Related papers: Superlinear stochastic heat equation on $\mathbb{R…
We indicate that the nonlinear Schr\"odinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods,…
This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence…
We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the…
In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This…
We consider the stochastic heat equation on $[0,\,1]$ with periodic boundary conditions and driven by space-time white noise. Under various natural conditions, we study small ball probabilities for the H\"older semi-norms of the solutions,…
This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the…
We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We…
We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the…
In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic…
In this note, we consider the stochastic heat and Schr\"odinger equation, and show that, at time $t$, the onset of the chaos occurs on the scale of $1/t$, and the Fourier spectrum of the solution is asymptotically Gaussian after centering…
In a recent work [DDRZ20], it has been developed a novel framework aimed at studying at a perturbative level a large class of non-linear, scalar, real, stochastic PDEs and inspired by the algebraic approach to quantum field theory. The main…
We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic…
We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and…
We consider the stochastic heat equation on the integer lattice $\mathbb{Z}^d$ in dimension $d \geq 3$ and with small coupling constant. We show uniqueness of global solutions within the class of positive functions that are stationary in…
In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such…
In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate…
We derive and analyze the ‘SHE' (Spherical Harmonics Expansion) type system of equations coupled in energy. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion tensor. The…
We study a stochastic version of the one-dimensional discrete nonlinear Schr{\"o}dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed…
The Swift-Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger…
There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in…