Abstract
This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.
Highlights
Gaussian noise and uWis interpreted both in the sense of Skorohod and Stratonovich
The recent paper [3] tackles the problem for a fractional noise in time, with some special examples of spatial covariance structures, within the landmark of Skorohod equations. In this case the results are confined to weak intermittency, with an upper bound on Lk moments obtained invoking hypercontractivity arguments and lower bounds computed only for the L2 norm
We show that the two notions coincide and some existence-uniqueness results which are the first link between pathwise and Malliavin calculus solutions to equation (1.1)
Summary
In this paper we are interested in the stochastic heat equation in Rd driven by a general multiplicative centered Gaussian noise. The recent paper [3] tackles the problem for a fractional noise in time, with some special (though important) examples of spatial covariance structures, within the landmark of Skorohod equations In this case the results are confined to weak intermittency, with an upper bound on Lk moments obtained invoking hypercontractivity arguments and lower bounds computed only for the L2 norm. With all those preliminary considerations in mind, the current paper proposes to study existence-uniqueness results, Feynman-Kac representations, chaos expansions and intermittency results for a very wide class of Gaussian noises W (including in particular those considered in [3, 16]), for both Skorohod and Stratonovich type equations (1.1).
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