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2022•arXiv (Cornell University)Open access

Stochastic Calculus in Infinite Dimensions and SPDEs

Daniel Goodair

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Abstract

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in infinite dimensions, with a particular focus on Stratonovich equations due to their physical importance as well as unbounded noise operators (with applications to transport noise). Furthermore we explore techniques in the existence theory for nonlinear stochastic partial differential equations.

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These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in infinite dimensions, with a particular focus on Stratonovich equations due to their physical importance as well as unbounded noise operators (with applications to transport noise). Furthermore we explore techniques in the existence theory for nonlinear stochastic partial differential equations.

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Available abstract

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in infinite dimensions, with a particular focus on Stratonovich equations due to their physical importance as well as unbounded noise operators (with applications to transport noise). Furthermore we explore techniques in the existence theory for nonlinear stochastic partial differential equations.

Key concepts: Stochastic calculus, Quantum stochastic calculus, Stochastic partial differential equation, Mathematics, Stochastic differential equation, Stratonovich integral, Time-scale calculus, Brownian motion

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