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Regular Random Field Solutions for Stochastic Evolution Equations

Antoni, Markus

In this thesis we investigate stochastic evolution equations for random fields X: Omega x [0; T] x U -> R, where [0; T] is a time interval, (Omega; F; P) a measure space representing the randomness of the system, and U is typically a domain in Rd (or again a measure space). More precisely, we concentrate on the parabolic situation where A is the generator of an analytic semigroup on Lp(U). We look for mild solutions so that X has values in Lp(U;Lq[0; T]) almost surely under appropriate Lipschitz and linear growth conditions on the nonlinearities. Compared to the classical semigroup approach, which gives X \in Lq([0; T];Lp(U)) almost surely, the order of integration is reversed. We show that this new approach together with a strong Doob and Burkholder-Davis-Gundy inequality leads to strong regularity results in particular for the time variable of the random field X, e.g. pointwise Hölder estimates for the paths, P-almost surely. For less-optimal regularity estimates we only need the relatively mild assumption that the resolvents of A extend uniformly to Lp(U;Lq[0; T]). However, in the maximal regularity case the difficulty of the reversed order of integration in time and space makes extended functional calculi results necessary. ... mehr

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DOI: 10.5445/IR/1000069854
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Jahr 2017
Sprache Englisch
Identifikator urn:nbn:de:swb:90-698548
KITopen-ID: 1000069854
Verlag KIT, Karlsruhe
Umfang 187 S.
Abschlussart Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 18.01.2017
Referent/Betreuer Prof. L. Weis
Schlagworte Stochastic integration, stochastic convolution, maximal regularity, stochastic evolution equations, pathwise Hölder regularity
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