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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

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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


Let U be an open set of ℝn, m be a positive Radon measure on U such that supp [m]=U, and (Pt)t>0 be a strongly continuous contraction sub-Markovian semigroup on L2(U;m). We investigate the structure of (Pt)t>0.

(i) Denote respectively by (A,D(A)) and (A^,D(A^)) the generator and the co-generator of (Pt)t>0. Under the assumption that C0(U)D(A)D(A^), we give an explicit Lévy–Khintchine type representation of A on C0(U).

(ii) If (Pt)t>0 is an analytic semigroup and hence is associated with a semi-Dirichlet form (,D()), we give an explicit characterization of ℰ on C0(U) under the assumption that C0(U)D().

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

Keywords: Dirichlet generator; semi-Dirichlet form; Markov process; Lévy–Khintchine type representation; LeJan type transformation rule; Beurling–Deny formula

MSC: 31C25; 60J25

About the article

Received: 2013-05-20

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, DOI: https://doi.org/10.1515/forum-2013-0082.

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