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Title: Some problems in abstract stochastic differential equations on Banach spaces
Author: Crewe, Paul
ISNI:       0000 0004 2743 9056
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2011
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This thesis studies abstract stochastic differential equations on Banach spaces. The well-posedness of abstract stochastic differential equations on such spaces is a recent result of van Neerven, Veraar and Weis, based on the theory of stochastic integration of Banach space valued processes constructed by the same authors. We study existence and uniqueness for solutions of stochastic differential equations with (possibly infinite) delay in their inputs on UMD Banach spaces. Such problems are also known as functional differential equations or delay differential equations. We show that the methods of van Neerven et al. extend to such problems if the initial history of the system lies in a space of a type introduced by Hale and Kato. The results are essentially of a fixed point type, both autonomous and non-autonomous cases are discussed and an example is given. We also study some long time properties of solutions to these stochastic differential equations on general Banach spaces. We show the existence of solutions to stochastic problems with almost periodicity in a weak or distributional sense. Results are again given for both autonomous and non-autonomous cases and depend heavily on estimates for R-bounds of operator families developed by Veraar. An example is given for a second order differential operator on a domain in ℝd. Finally we consider the existence of invariant measures for such problems. This extends recent work of van Gaans in Hilbert spaces to Banach spaces of type 2.
Supervisor: Batty, Charles Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Functional analysis (mathematics) ; stochastic differential equations ; stochastic integration ; Banach space ; functional analysis