Related papers: Resonances and computations
This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which…
We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear…
This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a…
Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate…
The formation of extreme localizations in nonlinear dispersive media can be explained and described within the framework of nonlinear evolution equations, such as the nonlinear Schr\"odinger equation (NLS). Within the class of exact NLS…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…
Probabilistic forecasting of multivariate time series is essential for various downstream tasks. Most existing approaches rely on the sequences being uniformly spaced and aligned across all variables. However, real-world multivariate time…
Latent neural stochastic differential equations (SDEs) have recently emerged as a promising approach for learning generative models from stochastic time series data. However, they systematically underestimate the noise level inherent in…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no…
Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing…
In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is…
We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random…
In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.
In this work, we provide a combinatorial formalism for dealing with the cancellations that have appeared recently in the context of dispersive PDEs with random initial data. The main idea is to transform iterated integrals encoded by…
We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM,…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only…
This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution…