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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / 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
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1435-5337
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Volume 28, Issue 2

Issues

Time inhomogeneous generalized Mehler semigroups and skew convolution equations

Shun-Xiang Ouyang / Michael Röckner
Published Online: 2015-01-11 | DOI: https://doi.org/10.1515/forum-2013-0192

Abstract

A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through ps,tf(x) = ∫ f(U(t,s)x + yt,s(dy), s,t ∈ ℝ, ts, x ∈ ℍ, for every bounded measurable function f on ℍ, where (U(t,s))ts is an evolution family of bounded operators on ℍ and (μt,s)ts is a family of probability measures on (ℍ,ℬ(ℍ)) satisfying the following time inhomogeneous skew convolution equations: μt,s = μt,r * (μr,sU(t,r)-1), trs. This kind of semigroups typically arise as the “transition semigroups” of non-autonomous (possibly non-continuous) Ornstein–Uhlenbeck processes driven by some proper additive process. Suppose that μt,s converges weakly to δ0 as ts or st. We show that μt,s has further weak continuity properties in t and s. As a consequence, we prove that for every ts, μt,s is infinitely divisible. Natural stochastic processes associated with (μt,s)ts are constructed and are applied to get probabilistic proofs for the weak continuity and infinite divisibility. Then we analyze the structure, existence and uniqueness of the corresponding evolution systems of measures (= space-time invariant measures) of (ps,t)ts. We also establish a dimension free Harnack inequality for (ps,t)ts and present some of its applications.

Keywords: Time inhomogeneous generalized Mehler semigroups; skew convolution equations; infinite divisibility; evolution system of measures; Harnack inequality

MSC: 60J75; 47D07

About the article

Received: 2013-12-04

Revised: 2014-06-19

Published Online: 2015-01-11

Published in Print: 2016-03-01


Funding Source: DFG

Award identifier / Grant number: SFB-701

Funding Source: DFG

Award identifier / Grant number: IRTG-1132


Citation Information: Forum Mathematicum, Volume 28, Issue 2, Pages 339–376, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0192.

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