SPDEs with coloured noise: Analytic and stochastic approaches

Issue		ESAIM: PS Volume 10, 2006


Page(s)		380 - 405
DOI		10.1051/ps:2006016
Published online		20 October 2006

ESAIM: P&S, October 2006, Vol. 10, pp. 380-405
DOI: 10.1051/ps:2006016

SPDEs with coloured noise: Analytic and stochastic approaches

Marco Ferrante¹ and Marta Sanz-Solé²

¹ Dipartimento di Matematica, Università di Padova, Via Belzoni 7, 35131 Padova, Italy; ferrante@math.unipd.it
² Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; marta.sanz@ub.edu

(Received February 22, 2005. Revised March 29, 2006. Published online 20 October 2006.)

Abstract
We study strictly parabolic stochastic partial differential equations on $\mathbb{R} ^d$ , $d\ge 1$ , 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.

Mathematics Subject Classification. 60H15, 60H25, 35R60.

Key words: Stochastic partial differential equations, mild and weak solutions, random noise.

© EDP Sciences, SMAI 2006

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