, Volume 18, Issue 6, pp 1113-1145
Date: 03 Aug 2012

Linear SPDEs Driven by Stationary Random Distributions

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Abstract

Using tools from the theory of stationary random distributions developed in Itô (Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math., 28:209–223, 1954) and Yaglom (Theory Probab. Appl., 2:273–320, 1957), we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (Electron. J. Probab. 4(6) 1999), and the fractional noise considered in Balan and Tudor (Stoch. Process. Appl., 120:2468–2494, 2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with this type of noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (Stoch. Process. Appl., 120:2468–2494, 2010) to the case H<1/2.

Communicated by Christian Houdré.
Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.