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Regular Random Field Solutions for Stochastic Evolution Equations

Antoni, Markus

Abstract:

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


Volltext §
DOI: 10.5445/IR/1000069854
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Publikationsjahr 2017
Sprache Englisch
Identifikator urn:nbn:de:swb:90-698548
KITopen-ID: 1000069854
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 187 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 18.01.2017
Schlagwörter Stochastic integration, stochastic convolution, maximal regularity, stochastic evolution equations, pathwise Hölder regularity
Referent/Betreuer Weis, L.
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