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In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems…
Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are…
In this paper, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations with ergodic structures. The limit function is represented as the viscosity solution…
In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the…
In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional It\^o (or path-dependent) calculus, the relationship between the systems and related path-dependent…
A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables…
Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE model is more…
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…
In this paper, we address the problem of existence and uniqueness of a global classical solution to a multidimensional stochastic Burgers equation without gradient-type assumptions on the force or the initial condition. The equation is…
In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss…
In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is…
The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
We develop a new generalized coupling approach to the study of stochastic delay equations with H\"older continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak…