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2009•Proceedings of the London Mathematical SocietyOpen access

Peano's theorem for rough differential equations in infinite-dimensional Banach spaces

Michael Caruana

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Abstract

We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

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We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

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Available abstract

We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

Key concepts: Mathematics, Peano existence theorem, Peano axioms, Picard–Lindelöf theorem, Eberlein–Šmulian theorem, Arzelà–Ascoli theorem, Brouwer fixed-point theorem, Banach space

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