ESAIM: Probability and Statistics

Table of Contents - October 2006 - Volume 10  

Research Article

SPDEs with coloured noise: Analytic and stochastic approaches

Ferrante, Marcoa1 and Sanz-Solé, Martaa2

a1 Dipartimento di Matematica, Università di Padova, Via Belzoni 7, 35131 Padova, Italy; ferrante@math.unipd.it

a2 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; marta.sanz@ub.edu

Abstract

We study strictly parabolic stochastic partial differential equations on $\mathbb{R}^d$ , d ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving path properties of the solution to a parabolic Cauchy problem in evolution form.


(Received February 22 2005)

(Revised March 29 2006)

(Online publication October 20 2006)

Key Words:

  • Stochastic partial differential equations;
  • mild and weak solutions;
  • random noise.

Mathematics Subject Classification:

  • 60H15;
  • 60H25;
  • 35R60
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