The Burgers equation with affine linear noise: Dynamics and stability

Salah Mohammed and Tusheng Zhang

Stochastic Processes and their Applications, 2012, vol. 122, issue 4, 1887-1916

Abstract: We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.

Keywords: Burgers equation; Affine linear noise; Perfect cocycle; Multiplicative ergodic theory; Lyapunov spectrum; Stationary solution; Hyperbolicity; Local stable manifold theorem; Invariant manifolds; Global invariant foliation (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:122:y:2012:i:4:p:1887-1916

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DOI: 10.1016/j.spa.2011.12.002

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