Questions tagged [dirichlet-forms]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
8 votes
1 answer
978 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 ...
Nate Eldredge's user avatar
6 votes
1 answer
225 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)\| \...
Adrián González Pérez's user avatar
6 votes
1 answer
525 views

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 ...
sharpe's user avatar
  • 701
5 votes
2 answers
582 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 ...
tuko's user avatar
  • 51
5 votes
2 answers
717 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 $X$ ...
Hans's user avatar
  • 438
4 votes
2 answers
1k views

Does a function exist which is not Riemann integrable and satisfies the given condition:

I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that $$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
Lakshmi Priya's user avatar
4 votes
1 answer
187 views

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 ...
sharpe's user avatar
  • 701
4 votes
1 answer
206 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 \...
sharpe's user avatar
  • 701
4 votes
1 answer
752 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 ...
sharpe's user avatar
  • 701
4 votes
1 answer
369 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 ...
sharpe's user avatar
  • 701
4 votes
1 answer
230 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 ...
wang mu's user avatar
  • 199
3 votes
1 answer
433 views

Product formula for Laplace de-Rham operator

Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &...
Nathanael Schilling's user avatar
3 votes
1 answer
347 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*}...
sharpe's user avatar
  • 701
3 votes
0 answers
64 views

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 ...
Curious's user avatar
  • 143
2 votes
1 answer
567 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 $\mathbb{R}^2$: \...
sharpe's user avatar
  • 701
2 votes
1 answer
192 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 ...
sharpe's user avatar
  • 701
2 votes
1 answer
126 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? $W^{1,2}(M)$ ...
wang mu's user avatar
  • 199
2 votes
0 answers
128 views

On a degenerate SDE in the unit ball

This is a question about a diffusion process on the unit ball. In this article J.S, the author considered the following SDE in the closed unit ball $E \subset \mathbb{R}^n$: \begin{align*} (1)\quad ...
sharpe's user avatar
  • 701
1 vote
1 answer
392 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 ...
sharpe's user avatar
  • 701
1 vote
1 answer
130 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 ...
sharpe's user avatar
  • 701
1 vote
1 answer
291 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 ${\...
yangmengqh's user avatar
1 vote
2 answers
909 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)<\...
wang mu's user avatar
  • 199
1 vote
0 answers
96 views

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 $\...
0xbadf00d's user avatar
  • 161
1 vote
0 answers
301 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 ...
yangmengqh's user avatar
0 votes
1 answer
151 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} \...
sharpe's user avatar
  • 701
0 votes
1 answer
359 views

Sufficient conditions for an asymptotic compactness

This question relates a theory of Mosco convergence. Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$. A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
sharpe's user avatar
  • 701