Parabolic SPDEs Research Articles

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Total Variation Error Bounds for the Accelerated Exponential Euler Scheme Approximation of Parabolic Semilinear SPDEs

Total Variation Error Bounds for the Accelerated Exponential Euler Scheme Approximation of Parabolic Semilinear SPDEs

Instantaneous everywhere-blowup of parabolic SPDEs

We consider the following stochastic heat equation ∂tu(t,x)=12∂x2u(t,x)+b(u(t,x))+σ(u(t,x))W˙(t,x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u(t,x) = \ frac{1}{2} \\partial ^2_x u(t,x) + b(u(t,x)) + \\sigma (u(t,x)) {\\dot{W}}(t,x), \\end{aligned}$$\\end{document}defined for (t,x)∈(0,∞)×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(t,x)\\in (0,\\infty )\ imes {\\mathbb {R}}$$\\end{document}, where W˙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\dot{W}}$$\\end{document} denotes space-time white noise. The function σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition ∫1∞dyb(y)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _1^\\infty \\frac{\ extrm{d}y}{b(y)}<\\infty \\end{aligned}$$\\end{document}implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that P{u(t,x)=∞for allt>0andx∈R}=1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}\\{ u(t,x)=\\infty \\quad \\hbox { for all } t>0 \\hbox { and } x\\in {\\mathbb {R}}\\}=1.$$\\end{document} The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022).

A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a $ \mathcal{C}^2 $ boundary

A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a $ \mathcal{C}^2 $ boundary

Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise

Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise

Dissipation in parabolic SPDEs II: Oscillation and decay of the solution

Dissipation in parabolic SPDEs II: Oscillation and decay of the solution

Deterministic Dynamics and Randomness in PDE

Over the last few years there has been spectacular progress in the study of parabolic SPDE, of nonlinear dispersive and wave equations and of probabilistic methods in PDE. An important direction connecting these three fields is the general question of how randomness affects the behavior of solutions to PDE. Research in recent years has been driven by the study of randomness in nonlinear evolution equations with a focus on the question of how to quantify the transport of such randomness under the nonlinear flow.

Efficient parameter estimation for parabolic SPDEs based on a log-linear model for realized volatilities

We construct estimators for the parameters of a parabolic SPDE with one spatial dimension based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime. The asymptotic variances are shown to be substantially smaller compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as a response of a log-linear model with spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate efficiency gains compared to previous estimation methods numerically and in Monte Carlo simulations.

CLT for approximating ergodic limit of SPDEs via a full discretization

In order to characterize quantitatively the fluctuations between the ergodic limit and the time-averaging estimator, we establish a central limit theorem for a full discretization of the parabolic SPDE, which shows that the normalized time-averaging estimator converges weakly to a normal distribution as the time stepsize tends to 0. A key ingredient in the proof is to extract an appropriate martingale difference series sum from the normalized time-averaging estimator via the Poisson equation, so that convergences of such a sum and the remainder are well balanced.

Absolute continuity of the solution to the stochastic Burgers equation

We prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts.

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove theL2-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degreekare used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.

Elliptic stochastic quantization

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\mathbb{R}^2$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the relation between an $\mathbb{R}^{d + 1}$ dimensional parabolic SPDE and its $\mathbb{R}^d$ dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our $d = 0$ context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the Gaussian case.

Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation

This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the one-dimensional stochastic Allen–Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to $$\frac{1}{2}$$ , like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.

Bayesian estimations for diagonalizable bilinear SPDEs

The main goal of this paper is to study the parameter estimation problem, using the Bayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We take the spectral approach by assuming that one path of the first N Fourier modes of the solution is continuously observed over a finite time interval. First, we show that the model is regular and fits into classical local asymptotic normality framework, and thus the MLE and the Bayesian estimators are weakly consistent, asymptotically normal, efficient, and asymptotically equivalent in the class of loss functions with polynomial growth. Secondly, and mainly, we prove a Bernstein–Von Mises type result, that strengthens the existing results in the literature, and that also allows to investigate the Bayesian type estimators with respect to a larger class of priors and loss functions than that covered by classical asymptotic theory. In particular, we prove strong consistency and asymptotic normality of Bayesian estimators in the class of loss functions of at most exponential growth. Finally, we present some numerical examples that illustrate the obtained theoretical results.

Optimal Berry–Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter θ appearing in parabolic SPDEs of the form duϵ(t,x)=(A0+θA1)uϵ(t,x)dt+ϵdW(t,x),where A0 andA1 are partial differential operators, W is a cylindrical Brownian motion (CBM) and ϵ↓0. We find an optimal Berry–Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein’s method.

SPDE limit of the global fluctuations in rank-based models

We consider systems of diffusion processes (“particles”) interacting through their ranks (also referred to as “rank-based models” in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof, we also derive quantitative propagation of chaos estimates for the particle system.

Ergodic Control for Lévy-Driven Linear Stochastic Equations in Hilbert Spaces

In this paper, controlled linear stochastic evolution equations driven by square integrable Levy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Ito formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied.

Weak Moment of a Class of Stochastic Heat Equation with Martingale-valued Harmonic Function

A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + \sigma(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a L\'{e}vy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise exponential rate for the case of $\mathcal{L}=-(-\Delta)^{\alpha/2},\,\,\alpha\in(1,2]$ generator of an alpha-stable process.

Stochastic evolution equations with Volterra noise

Volterra processes are continuous stochastic processes whose covariance function can be written in the form R(s,t)=∫0s∧tK(s,r)K(t,r)dr, where K is a suitable square integrable kernel. Examples of such processes are the fractional Brownian motion, multifractional Brownian motion or (in the non-Gaussian case) Rosenblatt process. In the first part, stochastic integral with respect to Volterra processes and cylindrical Volterra process in Hilbert spaces are defined and some of their properties are studied. In the second part, these results are applied to linear stochastic equations in Hilbert spaces driven by cylindrical Volterra processes. Measurability, mean-square continuity and paths continuity of their solutions are proved under various sets of conditions. The general results are illustrated by examples of parabolic and hyperbolic SPDEs.

Degenerate parabolic stochastic partial differential equations: Quasilinear case

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized It\^{o} formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.