Center manifolds for rough partial differential equations

Christian Kuehn, Alexandra Neamţu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for γ ∈ (1/3, 1/2]. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.

Original languageEnglish
Article number48
JournalElectronic Journal of Probability
Volume28
DOIs
StatePublished - 2023
Externally publishedYes

Keywords

  • Lyapunov-Perron method
  • center manifold
  • evolution equation, interpolation spaces
  • rough path

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