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The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a…
Stochastic differential equations (sdes) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes…
Recent years have seen spectacular progress in the mathematical study of hydrodynamic equations. Novel tools from convex integration in particular prove extremely versatile in establishing non-uniqueness results. Motivated by this…
We show the existence of non-unique stationary weak solutions for forced surface quasi-geostrophic (SQG) equation via a convex integration scheme. The scheme is implemented for the sum-difference system of two distinct solutions. Through…
Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few…
In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic…
We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as…
Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…
We study the surface quasi-geostrophic equation driven by a generic additive noise process $W$. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution $z$ associated with $W$…
We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This…
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…
We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy PDEs, fuzzy-stochastic PDEs offer powerful…
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
In this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the…
Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of…