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On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Mohammud FoondunDavar Khoshnevisan — 2010

Annales de l'I.H.P. Probabilités et statistiques

Consider a stochastic heat equation =  2+() for a space–time white noise and a constant >0. Under some suitable conditions on the initial function 0 and , we show that the quantities lim sup →∞−1sup ∈ln E(| ()|2) and lim sup →∞−1ln E(sup ∈| ()|2) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/. Our proof works by demonstrating quantitatively that the peaks of...

Initial measures for the stochastic heat equation

Daniel ConusMathew JosephDavar KhoshnevisanShang-Yuan Shiu — 2014

Annales de l'I.H.P. Probabilités et statistiques

We consider a family of nonlinear stochastic heat equations of the form t u = u + σ ( u ) W ˙ , where W ˙ denotes space–time white noise, the generator of a symmetric Lévy process on 𝐑 , and σ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u 0 . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that f = c f ' ' for some c g t ; 0 , we prove that if u 0 is a finite measure of compact support, then the solution is...

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