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SPDEs with coloured noise: Analytic and stochastic approaches

Published online by Cambridge University Press:  20 October 2006

Marco Ferrante  and
Marta Sanz-Solé
Marco Ferrante
Affiliation:
Dipartimento di Matematica, Università di Padova, Via Belzoni 7, 35131 Padova, Italy; ferrante@math.unipd.it
Marta Sanz-Solé
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; marta.sanz@ub.edu
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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 thecoefficients of the differential operator are random, we give sufficient conditions on thecorrelation of the noise ensuring Hölder continuity for the trajectories of thesolution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weakformulation of the equation are related, deriving path properties of the solution to aparabolic Cauchy problem in evolution form.


Keywords

Stochastic partial differential equationsmild and weak solutionsrandom noise.

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 380 - 405
Copyright
© EDP Sciences, SMAI, 2006

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