Related papers: Stochastic solutions and singular partial differen…
In this paper we establish a substitution formula for stochastic differential equation driven by generalized grey noise. We then apply this formula to investigate the absolute continuity of the solution with respect to the Lebesgue measure…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear…
We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by L\'evy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is…
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the…
The expressions of solutions for general $n\times m$ matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential…
The stochastic parabolic equations with random potentials, driving forces and initial conditions are considered. The Wick product is used to give sense to the product of two generalized stochastic processes, and the existence and uniqueness…
In differential equation discovery algorithms, numerical differentiation is usually a fixed preliminary step. Current methods improve robustness with data subsampling and sparsity but often ignore the variability from the differentiation…
In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a space-time white noise in dimension one. The equation we consider may also include…
A numerical approach for the approximation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a…
The past decades have seen increasing interest in modelling uncertainty by heterogeneous methods, combining probability and interval analysis, especially for assessing parameter uncertainty in engineering models. A unifying mathematical…
We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…
We introduce a concept of space-time holomorphic solutions of partial differential equations and construct a meromorphic solution of Navier-Stokes equations.
We study various solution behaviors of scale equations which are recently proposed in \cite{Kim}. On the contrary to conventional mathematical tools, scale equations are capable to accommodate various behaviors at different scale levels…
A parameter estimation problem is considered for a stochastic parabolic equation with multiplicative noise under the assumption that the equation can be reduced to an infinite system of uncoupled diffusion processes. From the point of view…
A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
Stochastic methods are ubiquitous to a variety of fields, ranging from Physics to Economy and Mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in…
The numerical evaluation of statistics plays a crucial role in statistical physics and its applied fields. It is possible to evaluate the statistics for a stochastic differential equation with Gaussian white noise via the corresponding…