Probability Theory and Related Fields

, Volume 153, Issue 1–2, pp 97–147

Non-linear rough heat equations

Article

Abstract

This article is devoted to define and solve an evolution equation of the form dyt = Δytdt + dXt (yt), where Δ stands for the Laplace operator on a space of the form \({L^p(\mathbb R^n)}\), and X is a finite dimensional noisy nonlinearity whose typical form is given by \({X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}\), where each x = (x(1), … , x(N)) is a γ-Hölder function generating a rough path and each fi is a smooth enough function defined on \({L^p(\mathbb R^n)}\). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.

Keywords

Rough paths theory Stochastic PDEs Fractional Brownian motion 

Mathematics Subject Classification (2000)

60H05 60H07 60G15 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut Élie Cartan NancyUniversité de NancyVanœduvre-lès-Nancy CedexFrance
  2. 2.CEREMADE, Université de Paris-DauphineParisFrance

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