# Questions tagged [dirichlet-forms]

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### Algebra core for generator of Dirichlet form

This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
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### Analyticity of the semigroup generated by a time-changed Brownian motion

Let $d$ be an integer. We denote by $m$ the Lebesgue measure on $\mathbb{R}^d$. We define $BL(\mathbb{R}^d)$ by \begin{align*} BL(\mathbb{R}^d)=\{f \in L^2_{\rm loc}(\mathbb{R}^d,m) \mid |\nabla f|\in ...
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### Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
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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 $\... 0 votes 1 answer 146 views ### Generators and Dirichlet forms I have a question about a Dirichlet form. Let$D$be a open subset of$\mathbb{R}^d$. Then, we can define$H^{1}(D)$by \begin{equation*} H^{1}(D)=\{f \in L^{2}(D,dx):\frac{\partial f}{\partial x_i} \... 1 vote 1 answer 367 views ### Hunt processes and its equivalence I have a question about Hunt processes and its equivalence. I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. The following theorem is stated in ... 1 vote 1 answer 128 views ### Identifying Dirichlet forms of part processes, how to prove I have a question about Dirichlet forms. Let$D$be a domain of$\mathbb{R}^d$and$H^{1}(D)$denotes$(1,2)$-Sobolev space on$D$with Neumann boundary condition. We define the following a Dirichlet ... 3 votes 1 answer 328 views ### Local upper estimates for Neumann heat kernels I have a question about Neumann heat kernels and its estimates. Let$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*}... 4 votes 1 answer 704 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 ... 4 votes 1 answer 358 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 ... 2 votes 1 answer 534 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$: \... 5 votes 2 answers 556 views ### Regular Dirichlet form and the associated transition kernel I am reading a paper by Fukushima "On a stochastic calculus related to Dirichlet forms and distorted Brownian motions" and support it by a book "Dirichlet forms and symmetric Markov processes" by ... 2 votes 1 answer 186 views ### Compactness of semigroups, boundary conditions I have a question about compactness of semigroups and boundary conditions. Let$\Omega$be an unbounded domain of$\mathbb{R}^d$with smooth boundary and$m(\Omega)=\infty$. Then we can define two ... 1 vote 0 answers 288 views ### A problem on Markov chains and Dirichlet forms Let$X$be a countable set. Let$c:X\times X\to[0,+\infty)$satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ... 1 vote 1 answer 290 views ### A problem about the quotient space of an extended Dirichlet space Let (\mathscr{E},\mathscr{F}) be a recurrent Dirichlet form on L^2(X;m) and \mathscr{F}_e the corresponding extended Dirichlet space, then 1\in\mathscr{F}_e and \mathscr{E}(1,1)=0. Let {\... 4 votes 1 answer 195 views ### Urysohn type cut off function I am looking for a cutoff function. The Urysohn's Lemma says Let X be a T_{4} space and A,B \subset X be two closed and disjoint subsets of X. Then there exists a continuous function f:X \... 6 votes 1 answer 220 views ### Examples of optimal ultracontractivity estimates for a Markovian semigroup T_t that do not depend polynomialy on t Let (X,\mu) be a measure space and T_t : L_2(\mu) \to L_2(\mu) for t \geq 0 a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:$$ \| T_t : L_p(\mu) \to L_q(\mu)\| \... 4 votes 1 answer 219 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 ... 2 votes 1 answer 124 views ### 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?$W^{1,2}(M)$... 1 vote 2 answers 906 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)<\...
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 $X$ ...