An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov

In this work, we discuss some asymptotic behavior of the stochastic two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model of order one, for the non-Newtonian fluid flows. We prove a central limit theorem and establish the moderate deviation principle for such models using a weak convergence method.

We derive non-asymptotic quantitative bounds for convergence to equilibrium of the exact preconditioned Hamiltonian Monte Carlo algorithm (pHMC) on a Hilbert space. As a consequence, explicit and dimension-free bounds for pHMC applied to high-dimensional distributions arising in transition path sampling and path integral molecular dynamics are given. Global convexity of the underlying potential energies

We give decay estimates of the solution to the linear Schrödinger equation in dimension \(d \ge 3\) with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé–Soffer–Sogge for decay of deterministic Schrödinger operators.

The 2D Euler equations with random initial condition has been investigates by Albeverio and Cruzeiro (Commun Math Phys 129:431–444, 1990) and other authors. Here we prove existence of solutions for the associated continuity equation in Hilbert spaces, in a quite general class with LlogL densities with respect to the enstrophy measure.

We investigate a stochastic transport equation driven by a multiplicative noise. For drift coefficients in \(L^q(0,T;{\mathcal {C}}^\alpha _b({\mathbb {R}}^d))\) (\(\alpha >2/q\)) and initial data in \(W^{1,r}({\mathbb {R}}^d)\), we show the existence and uniqueness of stochastic strong solutions. Opposite to the deterministic case where the same assumptions on drift coefficients and initial data induce

We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of \(L_{p}\) spaces of functions whose regularity is defined by a Lévy measure with O-regularly varying radial profile. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some probability density function estimates of the associated Lévy process are used as well.

We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter \(\varepsilon \) we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully

In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: \( du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum _{k = 1}^m g^k (u(t, x)) dw^k (t). \) We prove the convergence of a Wong–Zakai type approximation scheme of the above equation in the space \( C^{\theta } ([0, T], H^{\gamma }_p ({\mathbb {R}}^d)) \) in probability, for

Using a Besov topology on spaces of modelled distributions in the framework of Hairer’s regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type 2. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness

The claim of Lemma 3.7 in [1] needs to be modified. This lemma states that the sum of the absolute value of the cardinal functions is bounded with respect to the approximation parameters and the spatial variables.

Semilinear stochastic partial differential equations on bounded domains \({\mathscr {D}}\) are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are \(L^p\)-estimates for such equations. The \(L^p\)-estimates

In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit

It is well known in many settings that reversible Langevin diffusions in confining potentials converge to equilibrium exponentially fast. Adding irreversible perturbations to the drift of a Langevin diffusion that maintain the same invariant measure accelerates its convergence to stationarity. Many existing works thus advocate the use of such non-reversible dynamics for sampling. When implementing

Given a sequence \({\dot{L}}^\varepsilon \) of Lévy noises, we derive necessary and sufficient conditions in terms of their variances \(\sigma ^2(\varepsilon )\) such that the solution to the stochastic heat equation with noise \(\sigma (\varepsilon )^{-1} {\dot{L}}^\varepsilon \) converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive

Under a standard CFL condition, we prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise. In Dotti and Vovelle (Arch Ration Mech Anal 230(2):539–591, 2018), a framework for the analysis of the convergence of approximations to stochastic scalar first-order conservation

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for \(\alpha \)-Hölder continuous paths the convergence rate of the numerical

We prove the two dimensional KPZ equation with a logarithmically tuned nonlinearity and a small coupling constant, scales to the Edwards–Wilkinson equation with an effective variance.

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem

We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.