Related papers: Stochastic scalar conservation laws driven by roug…
We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which…
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on…
In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We show that under certain assumptions, a shock wave connecting two constant states emerges in…
This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under nonholonomic constraints. For this…
We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem adding a viscosity term and we provide $L^\infty$…
We consider a stochastic wave equation in spatial dimension three, driven by a Gaussian noise, white in time and with a stationary spatial covariance. The free terms are nonlinear with Lipschitz continuous coefficients. Under suitable…
We investigate the numerical approximation of the stochastic Allen--Cahn equation with multiplicative noise on a periodic domain. The considered scheme uses a recently proposed augmented variant of scalar auxiliary variable method for the…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…
Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance.…
We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic…
The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with…
In this paper we show the strong convergence of a fully explicit space-time discrete approximation scheme for the solution process of the two-dimensional incompressible stochastic Navier-Stokes equations on the torus driven by additive…
The averaging principle for slow-fast systems of various kind of stochastic (partial) differential equations has been extensively studied. An analogous result was shown for slow-fast systems of rough differential equations driven by random…
We indicate that the nonlinear Schr\"odinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods,…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic…
In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential…
Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood…