Questions tagged [dirichlet-forms]
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25
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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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1
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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,\...
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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 ...
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3
votes
1
answer
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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 &...
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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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6
votes
1
answer
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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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0
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92
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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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1
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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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1
vote
1
answer
367
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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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1
vote
1
answer
128
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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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3
votes
1
answer
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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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4
votes
1
answer
704
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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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4
votes
1
answer
358
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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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2
votes
1
answer
534
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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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5
votes
2
answers
556
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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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1
answer
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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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1
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0
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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
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...
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1
vote
1
answer
290
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A problem about the quotient space of an extended Dirichlet space
Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
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1
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Urysohn type cut off function
I am looking for a cutoff function.
The Urysohn's Lemma says
Let $X$ be a $T_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X \...
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1
answer
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Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$
Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:
$$
\| T_t : L_p(\mu) \to L_q(\mu)\| \...
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1
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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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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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vote
2
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906
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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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2
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704
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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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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 ...
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