cdot M_{\gamma}^{\rho}\in S_{00}^{E_{k}}$. Then, $F_{t}^{k}=F_{t}^{k+1},\;\forall t<\sigma_{E_{k}^{c}}$ ${\mathbb{P}}_{x}$-a.s. for all $x\in E_{k}$. In particular, $F_{t}:=\lim_{k\rightarrow\infty}F_{t}^{k}$, $t\geq 0$, is well defined in $A_{c,1}^{+,E}$, and related to $M_{\gamma}^{\rho}$ via the Revuz correspondence.

Proof.

We denote the set of all bounded, non-negative Borel measurable functions on $E_{k}$ by $\mathcal{B}_{b}^{+}(E_{k})$. Fix $f\in\mathcal{B}_{b}^{+}(E_{k})$ and for $x\in E_{k+1}$ define

[TABLE]

Since $f_{k}\in D({\mathcal{E}}^{E_{k+1}})$ and $f_{k}=0$ ${\mathcal{E}}^{E_{k+1}}$-q.e. on $E_{k}^{c}$, we have $f_{k}\in D({\mathcal{E}}^{E_{k}})$. For $x\in E_{k}$

[TABLE]

Then, for $g\in\mathcal{B}_{b}^{+}(E_{k})\cap L^{2}(E_{k},m)$

[TABLE]

Therefore, f_{k}=R_{1}^{E_{k}}\Big{(}f\,1_{E_{k}}\cdot M_{\gamma}^{\rho}\Big{)}\, $m$-a.e. Since R_{1}^{E_{k}}\Big{(}f\,1_{E_{k}}\cdot M_{\gamma}^{\rho}\Big{)} is 1-excessive for $(R_{\alpha}^{E_{k}})_{\alpha>0}$, we obtain for any $x\in E_{k}$

[TABLE]

Using in particular the strong Markov property, we obtain by direct calculation that the right hand limit equals $f_{k}(x)$ for any $x\in E_{k}$. Thus, we showed for all $x\in E_{k}$

[TABLE]

This implies that $F_{t}^{k}=F_{t}^{k+1},\;\forall t<\sigma_{E_{k}^{c}}$ ${\mathbb{P}}_{x}$-a.s. for all $x\in E_{k}$ (see e.g. [1, IV. (2.12) Proposition]). Then, using Proposition 3.2, $F_{t}:=\lim_{k\rightarrow\infty}F_{t}^{k}$, $t\geq 0$, is well defined in $A_{c,1}^{+,E}$. Moreover, $(F_{t})_{t\geq 0}$ is associated with $M_{\gamma}^{\rho}$ via the Revuz correspondence. $\Box$

Finally, almost surely in $X$, the time changed process $(Z_{t})_{t\geq 0}$ on $E$ can be defined as

[TABLE]

where $F_{t}^{-1}:=\inf\{s>0\mid F_{s}>t\}$, which is called the Liouville distorted Brownian motion.

Remark 3.8**.**

In [7] we considered the following assumptions: $\rho(x)=|x|^{\alpha}$, $\alpha\in[2,\infty)$ and a symmetric (possibly) degenerate (uniformly weighted) elliptic $2\times 2$ matrix $A=(a_{ij})_{1\leq i,j\leq 2}$, that is $a_{ij}\in L^{1}_{loc}({\mathbb{R}}^{2},dx)$ and there exists a constant $\lambda\geq 1$ such that for $dx$-a.e. $x\in{\mathbb{R}}^{2}$

[TABLE]

and the symmetric bilinear form

[TABLE]

The closure $({\mathcal{E}}^{A},D({\mathcal{E}}^{A}))$ of $({\mathcal{E}}^{A},C_{0}^{\infty}({\mathbb{R}}^{2}))$ is a strongly local, regular, symmetric Dirichlet form. It is known from [7] that there exists a Hunt process $(Y_{t})_{t\geq 0}$ starting from all points in ${\mathbb{R}}^{2}$ associated with the Dirichlet form $({\mathcal{E}}^{A},D({\mathcal{E}}^{A}))$. Following the methods and techniques as in this section, we can construct the positive continuous additive functional $(H_{t})_{t\geq 0}$ of $(Y_{t})_{t\geq 0}$ in the strict sense w.r.t. $M_{\gamma}^{\rho}$ and then the time changed process $Y_{H_{t}^{-1}}$, $t\geq 0$ on $E$ can be defined in the same way.