Non-linear rough heat equations
Deya, Aurélien; Gubinelli, Massimiliano; Tindel, Samy (2012), Non-linear rough heat equations, Probability Theory and Related Fields, 153, 1-2, p. 97-147. http://dx.doi.org/10.1007/s00440-011-0341-z
TypeArticle accepté pour publication ou publié
Journal nameProbability Theory and Related Fields
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Abstract (EN)This article is devoted to define and solve an evolution equation of the form dy t = Δy t dt + dX t (y t ), where Δ stands for the Laplace operator on a space of the form Lp(\mathbb Rn)Lp(Rn), and X is a finite dimensional noisy nonlinearity whose typical form is given by Xt(j)=åi=1N xit fi(j)Xt()=Ni=1xitfi() , where each x = (x (1), … , x (N)) is a γ-Hölder function generating a rough path and each f i is a smooth enough function defined on Lp(\mathbb Rn)Lp(Rn). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.
Subjects / KeywordsRough paths theory; Stochastic PDEs; Fractional Brownian motion
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