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Correlated with the trend of increasing degrees of freedom in robotic systems is a similar trend of rising interest in Spatio-Temporal systems described by Partial Differential Equations (PDEs) among the robotics and control communities.…
We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and…
We study parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a $C_0-$semigroup and one linear bounded operator with Wick-type multiplication, all of them set…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
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…
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…
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by…
The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space.…
We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our…
This paper investigates the solvability and optimal control of a class of impulsive stochastic differential equations (SDEs) within a Hilbert space setting. First, we establish the existence and uniqueness of mild solutions for the proposed…
We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A…
This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a…
It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the…
There is a rising interest in Spatio-temporal systems described by Partial Differential Equations (PDEs) among the control community. Not only are these systems challenging to control, but the sizing and placement of their actuation is an…
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully…
We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth…
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of…
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give…
In this article, we construct weak solutions for a class of Stochastic PDEs in the space of tempered distributions via Girsanov's theorem. It is to be noted that our drift and diffusion coefficients $(L,A)$ of the considered Stochastic PDE…
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…