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Stochastic Hamilton-Jacobi equations

Published online by Cambridge University Press:  04 August 2010

A. Truman
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
H. Z. Zhao
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
Alison Etheridge
Affiliation:
University of Edinburgh
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Summary

Abstract

In this paper we describe the stochastic Hamilton Jacobi theory and its applications to stochastic heat equations, Schrodinger equations and stochastic Burgers’ equations.

Keywords. Stochastic Hamilton Jacobi equation, Stochastic heat equation, Stochastic Burgers’ equation, Stochastic harmonic oscillator.

Introduction

It is well-known that the leading term in Varadhan's and Wentzell-Freidlin's large deviation theories (Varadhan (1967), Wentzell and Freidlin (1970)) and Maslov's quasi-classical asymptotics of quantum mechanics (Maslov (1972)) involves the solution of a variational problem which gives a Lipschitz continuous solution of the Hamilton Jacobi equation if it exists (Fleming (1969, 1986)). It was proved by Truman (1977) and Elworthy and Truman (1981, 1982) that before the caustic time the Hamilton Jacobi function which is C1,2 gives the exact solutions of the diffusion equations geared to small time asymptotics. The main tools in the theory are classical mechanics and the Maruyama-Girsanov-Cameron-Martin formula. The philosophy of the theory is to choose a suitable drift for a Brownian motion on the configuration space manifold and to employ the MGCM theorem to simplify the Feynman-Kac representation of the solutions for the heat equations. An extended version of this theory to degenerate diffusion equations was obtained in Watling (1992). The Brownian Riemannian bridge process was obtained by this means in Elworthy and Truman (1982). The extension to more general Riemannian manifolds was obtained in Elworthy (1988), Ndumu (1986,1991). The same methods have been applied to travelling waves for nonlinear reaction diffusion equations in Elworthy, Truman and Zhao (1994).

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Publisher: Cambridge University Press
Print publication year: 1995

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  • Stochastic Hamilton-Jacobi equations
    • By A. Truman, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K., H. Z. Zhao, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
  • Edited by Alison Etheridge, University of Edinburgh
  • Book: Stochastic Partial Differential Equations
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526213.018
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  • Stochastic Hamilton-Jacobi equations
    • By A. Truman, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K., H. Z. Zhao, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
  • Edited by Alison Etheridge, University of Edinburgh
  • Book: Stochastic Partial Differential Equations
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526213.018
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Stochastic Hamilton-Jacobi equations
    • By A. Truman, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K., H. Z. Zhao, Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
  • Edited by Alison Etheridge, University of Edinburgh
  • Book: Stochastic Partial Differential Equations
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526213.018
Available formats
×