Short-time asymptotic expansions of semilinear evolution equations

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Abstract

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.
Original languageEnglish
Pages (from-to)141-167
Number of pages27
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume146
Issue number1
DOIs
StatePublished - Jan 2016

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Semilinear evolution equation
Asymptotic expansion
Symbolic calculus
Backward stochastic differential equation
Kernel methods
Algebraic approach
Heat kernel
Pseudodifferential operators
Banach algebra
Approximate solution
Valid
Gradient
Perturbation
Arbitrary
Algebra
Differential equations
perturbation

Cite this

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abstract = "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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