We consider nonlinear parabolic SPDEs of the form ∂tu = Lu+σ(u)ẇ, whereẇ denotes space-time white noise, σ : R → R is [globally] Lipschitz continuous, and L is the L 2 -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 [25,29,30], and also probability theory [1,5] in the two exactly-solvable cases where u0 = δ0 and u0 ≡ 1.
We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y), u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t, x)} t∈R + ,x∈ [0,1] , in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes t → u(t, x) and x → u(t, x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.
We consider a nonlinear stochastic heat equation ∂tu = 1 2 ∂xxu + σ(u)∂xtW , where ∂xtW denotes space-time white noise and σ : R → R is Lipschitz continuous. We establish that, at every fixed time t > 0, the global behavior of the solution depends in a critical manner on the structure of the initial function u0: under suitable conditions on u0 and σ, sup x∈R ut(x) is a.s. finite when u0 has compact support, whereas with probability one, lim sup |x|→∞ ut(x)/(log|x|) 1/6 > 0 when u0 is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.
The primary goal of this paper is to study the range of the random field X(t) = N j =1 X j (t j ), where X 1 , . . . , X N are independent Lévy processes in R d .To cite a typical result of this paper, let us suppose that i denotes the Lévy exponent of X i for each i = 1, . . . , N. Then, under certain mild conditions, we show that a necessary and sufficient condition for X(R N + ) to have positive d-dimensional Lebesgue measure is the integrability of the function. This extends a celebrated result of Kesten and of Bretagnolle in the one-parameter setting. Furthermore, we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion. This extends a theorem of Hawkes to the present random fields setting and completes the analysis of local times for additive Lévy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.
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