The aim of this work is to study a stochastic obstacle problem governed by a T-monotone operator, a random force and a multiplicative stochastic reaction in the frame of Sobolev spaces. After proving a result of existence and uniqueness of the variational solution, by using an ad hoc perturbation of the stochastic reaction and a penalization of the constraint, we prove Lewy–Stampacchia’s inequalities

In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our

We consider a stochastic discretization of the stationary viscous Hamilton-Jacobi equation on the flat d–dimensional torus \(\mathbb {T}^d\) associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on \(\mathbb {T}^d\), up to additive constants. By additionally assuming a technical condition on the associated

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate \(\smash {O(t^\frac{1}{2})}\) whereas the support of solutions to the deterministic

We study the focusing stochastic nonlinear Schrödinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the \(L^2\)-critical and supercritical cases. The mass (\(L^2\)-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both

We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel–Riesz capacity and the \(\gamma \)-dimensional Hausdorff measure. We

We present the existence, uniqueness, and regularity of a strong solution to a super-linear stochastic partial differential equation (SPDE) with the random fractional Laplacians: $$\begin{aligned} u_t = a\Delta ^{\alpha /2}u + b u_x + cu + \xi |u|^{1+\lambda } \dot{W},\quad t>0\,;\quad u(0,\cdot ) = u_0(\cdot ), \end{aligned}$$ where \(\dot{W}\) is a space-time white noise, \(\alpha \in (1,2)\), and

We consider a damped/driven nonlinear Schrödinger equation in \(\mathbb {R}^n\), where n is arbitrary, \({\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,\) under odd periodic boundary conditions. Here \(\eta (t,x)\) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy \( \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, \)

Following the ideas of Zhikov and Piatnitski (Izv Math 70(1):19–67, 2006), and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations: the equation of harmonic maps into the sphere and the Landau–Lifschitz–Gilbert equation. These equations have strong nonlinear features, and in general their solutions

We construct a K-rough path [along the terminology of Deya (Probab Theory Relat Fields 166:1–65, 2016)] above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise

This article extends the celebrated Sewing lemma, known from the theory of rough paths, to multi-parameter fields on hyper-cubes. We use this to construct Young integrals for multi-parameter Hölder fields on general domains \(\left[ 0,T\right] ^{k}\) with \(k\ge 1\) taking values in \({\mathbb {R}}^{d}\). Moreover, we show existence, uniqueness and stability of some particular types of hyperbolic SPDEs

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 prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of type $$\begin{aligned} \text {d}X_t=b(t,X)\text {d}t+\sigma (t,X)\text {d}W_t, \quad t\ge 0, \end{aligned}$$ and show that for such equations uniqueness in law is equivalent to joint uniqueness in law for deterministic initial

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter \(\beta ^2 > 0\) in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and

The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature

We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by Bényi et al. (Trans Am Math Soc Ser B 2:1–50, 2015) to the current stochastic PDE setting, we present a concise argument

We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.

A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension \(d\ge 2\); the case

We study the iterative algorithm proposed by Armstrong et al. (An iterative method for elliptic problems with rapidly oscillating coefficients, 2018. arXiv preprint arXiv:1803.03551) to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals

For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in both time and space leads to various modifications of

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.

For a class of finite elements approximations for linear stochastic parabolic PDEs it is proved that one can accelerate the rate of convergence by Richardson extrapolation. More precisely, by taking appropriate mixtures of finite elements approximations one can accelerate the convergence to any given speed provided the coefficients, the initial and free data are sufficiently smooth.

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