Linear Stochastic Heat Equation Research Articles

Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs

Abstract We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${\mathcal O}(h^{2}+\tau )$ for its solution, and conclude from it a rate of almost ${\mathcal O}(h^{2}+\tau )$ resp. ${\mathcal O}(h^{2}+\tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.

Local times for systems of non-linear stochastic heat equations

We consider \(u(t,x)=(u_1(t,x),\cdots ,u_d(t,x))\) the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a d-dimensional space-time white noise. We prove that, when \(d\le 3\), the local time \(L(\xi ,t)\) of \(\{u(t,x)\,,\;t\in [0,T]\}\) exists and \(L(\cdot ,t) \) belongs a.s. to the Sobolev space \( H^{\alpha }({\mathbb {R}}^d)\) for \(\alpha <\frac{4-d}{2}\), and when \(d\ge 4\), the local time does not exist. We also show joint continuity and establish Hölder conditions for the local time of \(\{u(t,x)\,,\;t\in [0,T]\}\). These results are then used to investigate the irregularity of the coordinate functions of \(\{u(t,x)\,,\;t\in [0,T]\}\). Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of \((u(t_1,x)-u(t_0,x),\cdots ,u(t_n,x)-u(t_{n-1},x))\), which is a new result and of independent interest.

A linear stochastic biharmonic heat equation: hitting probabilities

Consider the linear stochastic biharmonic heat equation on a d–dimen- sional torus (d=1,2,3), driven by a space-time white noise and with periodic boundary conditions: 0.1∂∂t+(-Δ)2v(t,x)=σW˙(t,x),(t,x)∈(0,T]×Td,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{\\partial }{\\partial t}+(-\\varDelta )^2\\right) v(t,x)= \\sigma \\dot{W}(t,x),\\ (t,x)\\in (0,T]\ imes {\\mathbb {T}}^d, \\end{aligned}$$\\end{document}v(0,x)=v_0(x). We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for d=2, they include a z(log tfrac{c}{z})^{1/2} term. Consider D independent copies of the random field solution to (0.1). Applying the criteria proved in Hinojosa-Calleja and Sanz-Solé (Stoch PDE Anal Comp 2021. https://doi.org/10.1007/s40072-021-00190-1), we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets.This yields results on the polarity of sets and on the Hausdorff dimension of the path process.

A posteriori error estimation and space-time adaptivity for a linear stochastic PDE with additive noise

Abstract We present a strong residual-based a posteriori error estimate for a finite element-based space-time discretization of the linear stochastic convected heat equation with additive noise. This error estimate is used for an adaptive algorithm that automatically selects deterministic mesh parameters in space and time. For every $n \geq 0$, we find a new time-step $\tau _n$, a new spatial mesh ${\mathcal M}_{n}$ terminating within finitely many iterations and a finite element value approximation $Y^n_h$ on this spatial mesh, which then approximates strongly the solution of the stochastic partial differential equation (SPDE) within a prescribed tolerance.

L2-regularity of solutions to linear backward stochastic heat equations, and a numerical application

In this work, we mainly explore the L2-regularity for the second component of the solutions to linear backward stochastic heat equations, which is crucial to obtain the convergence of the numerical solutions. As an application, we provide the convergence rate for time-discretized Galerkin approximation of these equations.

Averaging Gaussian functionals

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati's Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $\gamma_0$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits Gaussian fluctuation; with some extra mild integrability condition on $\gamma_0$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

Posterior contraction rates for non-parametric state and drift estimation

<p style='text-indent:20px;'>We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman–Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upon scale separation and a slow manifold approximation.

On the distribution and q-variation of the solution to the heat equation with fractional Laplacian

We study the probability distribution of the solution to the linear stochastic heat equation with fractional Laplacian and white noise in time and white or correlated noise in space. As an application, we deduce the behavior of the q-variations of the solution in time and in space.

On the density of the supremum of the solution to the linear stochastic heat equation

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an interval (at a fixed spatial position), and the smoothness of the density of the supremum of the solution over a space-time rectangle that touches the $$t = 0$$ axis. Applying the properties of the divergence operator, we establish a Gaussian-type upper bound on these two densities respectively, which presents a close connection with the Holder-continuity properties of the solution.

The linear stochastic heat equation with Hermite noise

We analyze the solution to the linear stochastic heat equation driven by a multiparameter Hermite process of order [Formula: see text]. This solution is an element of the [Formula: see text]th Wiener chaos. We discuss various properties of the solution, such as the necessary and sufficient condition for its existence, self-similarity, [Formula: see text]-variation and regularity of its sample paths. We will also focus on the probability distribution of the solution, which is non-Gaussian when [Formula: see text].

Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on [formula omitted

We study the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S2. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.

On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian

Consider stochastic heat equations with fractional Laplacian on Rd. The driving noise is generalized Gaussian which is white in time but spatially homogeneous. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the peaks and compute the macroscopic Hausdorff dimensions of the peaks for (i) and (ii). These result imply that both (i) and (ii) exhibit multi-fractal behavior even though only (ii) is intermittent. This is an extension of a result of Khoshnevisan et al. (2017) to a wider class of stochastic heat equations.

Existence and space-time regularity for stochastic heat equations on p.c.f. fractals

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly H\"older continuous in space and time. We calculate the respective H\"older exponents, which extend the well-known results on the H\"older exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.

Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise

In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in (\frac 14, \frac 12]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model.

Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

We consider the linear stochastic heat equation on $$\mathbb {R}^\ell $$ , driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $$H\in (\frac{1}{4}, \frac{1}{2}]$$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.

Sample paths of the solution to the fractional-colored stochastic heat equation

Let [Formula: see text] be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process [Formula: see text] with respect to the time and to the space variable, respectively. In particular, we derive exact uniform moduli of continuity and Chung-type laws of iterated logarithm.

KPZ Reloaded

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Goncalves and Jara and paracontrolled solutions. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation which does not rely on the Cole-Hopf transform.

A Weak Space-Time Formulation for the Linear Stochastic Heat Equation

We apply the well-known Banach-Necas-Babuska inf-sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the the data and on the covariance operator associated to the driving Wiener process, in order to have existence and uniqueness of the solution. We show the relation of the obtained solution to the so-called mild solution and to the variational solution of the same problem. The spatial regularity of the solution is also discussed. Finally, an extension to the case of linear multiplicative noise is presented.

KPZ equation, its renormalization and invariant measures

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because the nonlinearity is marginally defined with respect to the roughness of the forcing noise. However, its Cole-Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer [13] has recently proved that the solution of SHE can actually be derived through the Cole-Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line $\mathbb{R}$ or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole-Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient 1/24. The methods are essentially stochastic analytic: The Wiener-It\^o expansion and a similar method for establishing the Boltzmann-Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on $\mathbb{R}$.