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.
We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on p-th moments of the solution (p 1), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal .
We study the stochastic heat equation ∂ t u = Lu + σ(u)Ẇ in (1 + 1) dimensions, whereẆ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a symmetric Lévy process that has finite exponential moments, and u 0 has exponential decay at ±∞. We prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting. [See, however,[19,20] for the analysis of the location of the peaks in a different model.]Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.
Consider the stochastic heat equation ∂tu = ( κ /2)∆u + σ(u)Ḟ , where the solution u := ut(x) is indexed by (t , x) ∈ (0 , ∞) × R d , andḞ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-x fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is f (x) ∝ x −α , then the "fluctuation exponents" of the solution are ψ for the spatial variable and 2ψ − 1 for the time variable, where ψ := 2/(4 − α). Moreover, these exponent relations hold as long as α ∈ (0 , d ∧ 2); that is precisely when Dalang's theory [12] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [22,23].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.