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Evolution equation with Gaussian potential. (English) Zbl 0861.35149
From the author’s summary: “We study the stochastic partial differential equation \[ du_t=\textstyle{{1\over 2}} \Delta u_tdt+u_td\zeta_t,\quad u_0=1, \] where \(\zeta:\mathbb{R}_+\times\mathbb{R}^d\times\Omega\to\mathbb{R}\) is a space-time Gaussian random field, white noise in time and with certain assumptions on the spatial covariance structure. We show existence and uniqueness of solutions under certain hypotheses on the space-covariance. We then prove regularity results for the solutions \(u\). We show that solutions exist with the usual Dirac-delta space-correlation for one space dimension and show that a solution may be constructed with delta space-correlation of Peierl’s type for \(d=2\). In the last section, we examine the moments of the solution and show that the Zeldovich-Molchanov intermittent property is satisfied”.
Reviewer: N.Jacob (Erlangen)

35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
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