One-dimensional Stochastic Heat Equation Research Articles

A Feynman–Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise

We consider the (unique) mild solution [Formula: see text] of a one-dimensional stochastic heat equation on [Formula: see text] driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of [Formula: see text], denoted by [Formula: see text], and its representation as a Feynman–Kac type closed form. The chaos expansion of [Formula: see text] makes it possible to find its (optimal) Hölder regularity especially in space.

Superlinear stochastic heat equation on ℝ^{𝕕}

In this paper, we study the stochastic heat equation (SHE) on R d \mathbb {R}^d subject to a centered Gaussian noise that is white in time and colored in space. We establish the existence and uniqueness of the random field solution in the presence of locally Lipschitz drift and diffusion coefficients, which can have certain superlinear growth. This is a nontrivial extension of the recent work by Dalang, Khoshnevisan and Zhang [Ann. Probab. 47 (2019), pp. 519–559], where the one-dimensional SHE on [ 0 , 1 ] [0,1] subject to space-time white noise has been studied.

Convergence of densities of spatial averages of stochastic heat equation

In this paper, we consider the one-dimensional stochastic heat equation driven by a space–time white noise. In two different scenarios, (i) initial condition u0=1 and general nonlinear coefficient σ, (ii) initial condition u0=δ0 and σ(x)=x (Parabolic Anderson Model), we establish rates of convergence with respect to the uniform distance between the density of spatial averages of (renormalized) solution and the density of the standard normal distribution. These results are based on a combination of Stein’s method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the Lp-norm of the second Malliavin derivative.

A central limit theorem for the stochastic heat equation

We consider the one-dimensional stochastic heat equation driven by a multiplicative space–time white noise. We show that the spatial integral of the solution from −R to R converges in total variance distance to a standard normal distribution as R tends to infinity, after renormalization. We also show a functional version of this central limit theorem.

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space–time white noise.

Averaging principle for the heat equation driven by a general stochastic measure

We study the one-dimensional stochastic heat equation in the mild form driven by a general stochastic measure μ, for μ we assume only σ-additivity in probability. The time averaging of the equation is considered, uniform a. s. convergence to the solution of the averaged equation is obtained.

A fully discrete approximation of the one-dimensional stochastic heat equation

Abstract A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in $L^q(\varOmega )$, for all $q\geqslant 2$, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

One-dimensional stochastic heat equation with discontinuous conductance

ABSTRACTWe introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.

Large deviations for stochastic heat equation with rough dependence in space

In this paper, we establish a large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable.

Intermittency for stochastic partial differential equations driven by strongly inhomogeneous space–time white noises

In this paper, the main topic is to investigate the intermittent property of the one-dimensional stochastic heat equation driven by an inhomogeneous Brownian sheet, which is a noise deduced from the study of the catalytic super-Brownian motion. Under some proper conditions on the catalytic measure of the inhomogeneous Brownian sheet, we show that the solution is weakly full intermittent based on the estimates of moments of the solution. In particular, it is proved that the second moment of the solution grows at the exponential rate. The novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the Lebesgue measure on R.

A Central Limit Theorem for Stochastic Heat Equations in Random Environment

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in $L^1$ sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. It concentrates only on the space of constant functions.

The Continuum Directed Random Polymer

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Holder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).

Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens in [17]. In particular, we deal with the one-dimensional stochastic heat equation in [0, 1] driven by the space-time white noise, and the stochastic heat and wave equations in R d ( d ≥ 1 and d ≤ 3 , respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.