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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
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Volume 27, Issue 5

# Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms

Wei Sun
/ Jing Zhang
Published Online: 2014-01-25 | DOI: https://doi.org/10.1515/forum-2013-0082

## Abstract

Let U be an open set of ℝn, m be a positive Radon measure on U such that $\mathrm{supp}\left[m\right]=U$, and ${\left({P}_{t}\right)}_{t>0}$ be a strongly continuous contraction sub-Markovian semigroup on ${L}^{2}\left(U;m\right)$. We investigate the structure of ${\left({P}_{t}\right)}_{t>0}$.

(i) Denote respectively by $\left(A,D\left(A\right)\right)$ and $\left(\stackrel{^}{A},D\left(\stackrel{^}{A}\right)\right)$ the generator and the co-generator of ${\left({P}_{t}\right)}_{t>0}$. Under the assumption that ${C}_{0}^{\infty }\left(U\right)\subset D\left(A\right)\cap D\left(\stackrel{^}{A}\right)$, we give an explicit Lévy–Khintchine type representation of A on ${C}_{0}^{\infty }\left(U\right)$.

(ii) If ${\left({P}_{t}\right)}_{t>0}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form $\left(ℰ,D\left(ℰ\right)\right)$, we give an explicit characterization of ℰ on ${C}_{0}^{\infty }\left(U\right)$ under the assumption that ${C}_{0}^{\infty }\left(U\right)\subset D\left(ℰ\right)$.

We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

MSC: 31C25; 60J25

Revised: 2013-11-26

Published Online: 2014-01-25

Published in Print: 2015-09-01

Funding Source: NSERC

Award identifier / Grant number: 311945-2013

Funding Source: NSFC

Award identifier / Grant number: 11361021

Funding Source: NSFC

Award identifier / Grant number: 11201102

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 3111–3148, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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