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In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of…
The construction of stochastic solutions is a powerful method to obtain localized solutions in configuration or Fourier space and for parallel computation with domain decomposition. Here a stochastic solution is obtained for the…
We study one-dimensional linear hyperbolic systems with $L^{\infty}$-coefficients subjected to periodic conditions in time and reflection boundary conditions in space. We derive a priori estimates and give an operator representation of…
We deal with parameter estimation for a linear parabolic second-order stochastic partial differential equation in two space dimensions driven by two types of $Q$-Wiener processes based on high frequency data with respect to time and space.…
We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively…
In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = \Delta^{\alpha/2} u + b \cdot \nabla u + f, $$ where $\Delta^{\alpha/2}$ denotes the fractional Laplacian, $b$ is a…
The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a…
We study a class of Tricomi-type partial differential equations previously investigated in [28]. Firstly, we generalize the representation formula for the solution obtained there by allowing the coefficient in front of the second-order…
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone…
In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general…
In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the…
We provide the details of an implementation of Fourier techniques for solving second-order linear partial differential equations (with constant coefficients) using a computer algebra system. The general Sturm-Liouville problem for the heat,…
A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to investigate a stochastic porous medium…
This paper addresses the direct and inverse source problems for the stochastic acoustic, biharmonic, electromagnetic, and elastic wave equations in a unified framework. The driven source is assumed to be a centered generalized microlocally…
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting…
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
This paper is devoted to studying stochastic parabolic evolution equations with additive noise in Banach spaces of M-type 2. We construct both strict and mild solutions possessing very strong regularities. First, we consider the linear…
This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and…
It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…