Martingale Solutions Research Articles

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, the stopping times, the stochastic compactness method and the Jakubowski–Skorokhod theorem.

Stochastic one layer shallow water equations with Lévy noise

This work investigates the existence of both martingale and pathwise solutions of the single layer shallow water equations on a bounded domain \begin{document}$ \mathcal{M} \subset \mathbb{R}^2 $\end{document} perturbed by a Levy noise which may represent bursts of surface winds. The construction of both solutions are based on some truncation, the classical Faedo-Galerkin approximation scheme, a modified version of the Skorokhod representation theorem, stopping time arguments and anisotropic estimates.

Fractionally Dissipative Stochastic Quasi-Geostrophic Type Equations on ${\mathbb{R}}^{d}$

A stochastic fractionally dissipative quasi-geostrophic type equation on ${\mathbb{R}}^{d}$ with a multiplicative Gaussian noise is considered. We prove the existence of a martingale solution. The ...

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

AbstractIn this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.

Global martingale solutions for a stochastic population cross-diffusion system

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski’s generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia’s truncation method due to Chekroun, Park, and Temam.

Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.

Martingale solutions for the stochastic nonlinear Schr\xf6dinger equation in the energy space

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H^1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.

Strong solutions for a stochastic model of two-dimensional second grade fluids driven by Lévy noise

In this paper, we consider stochastic second grade fluids driven by Lévy noise on a bounded domain of R 2 . Global existence and uniqueness of strong probabilistic solutions are established, which improves the existing corresponding results for martingale solutions in literature.

On the convergence of a stochastic 3D globally modified two-phase flow model

We study in this article a stochastic 3D globally modified Allen-Cahn-Navier-Stokes model in a bounded domain. We prove the existence and uniqueness of a strong solutions. The proof relies on a Galerkin approximation, as well as some compactness results. Furthermore, we discuss the relation between the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations and the stochastic 3D Allen-Cahn-Navier-Stokes equations, by proving a convergence theorem. More precisely, as a parameter $N$ tends to infinity, a subsequence of solutions of the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations converges to a weak martingale solution of the stochastic 3D Allen-Cahn-Navier-Stokes equations.

On a doubly nonlinear PDE with stochastic perturbation

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. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an $$L^1$$ -method. Using this result, we are able to show existence and uniqueness of strong solutions.

Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces

We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise.

Well-posedness for a pseudomonotone evolution problem with multiplicative noise

Our aim is the study of well-posedness for the stochastic evolution equation $$\begin{aligned} \mathrm{d}u -\mathrm {div}\,(|\nabla u|^{p-2}\nabla u +F(u)) \,\mathrm{d} t = H(u) \,\mathrm{d}W, \end{aligned}$$ for $$T>0$$ , on a bounded Lipschitz domain $$D \subset \mathbb {R}^d$$ with homogeneous Dirichlet boundary conditions, initial values $$u_0\in L^2(D)$$ , $$p>2$$ and $$F:\mathbb {R}\rightarrow \mathbb {R}^d$$ Lipschitz continuous. W(t) is a cylindrical Wiener process in $$L^2(D)$$ with respect to a filtration $$(\mathcal {F}_t)$$ satisfying the usual assumptions on a complete, countably generated probability space $$(\varOmega ,\mathcal {F},\mathbb {P})$$ . We consider the case of multiplicative noise satisfying appropriate regularity conditions. By a semi-implicit time discretization, we obtain approximate solutions. Using the theorems of Skorokhod and Prokhorov, we are able to pass to the limit and show existence of martingale solutions. We establish pathwise $$L^1$$ -contraction and uniqueness and obtain existence and uniqueness of strong solutions.

Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation

The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere S2. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

Feedback Optimal Controllers for the Heston Model

We prove the existence of an optimal feedback controller for a stochastic optimization problem constituted by a variation of the Heston model, where a stochastic input process is added in order to minimize a given performance criterion. The stochastic feedback controller is found by solving a nonlinear backward parabolic equation for which one proves the existence and uniqueness of a martingale solution.

Quantitative stability estimates for Fokker–Planck equations

We derive quantitative stability estimates for solutions of Fokker–Planck equations with irregular coefficients. We are mainly concerned with two different situations: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift coefficient satisfies only the Ladyzhenskaya–Prodi–Serrin condition. Our method is based on Trevisan's superposition principle, which represents the solution to the Fokker–Planck equation as the marginal distribution of the martingale solution of the associated stochastic differential equation.

Time-splitting methods to solve the Hall-MHD systems with Lévy noises

In this paper, we establish the existence of a martingale solution to the stochastic incompressible Hall-MHD systems with Levy noises in a bounded domain. The proof is based on a new method, i.e., the time splitting method and the stochastic compactness method.

Modified Massive Arratia Flow and Wasserstein Diffusion

Extending previous work by the first author we present a variant of the Arratia flow, which consists of a collection of coalescing Brownian motions starting from every point of the unit interval. The important new feature of the model is that individual particles carry mass that aggregates upon coalescence and that scales the diffusivity of each particle in an inverse proportional way. In this work we relate the induced measure‐valued process to the Wasserstein diffusion of von Renesse and Sturm. First, we present the process as a martingale solution to an SPDE similar to that of von Renesse and Sturm. Second, as our main result we show a Varadhan formula 42 for short times that is governed by the quadratic Wasserstein distance. © 2018 Wiley Periodicals, Inc.

Stochastic generalized magnetohydrodynamics equations: well-posedness

In this paper, we study a stochastic magnetohydrodynamics system with fractional diffusion , , in , . Our main goal is to identify the regularity of the driving noises and the conditions on under which we can prove the existence of a martingale solution.

Stochastic heat equations with values in a Riemannian manifold

The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.