Related papers: A Solution Theory for a General Class of SPDEs
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
The solutions of parabolic and hyperbolic stochastic partial differential equations (SPDEs) driven by an infinite dimensional Brownian motion, which is a martingale, are in general not semi-martingales any more and therefore do not satisfy…
This paper investigates a Stochastic Partial Differential Equation (SPDE) derived from the Fokker-Planck equation associated with Score-based Generative Models. We modify the standard Fokker-Planck equation to better represent practical…
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential…
In this paper we investigate a nonlinear stochastic partial differential equation (spde in short) perturbed by a space-correlated Gaussian noise in arbitrary dimension $d\geq1$, with a non-Lipschitz coefficient noisy term. The equation…
Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we…
In this paper, we are interested in the analytical study of a nonlinear Stochastic Partial Differential Equation (SPDE) arising as a model of phytoplankton aggregation. This SPDE consists in a diffusion equation with a chemotaxis term…
Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial…
In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dot{\xi}$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric…
In this article, we introduce a time-independent version of the L\'evy colored noise considered in Balan (2015) and Balan and Jim\'enez (2026). We study the existence of the solution of a linear stochastic partial differential equation with…
We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for…
In a recent work [DDRZ20], it has been developed a novel framework aimed at studying at a perturbative level a large class of non-linear, scalar, real, stochastic PDEs and inspired by the algebraic approach to quantum field theory. The main…
We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
This paper is devoted to studying abstract stochastic semilinear evolution equations with additive noise in Hilbert spaces. First, we prove the existence of unique local mild solutions and show their regularity. Second, we show the regular…
A general approach to provide approximate parameterizations of the "small" scales by the "large" ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of…
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
This paper is concerned with the problem of regularization by noise of systems of reaction-diffusion equations with mass control. It is known that $\textit{strong}$ solutions to such systems of PDEs may blow-up in finite time. Moreover, for…
In Rajeev (2013), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one to one correspondence between solutions of a class of finite dimensional SDEs and solutions of a class of SPDEs in…