Short-time asymptotic expansions of semilinear evolution equations

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 calc...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 146; no. 1; pp. 141 - 167
Main Author Fahrenwaldt, Matthias A.
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.02.2016
Cambridge University Press
Subjects
Algebra
Approximation
Asymptotic expansions
Construction
Differential equations
Evolution
Linear equations
Mathematical analysis
Mathematics
Perturbation methods
asymptotic analysis
backward stochastic differential equations
semilinear equations
Banach algebras
Secondary 35B40
Primary 35K58
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Summary: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.
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ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210515000372