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4
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1answer
87 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
1answer
91 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 ...
1
vote
1answer
69 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
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2answers
876 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 ...
5
votes
2answers
468 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 $X$ ...
5
votes
1answer
350 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 ...