Related papers: Averaging along irregular curves and regularisatio…

We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed as…

We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a…

We consider the stationary Boltzmann equation with the angular cutoff cross section in a bounded convex domain under the incoming boundary condition. In this article, we discuss the fractional Sobolev regularity of the solution without…

One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating…

Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary H\"older-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the…

Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral…

This paper establishes the averaging method to a coupled system consisting of two stochastic differential equations which has a slow component driven by fractional Brownian motion (FBM) with less regularity $1/3< H \leq 1/2$ and a fast…

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^{\alpha}\partial_{x}u+u\partial_{x}u=0, &…

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the…

This paper investigates a non-autonomous slow-fast system, which is generalized by stochastic differential equations (SDEs) with locally Lipschitz coefficients, subjected to standard Brownian motion (Bm) and fractional Brownian motion (fBm)…

We show that generic H\"older continuous functions are $\rho$-irregular. The property of $\rho$-irregularity has been first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and plays a key role in the study of…

We prove averaging theorems for ordinary differential equations and retarded functional differential equations. Our assumptions are weaker than those required in the results of the existing literature. Usually, we require that the…

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains…

Training Neural Ordinary Differential Equations (ODEs) is often computationally expensive. Indeed, computing the forward pass of such models involves solving an ODE which can become arbitrarily complex during training. Recent works have…

This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form $dX(t)=AX(t)dt+F(X(t))dt+dW(t)$, driven by space-time white noise. In previous…

A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of…

Classical neural ordinary differential equations (ODEs) are powerful tools for approximating the log-density functions in high-dimensional spaces along trajectories, where neural networks parameterize the velocity fields. This paper…

Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \] where…

We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial…

This article investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic reaction-diffusion-advection equations (SRDAEs) with spatially homogeneous colored noises and infinitesimal generators of subordinate…