2007/10/29 by Marta Sanz-Solé, Sanz-Solé, Marta, Pierre-A. Vuillermot +1
Mathematics · #35K55 #35R60 #60G15 #60G17 #60G18 #60H15 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:35K55 #msc:35R60 #msc:60G15 #msc:60G17 #msc:60G18 #msc:60H15
paper · pdf · doi:10.48550/arxiv.0710.5485
37 pages
arxiv created 2009/02/19 · arxiv updated 2009/12/01
In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset D⊂ℝd and driven by an infinite-dimensional fractional noise. The noise is derived from an L2(D)-valued fractional Wiener process WH whose covariance operator satisfies appropriate restrictions; moreover, the Hurst parameter H is subjected to constraints formulated in terms of d and the Hölder exponent of the derivative h^′ of the noise nonlinearity in the equations. We prove the existence of such solution, establish its relation with the variational solution introduced in \citenuavu and also prove the Hölder continuity of its sample paths when we consider it as an L2(D)--valued stochastic processes. When h is an affine function, we also prove uniqueness. The proofs are based on a relation between the notions of mild and variational solution established in Sanz-Solé and Vuillermot 2003, and adapted to our problem, and on a fine analysis of the singularities of Green's function associated with the class of parabolic problems we investigate. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness.