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Short-time asymptotic expansions of semilinear evolution equations

Part of: Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  07 January 2016

Matthias A. Fahrenwaldt
Matthias A. Fahrenwaldt*
Affiliation:
Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany ([email protected])
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Extract

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

Keywords

semilinear equationsasymptotic analysisBanach algebrasbackward stochastic differential equations

MSC classification

Secondary: 35K58: Semilinear parabolic equations 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 1 , February 2016 , pp. 141 - 167
Copyright
Copyright © Royal Society of Edinburgh 2016 

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