Abstract. We consider nonlinear parabolic SPDEs of the form ∂tu = Lu + σ(u)ϖ, where ϖ denotes space-time white noise, σ: R→R is [globally] Lipschitz continuous, and L is the L2-generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is "weakly intermittent," provided that the symmetrization of L is recurrent and the initial data is sufficiently large.
Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for L in dimension (1+1). When L=κ∂xx for κ>0, these formulas agree with the earlier results of statistical physics (Kardar, 1987; Krug and Spohn, 1991; Lieb and Liniger, 1963), and also probability theory (Bertini and Cancrini, 1994; Carmona and Molchanov, 1994), in the two exactly-solvable cases where u0=δ0 and u0=1.
Keywords. Stochastic partial differential equations, Lévy processes, Liapounov exponents, weak intermittence, the Burkholder-Davis-Gundy inequality.
AMS Classification (2000). Primary. 60H15, 60J55; Secondary. 82B44
Support. The research of DK was supported in part by a grant from the U.S. National Science Foundation.
Pre/E-Prints. This paper is available in
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Mohammud Foondun Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. mohammud@math.utah.edu |
Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |
Last Update: April 18, 2008
©
2008 - Mohammud Foondun and Davar Khoshnevisan