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Questions tagged [dirichlet-forms]

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How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?

Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...
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Spectral gap inequalities and hitting times

I have a question about spectral gaps for generators of continuous -time Markov processes. Let $(\mathcal{E},\mathcal{F})$ be a regular Dirichlet form on metric measure space $(E,\mu)$. We assume $m$ ...
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1answer
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Generators and Dirichlet forms

I have a question about a Dirichlet form. Let $D$ be a open subset of $\mathbb{R}^d$. Then, we can define $H^{1}(D)$ by \begin{equation*} H^{1}(D)=\{f \in L^{2}(D,dx):\frac{\partial f}{\partial x_i} \...
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1answer
116 views

Hunt processes and its equivalence

I have a question about Hunt processes and its equivalence. I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. The following theorem is stated in ...
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1answer
109 views

Identifying Dirichlet forms of part processes, how to prove

I have a question about Dirichlet forms. Let $D$ be a domain of $\mathbb{R}^d$ and $H^{1}(D)$ denotes $(1,2)$-Sobolev space on $D$ with Neumann boundary condition. We define the following a Dirichlet ...
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1answer
178 views

Local upper estimates for Neumann heat kernels

I have a question about Neumann heat kernels and its estimates. Let $D$ be a domain of $\mathbb{R}^d$. We define the Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^{2}(D)$ as follows: \begin{align*}...
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1answer
240 views

Dynkin Hunt formula

I have a question about Dynkin Hunt formula. Last day, I found a formula in this paper enter link description here. The formula is the equation (2.5) in this paper, which is called Dynkin Hunt ...
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1answer
283 views

On Brownian motions

I have a question about Brownian motions and its heat kernel. Using Dirichlet form theory, we can construct Brownian motions on manifolds, domains of Euclidean space under mild assumptions. For ...
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1answer
284 views

Reflecting Brownian motion and its transition probability density

I have a question about reflecting Brownian motion on an unbounded domain. Let us consider the reflecting Brownian motion $\{X_t\}_{t \ge 0}$ on the following domain $\bar{D}$ of $\mathbb{R}^2$: \...
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2answers
223 views

Regular Dirichlet form and the associated transition kernel

I am reading a paper by Fukushima "On a stochastic calculus related to Dirichlet forms and distorted Brownian motions" and support it by a book "Dirichlet forms and symmetric Markov processes" by ...
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1answer
146 views

Compactness of semigroups, boundary conditions

I have a question about compactness of semigroups and boundary conditions. Let $\Omega$ be an unbounded domain of $\mathbb{R}^d$ with smooth boundary and $m(\Omega)=\infty$. Then we can define two ...
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A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
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1answer
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A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
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1answer
133 views

Urysohn type cut off function

I am looking for a cutoff function. The Urysohn's Lemma says Let $X$ be a $T_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X \...
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Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$

Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form: $$ \| T_t : L_p(\mu) \to L_q(\mu)\| \...
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1answer
155 views

generator of Dirichlet form coincide with the absolute part of the “Laplacian”

Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the ...
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1answer
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Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?

M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$. Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form? $W^{1,2}(M)$ ...
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2answers
896 views

Heat flow $P_tf \to f$ in $W^{1,2}$ for $f \in W^{1,2}$?

$\varepsilon:L^2(X,m) \to [0,\infty]$ is a strongly local, symmetric Dirichlet form generating a Markov semigroup $(P_t)_{t\ge0}$ in $L^2(X,m)$. Let $D(\varepsilon)=\{f\in L^2(X,m):\varepsilon(f)<\...
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2answers
601 views

Symmetric Feller processes and Dirichlet Forms

Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ ...
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750 views

Is there a regular Dirichlet form with no associated Feller process?

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...