Related papers: L\'{e}vy driven linear and semilinear stochastic p…
In this paper we study the Large Deviation Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space $\mathbb{R}^d$, with arbitrary dimension $d\geq 1$, under random influence which…
This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts the problem reduces to stochastic…
We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional L{\'e}vy process so that it admits a unique local mild solution in spaces of functions…
Semilinear stochastic evolution equations with multiplicative Poisson noise and monotone nonlinear drift are considered. We do not impose coercivity conditions on coefficients. A novel method of proof for establishing existence and…
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.
We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb…
We consider the effect of replacing in stochastic differential equations leading to the dynamical collapse of the statevector, white noise stochastic processes with non white ones. We prove that such a modification can be consistently…
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution…
The numerical analysis of stochastic parabolic partial differential equations of the form $$ du + A(u) = f \,dt + g \, dW, $$ is surveyed, where $A$ is a partial operator and $W$ a Brownian motion. This manuscript unifies much of the theory…
The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained…
For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space.…
We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable L\'evy noise. To this end, the Malliavin regularity of the…
In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in…
We establish the Malliavin differentiability of McKean-Vlasov stochastic differential equations (MV-SDEs) with common noise under the global Lipschitz assumption in the space variable and the measure variable. Our result gives also meaning…
We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and L\'evy noises. The fast variables evolve at much faster time scale than the slow variables, but they are…
In this paper, we aim to develop a new weak formulation that ensures well-posedness for a broad range of stochastic partial differential equations with pseudo-differential operators whose symbols depend only on time and spatial frequencies.…
In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued…
A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously…
In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our…
This work is concerned with the existence of mild solutions to non-linear Fokker-Planck equations with fractional Laplace operator $(-\Delta)^s$ for $s\in\left(\frac12,1\right)$. The uniqueness of Schwartz distributional solutions is also…