Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition

Raluca M. Balan () and Le Chen ()
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Raluca M. Balan: University of Ottawa
Le Chen: University of Kansas

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 2216-2265

Abstract: Abstract In this article, we study the parabolic Anderson model driven by a space-time homogeneous Gaussian noise on $$\mathbb {R}_{+} \times \mathbb {R}^d$$ R + × R d , whose covariance kernels in space and time are locally integrable nonnegative functions which are nonnegative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on $$\mathbb {R}^d$$ R d , and the spectral measure of the noise satisfies Dalang’s (Electron J Probab 4(6):29, 1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the p-th moments of the solution, for any $$p \ge 2$$ p ≥ 2 . In addition, we show that this solution has a Hölder continuous modification with the same regularity and under the same condition as in the case of the white noise in time, regardless of the temporal covariance function of the noise.

Keywords: Stochastic partial differential equations; Rough initial conditions; Parabolic Anderson model; Malliavin calculus; Wiener chaos expansion; Primary 60H15; Secondary 60H07; 37H15 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-017-0772-2

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