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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)

MSC:
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Carmona, R.; Molchanov, S.A., ()
[2] Carmona, R.; Simon, B., Pointwise bounds on eigenfunctions and wavepackets in N-body quantum systems, Comm. math. phys., (1981) · Zbl 0464.35085
[3] Dawson, D.A.; Salehi, H., Spatially homogeneous random evolutions, J. multivariate analysis, 10, 141-180, (1980) · Zbl 0439.60051
[4] Flandoli, F.; Schaumlöffel, K.-U., Stochastic parabolic equations in bounded domain. random evolution operator and Lyapunov exponents, Stochastics and stochastic reports, 29, 461-485, (1990) · Zbl 0704.60060
[5] Kunita, H., ()
[6] Noble, J.M., Evolution equation with Lévy potential, Integral and differential equations, 9, 2, (1996) · Zbl 0861.35149
[7] Noble, J.M., Directed polymer in random medium, Stochastic analysis and applications, 15, 4, (1997)
[8] Reed, M.; Simon, B., ()
[9] Reed, M.; Simon, B., ()
[10] Revuz, D.; Yor, M., ()
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