Intermittence and nonlinear parabolic stochastic partial differential equations

By Mohammud Foondun, Davar Khoshnevisan. 2008

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We consider nonlinear parabolic SPDEs of the form $\partial_t u={\cal L} u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:R\to R$ is [globally] Lipschitz continuous, and $\cal L$ is the $L^2$-generator of a L'evy 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 $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\cal L$ 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 $\cal L$ in dimension $(1+1)$. When ${\cal L}=\kappa\partial_{xx}$ for $\kappa>0$, 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 (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is when $u_0=\delta_0$ or $u_0\equiv 1$; in those cases the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d81453854ac912b69b4d525a6d00f8a012a23a51

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Citations in the corpus

Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions (2013, background): “4, to allow all adapted, jointly measurable and integrable random fields (see also [19], Lemma 3.”
Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise (2019, background): “We now follow Foondun and Khoshnevisan [13] and consider a two-parameter family {Nβ,k}β>0,k≥1 of norms—each defined on the space of space–time random fields—as follows: For all real numbers β > 0 and k ≥ 1, and for every space–time random field := { (t, x); t ≥ 0, x ∈ [0,1]}, (3.6) Nβ,k( ) := sup t≥”
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency (2014, background): “g [12, 13, 14, 18, 31]), and as a localization property for the polymer measure [7].”
Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions (2013, methodology): “The use of non-smooth initial data is initially motivated by the parabolic Anderson model (in which ρ(u) = u) with initial condition given by the Dirac delta function μ = δ0 (see [2], and more recently, [6,7,13]).”
On the existence and position of the farthest peaks of a family of stochastic heat and wave equations (2010, background): “Foondun and Khoshnevisan [ 17 ] have shown that:”