arXiv subjects

Nicholas Eisenberg

Publications and source records attributed to Nicholas Eisenberg.

2 recordsLinked to original sources

arXiv2022

Invariant measures for the nonlinear stochastic heat equation with no drift term

This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}Δu = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function $ρ$, which will together guarantee the existence of an invariant measure in the weighted space $L^2_ρ(\mathbb{R}^d)$. In particular, our result includes the parabolic Anderson model (i.e., the case when $b(u) = λu$) starting from the Dirac delta measure.

math.PR↗

arXiv2021

Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $L^p(Ω)$-solution exits for all $p\ge 2$. In this case, we derive exact moment asymptotics following the same strategy in a recent work by Balan et al [1]. In the case when there exits only a local solution, we determine the precise deterministic time, $T_2$, before which a unique $L^2(Ω)$-solution exits, but after which the series corresponding to the $L^2(Ω)$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.

math.PR↗