A pde approach to small stochastic perturbations of Hamiltonian flows

Hitoshi Ishii*, Panagiotis E. Souganidis

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    9 Citations (Scopus)


    In this note we present a unified approach, based on pde methods, for the study of averaging principles for (small) stochastic perturbations of Hamiltonian flows in two space dimensions. Such problems were introduced by Freidlin and Wentzell and have been the subject of extensive study in the last few years using probabilistic arguments. When the Hamiltonian flow has critical points, it exhibits complicated behavior near the critical points under a small stochastic perturbation. Asymptotically the slow (averaged) motion takes place on a graph. The issues are to identify both the equations on the sides and the boundary conditions at the vertices of the graph. Our approach is very general and applies also to degenerate anisotropic elliptic operators which could not be considered using the previous methodology.

    Original languageEnglish
    Pages (from-to)1748-1775
    Number of pages28
    JournalJournal of Differential Equations
    Issue number2
    Publication statusPublished - 2012 Jan 15


    • Averaging
    • Hamiltonian flows
    • Pde approach
    • Primary
    • Secondary
    • Stochastic perturbations

    ASJC Scopus subject areas

    • Analysis


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