Stochastic Heat Equation Research Articles

Wellposedness and stability of fractional stochastic nonlinear heat equation in Hilbert space

In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space $$ L^{2}(0,1) $$ ; perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the $$ p^{th} $$ -moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.

Convergences of the rescaled Whittaker stochastic differential equations and independent sums

We study some SDEs derived from the q→1 limit of a 2D surface growth model called the q-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that “come down from infinity”: After rescaling and re-centering, convergences to the time-inverted stationary additive stochastic heat equation (SHE) hold. The point of view in this paper is a novel probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations, both in diverging integrated forms, explain the convergence of the re-centered covariance functions. The proof of the process-level convergence identifies additional divergent terms in the dynamics and considers nontrivial cancellations.

Limiting dynamics of stochastic heat equations with memory on thin domains

This paper is concerned with the limiting behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an -dimensional narrow domain. We also establish the upper-semicontinuity of these attractors when a family of -dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Transportation Inequalities for Stochastic Heat Equation with Rough Dependence in Space

Transportation Inequalities for Stochastic Heat Equation with Rough Dependence in Space

Weak intermittency of stochastic heat equation under discretizations

Intermittency arises typically in random fields of multiplicative type like the solution of stochastic heat equation. This paper investigates whether the stochastic heat equation with multiplicative noise under a discretization could inherit the dynamical behavior in particular the weak intermittency of the original equation. The exact solution of the stochastic heat equation with multiplicative noise is proved to admit weak intermittency with a specific index of Lyapunov exponents. We prove the existence of the weak intermittency for stochastic heat equation under a class of discretizations, and further the preservation of the index of Lyapunov exponents of the exact solution by a renewal approach, provided in addition that the initial datum is a positive constant and the spatial partition number is large.

Small ball probability estimates for the Hölder semi-norm of the stochastic heat equation

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 Hölder semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these Hölder semi-norms.

Stochastic heat equation: Numerical positivity and almost surely exponential stability

In this paper, the numerical positivity and almost surely exponential stability of the stochastic heat equation are discussed. The finite difference method and the split-step backward Euler are considered for spatial and temporal, respectively. Motivated from physical applications such as temperature, finance and so on, positivity has real significance, which is volatilized by some common numerical treatments. To this end the numerical positivity is obtained by the properties of M-matrix and the truncated random variables, which overcomes the unboundedness of the random variables. For the investigation of the almost surely exponential stability, a stochastic stability matrix is introduced, and then the stability analysis reduces to the estimation of the eigenvalues and martingales thorough a family of matrices and perturbation theorems. The stabilization ability of the multiplicative noise is verified again from a generalization of the stochastic heat equation. Finally, some numerical experiments are given to validate our numerical results.

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.

Law of large numbers and fluctuations in the sub-critical and [formula omitted] regions for SHE and KPZ equation in dimension [formula omitted

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) equation in dimension d≥3 as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in Mukherjee et al. (2016) to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards–Wilkinson regime of the SHE and KPZ equation studied in Gu et al. (2018), Magnen and Unterberger (2018), Dunlap et al. (2020) to the full L2-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments {ω(t)−ω(s)}s<t of d-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form H(t−s,ω(t)−ω(s)) that are invariant under uniform translations of paths. In such interactions, we allow long-range dependence in the time variable (including power law decay up to t↦(1+t)−(2+ε) for ε>0) and unbounded (singular) interactions (including singularities of the form x↦1/|x|p in d≥3 or x↦δ0(x) in d=1) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Fröhlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance, which is strictly positive. As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions d≥3. When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.

Parameter-dependent filtering of Gaussian processes in Hilbert spaces

The filtering problem for non-Markovian Gaussian processes on rigged Hilbert spaces is considered. Continuous dependence of the filter and observation error on parameters which may be present both in the signal and observation processes is proved. The general results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter.

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 $$L^p(\Omega )$$ -solution exists for all $$p\ge 2$$ . In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al. (Inst Henri Poincaré Probab Stat. To appear, 2021). In the case when there exists only a local solution, we determine the precise deterministic time, $$T_2$$ , before which a unique $$L^2(\Omega )$$ -solution exists, but after which the series corresponding to the $$L^2(\Omega )$$ 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.

The cutoff phenomenon for the stochastic heat and wave equation subject to small Lévy noise

This article generalizes the small noise cutoff phenomenon obtained recently by Barrera, Högele and Pardo (JSP2021) to the mild solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. The methods rely on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the multiplicative stochastic solution flows in terms of stochastic exponentials.

A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

Accelerated exponential Euler scheme for stochastic heat equation: convergence rate of the density

AbstractThis paper studies the numerical approximation of the density of the stochastic heat equation driven by space-time white noise via the accelerated exponential Euler scheme. The existence and smoothness of the density of the numerical solution are proved by means of Malliavin calculus. Based on a priori estimates of the numerical solution we present a test-function-independent weak convergence analysis, which is crucial to show the convergence of the density. The convergence order of the density in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear drift case and nearly $1$ in the affine drift case. As far as we know, this is the first result on the existence and convergence of density of the numerical solution to the stochastic partial differential equation.

Mixed stochastic heat equation with fractional Laplacian and gradient perturbation

We introduce a new stochastic heat equation with a mixed operator, which is a combination of the standard Laplacian, a fractional Laplacian and the gradient operator, driven by an additive Gaussian noise which is white in time and in space. We establish the existence of the solution and we study its behavior with respect to the time variable. In particular, we establish sharp two-sided interesting estimates of its variance increments, which allow us to get some regularity characteristics of the solution sample paths, among many other possible applications.

Optimal Hölder continuity and hitting probabilities for SPDEs with rough fractional noises

We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index 12 and spatial index H≤1/2. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. Based on this sharp regularity, we obtain lower and upper bounds of hitting probabilities of the solutions in terms of Bessel–Riesz capacity and Hausdorff measure, respectively.