In this paper we prove almost sure local (in time) well-posedness in both adapted atomic spaces and Fourier restriction spaces for the periodic cubic nonlinear Schrödinger equation on \(\mathbb {T}^d\) (\(d\ge 3\)) in the super-critical regime.

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that

This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Itô’s formulation for bounded domains. Moreover, the uniqueness is established

One proves the uniqueness of distributional solutions to nonlinear Fokker–Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean–Vlasov stochastic differential equation (SDE).

Feynman–Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrödinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrödinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural

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

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order \(\frac{d}{2}\), where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise

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.

We show that for a family of randomly kicked Hamilton–Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the earlier results of the authors, this completes the program in the multi-dimensional setting started by E, Khanin, Mazel and Sinai in the one-dimensional case.

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an

We prove a Stroock–Varadhan’s type support theorem for a stochastic partial differential equation on the real line with a noise term driven by a cylindrical Wiener process on \(L_2 ({\mathbb {R}})\). The main ingredients of the proof are V. Mackevičius’s approach to support theorem for diffusion processes and N.V. Krylov’s \(L_p\)-theory of SPDEs.

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.

We construct a martingale solution of the stochastic nonlinear Schrödinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in \(H^1\) on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of

We study a large deviation principle for a system of stochastic reaction–diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the

We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and show that the solution

We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent \(p=p(\omega ,t,x)\) (as a result of a random electric field)

We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external

We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang’s condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses

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 progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain \(L^{p}(L^{q})\) estimates for all \(p>2\) and \(q\ge 2\)

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations

In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly,

We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the

In this paper we construct an invariant probability measure concentrated on \(H^2(K)\times H^1(K)\) for a general cubic Klein–Gordon equation (including the case of the wave equation). Here K represents both the 3-dimensional torus or a bounded domain with smooth boundary in \({\mathbb {R}}^3\). That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L\(\acute{\mathrm {e}}\)vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability

We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions

We study the random conductance model on the lattice \({\mathbb {Z}}^d\), i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301)

We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in \(L^2(\mathbb {T})\). As for other power-type nonlinearities, namely (i) (super)quintic when \(d = 1\) and (ii) (super)cubic when \(d \ge 2\), we prove local well-posedness

We consider sample path properties of the solution to the stochastic heat equation, in \({\mathbb {R}}^d\) or bounded domains of \({\mathbb {R}}^d\), driven by a Lévy space–time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than \(-\frac{d}{2}\). Concerning

We develop a solution theory in Hölder spaces for a quasi-linear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE lemmas which establish a priori Hölder bounds for a parabolic equation in divergence form with irregular right-hand-side term. We apply these bounds to the case of a right-hand-side noise term which is white in time and trace class in space, to obtain

We show that the solutions to the damped stochastic wave equation converge pathwise to the solution of a stochastic heat equation. This is called the Smoluchowski–Kramers approximation. Cerrai and Freidlin have previously demonstrated that this result holds in the cases where the system is exposed to additive noise in any spatial dimension or when the system is exposed to multiplicative noise and the

We present in this note a local in time well-posedness result for the singular 2-dimensional quasilinear generalized parabolic Anderson model equation$$\begin{aligned} \partial _t u - a(u)\Delta u = g(u)\xi . \end{aligned}$$The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting

So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Lévy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it

We study the ergodicity properties of weak solutions to a relativistic generalization of Burgers equation posed on a curved background and, specifically, a Schwarzschild black hole. We investigate the interplay between the dynamics of shocks, a curved geometric background, and a random boundary forcing, and solve three problems of independent interest. First of all, we consider the standard Burgers

We approximate the white-noise driven stochastic heat equation by replacing the fractional Laplacian by the generator of a discrete time random walk on the one dimensional lattice, and approximating white noise by a collection of i.i.d. mean zero random variables. As a consequence, we give an alternative proof of the weak convergence of the scaled partition function of directed polymers in the intermediate

We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Lévy processes in this context.

The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, \(\pi ^N\), with Lebesgue density on \({\mathbb {R}}^N\); it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm

One introduces a new variational concept of solution for the stochastic differential equation \(dX+A(t)X\,dt+{\lambda }X\,dt=X\,dW, t\in (0,T)\); \(X(0)=x\) in a real Hilbert space where \(A(t)={\partial }{\varphi }(t), t\in (0,T)\), is a maximal monotone subpotential operator in H while W is a Wiener process in H on a probability space \(\{{\Omega },{\mathcal {F}},\mathbb {P}\}\). In this new context

We construct renormalised models of regularity structures by using a recursive formulation for the structure group and for the renormalisation group. This construction covers all the examples of singular SPDEs which have been treated so far with the theory of regularity structures and improves the renormalisation procedure based on Hopf algebras given in Bruned–Hairer–Zambotti (Algebraic renormalisation

We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in \(\mathbb {R}^2\). Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian