Parabolic Stochastic Equations Research Articles

Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

Abstract In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Abstract In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. We then apply these Carleman estimates to investigate two inverse problems for the space semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the H"{o}lder stability for the discrete Cauchy problem.

Null controllability for stochastic coupled systems of fourth order parabolic equations

This paper aims to establish null controllability for systems coupled by two backward fourth order stochastic parabolic equations. The main goal is to control both equations with only one control act on the drift term. To achieve this, we develop a new global Carleman estimate for fourth order stochastic parabolic equations, which allows us to deduce a suitable observability inequality for the adjoint systems.

Smooth center-stable/unstable manifolds and foliations of stochastic evolution equations with non-dense domain

The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples, a stochastic age-structured model and a stochastic parabolic equation, to illustrate our results.

Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *

Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.

Feynman–Kac formula for parabolic Anderson model in Gaussian potential and fractional white noise

In this paper, we establish a Feynman–Kac formula for the stochastic parabolic Anderson model with Gaussian potential in space and fractional white noise in time with Hurst parameter H > 1/2. We obtain the necesscary and suffcient condition for the integrability of the Gaussian potential and the exponential integrability of the solution which is defined by Feynman–Kac formula. By the smoothing of the fractional white noise and techniques from Malliavin calculus, we prove that the Feynman–Kac representation is a mild solution of the stochastic parabolic Anderson equation.

Observability Inequality from Measurable Sets and the Shape Design Problem for Stochastic Parabolic Equations

Observability Inequality from Measurable Sets and the Shape Design Problem for Stochastic Parabolic Equations

Inverse problems for stochastic partial differential equations: Some progresses and open problems

The aim of this paper is to survey some recent works on inverse problems for stochastic partial differential equations (SPDEs for short), which were solved by means of Carleman estimate. For simplicity, we focus on two typical SPDEs, i.e, stochastic hyperbolic equations and stochastic parabolic equations. Our particular concerns are those inverse problems which are genuinely stochastic and therefore cannot be reduced to any deterministic ones. Also, some open problems are presented.

Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

Regularity of Wong-Zakai approximations for a class of stochastic degenerate parabolic equations with multiplicative noise

Regularity of Wong-Zakai approximations for a class of stochastic degenerate parabolic equations with multiplicative noise

Dynamically orthogonal narrow-angle parabolic equations for stochastic underwater sound propagation. Part I: Theory and schemes.

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle parabolic equation and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.

Fractional stochastic parabolic equations with fractional noise

Fractional stochastic parabolic equations with fractional noise

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Unique continuation for a fourth-order stochastic parabolic equation

In this paper, we prove a boundary unique continuation property for a fourth-order stochastic parabolic equation evolving in a domain G⊂Rn. Our result shows that the value of the solution can be determined by the observation on an arbitrary open subset of the boundary. The quantitative version of this property can be derived by the global Carleman estimate, which is deduced from a weighted identity for a fourth-order stochastic parabolic operator. The results in this paper are also new even if the fourth-order stochastic parabolic equation reduces to the corresponding fourth-order deterministic parabolic equation.

Stability of Stochastic Partial Differential Equations

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.

Optimal control for a nonlinear stochastic PDE model of cancer growth

ABSTRACT We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control.

Null Controllability for Stochastic Parabolic Equations with Dynamic Boundary Conditions

Null Controllability for Stochastic Parabolic Equations with Dynamic Boundary Conditions

Dynamics of Non-Autonomous Stochastic Semi-Linear Degenerate Parabolic Equations with Nonlinear Noise

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to the equation with nonlinear noise, we first establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong–Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong–Zakai approximate system of the equation under consideration driven by a linear multiplicative noise.

A numerical solution for a quasi solution of the time-fractional stochastic backward parabolic equation

In the present study, we consider a time-fractional stochastic backward parabolic equation driven by standard Brownian motion. In this problem, the fractional derivative is considered in the Caputo sense. Using the minimization of a least-squares functional, stochastic variational formulation, Fréchet differentiability and utility theorems adopted directly from deterministic fractional backward equations, the existence and uniqueness theorems for a quasi solution of the proposed problem are proved. To approximate the quasi solution, a numerical technique based on 2D Chebyshev wavelets is applied. We employ the Levenberg–Marquardt regularization technique since the derived equivalent system of linear equations is ill-posed. Also, the convergence analysis for this numerical algorithm is investigated. Our results provide a new insight to find quasi solutions and apply adapted deterministic methods for some fractional stochastic backward equations. Moreover, a numerical example is provided to indicate the accuracy and efficiency of the Chebyshev wavelet method in solving the mentioned problem.

A Reliable Computational Scheme for Stochastic Reaction–Diffusion Nonlinear Chemical Model

The main aim of this contribution is to construct a numerical scheme for solving stochastic time-dependent partial differential equations (PDEs). This has the advantage of solving problems with positive solutions. The scheme provides conditions for obtaining positive solutions, which the existing Euler–Maruyama method cannot do. In addition, it is more accurate than the existing stochastic non-standard finite difference (NSFD) method. Theoretically, the suggested scheme is more accurate than the current NSFD method, and its stability and consistency analysis are also shown. The scheme is applied to the linear scalar stochastic time-dependent parabolic equation and the nonlinear auto-catalytic Brusselator model. The deficiency of the NSFD in terms of accuracy is also shown by providing different graphs. Many observable occurrences in the physical world can be traced back to certain chemical concentrations. Examining and understanding the inter-diffusion between chemical concentrations is important, especially when they coincide. The Brusselator model is the gold standard for describing the relationship between chemical concentrations and other variables in chemical systems. A computational code for the proposed model scheme may be made available to readers upon request for convenience.