Martingale Solutions Research Articles

Convergence of the solution of the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes equations

We study in this article the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes model in a 3D dimensional bounded domain. We prove the existence and uniqueness of strong solutions. Furthermore, we discuss the relation of the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes equations to the stochastic 3D Cahn–Hilliard–Navier–Stokes equations by proving a convergence theorem, that as the parameter N tends to infinity, a subsequence of solutions of the stochastic 3D globally modified Cahn–Hilliard–Navier–Stokes equations converges to a weak martingale solution of the stochastic 3D Cahn–Hilliard–Navier–Stokes equations.

On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions

This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in $L^1$.

Existence of Positive Solutions to Stochastic Thin-Film Equations

We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size $h$. Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Holder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence.

Martingale Solution to Stochastic Extended Korteweg-de Vries Equation

We study a stochastic extended Korteweg - de Vries equation driven by a multiplicative noise. We prove the existence of a martingale solution to the equation studied. The proof of the solution is based on two approximations of the problem considered and the compactness method.

Local martingale solutions to the stochastic two layer shallow water equations with multiplicative white noise

We study the two layers shallow water equations on a bounded domain M⊂R2 driven by a multiplicative white noise, and obtain the existence and uniqueness of a maximal pathwise solution for a limited period of time, the time of existence being strictly positive almost surely. The proof makes use of anisotropic estimates and stopping time arguments, of the Skorohod representation theorem, and the Gyöngy–Krylov theorem which is an infinite dimensional analogue of the Yamada–Watanabe theorem.

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.

The stochastic Navier\u2013Stokes equations for heat-conducting, compressible fluids: global existence of weak solutions

We investigate the well posedness of the stochastic Navier–Stokes equations for viscous, compressible, non-isentropic fluids. The global existence of finite-energy weak martingale solutions for large initial data within a bounded domain of mathbb {R}^d is established under the condition that the adiabatic exponent gamma > d/2. The flow is driven by a stochastic forcing of multiplicative type, white in time and colored in space. This work extends recent results on the isentropic case, the main contribution being to address the issues which arise from coupling with the temperature equation. The notion of solution and corresponding compactness analysis can be viewed as a stochastic counterpart to the work of Feireisl (Dynamics of viscous compressible fluids, vol 26. Oxford University Press, Oxford, 2004).

The stochastic Swift–Hohenberg equation

In this paper, we will study the stochastic Swift–Hohenberg equation. The weak martingale solution, stationary martingale solution, invariant measures, mild solution, large deviation principle and random attractors for the stochastic Swift–Hohenberg equation will be considered.

The Kardar–Parisi–Zhang Equation as Scaling Limit of Weakly Asymmetric Interacting Brownian Motions

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.

Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

In this paper we present a systematic study of a stochastic PDE with multiplicative noise modeling the motion of viscous and inviscid grade-two fluids on a bounded domain O of R2. We aim to identify the minimal conditions on the boundary smoothness of the domain for the well-posedness and time regularity of the solution. In particular, we found out that the existence of a H1(O) weak martingale solution holds for any bounded Lipschitz domain O. When O is a convex polygon the solution u lives in the Sobolev space W2,r(O) for some r>2 and rot(u−αΔu) is continuous in L2(O) with respect to the time variable. Moreover, pathwise uniqueness of solution holds. The existence result is new for the stochastic inviscid model and improves previous results for the viscous one. The time continuity result is new, even for the deterministic case when the domain O is a convex polygon.

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

We give an existence and asymptotic result for the so-called finite energy weak martingale solution of the compressible isentropic Navier–Stokes system driven by some random force in the whole spatial region. In particular, given a general nonlinear multiplicative noise, we establish the convergence to the incompressible system as the Mach number, representing the ratio between the average flow velocity and the speed of sound, approaches zero.

The stochastic Korteweg–de Vries equation on a bounded domain

This paper considers the stochastic Korteweg–de Vries equation on a bounded domain. The existence of a weak martingale solution is discussed by the Galerkin’s approximation and compactness method. Based on this result, we establish the Unique Continuation Property for the stochastic Korteweg–de Vries equation.

Restricted Markov uniqueness for the stochastic quantization of P(Φ)2 and its applications

In this paper we obtain restricted Markov uniqueness of the generator and uniqueness of martingale (probabilistically weak) solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the Dirichlet form approach and those obtained in [10] and in [24]. We prove that the solution X−Z, where X is obtained by the Dirichlet form approach in [4] and Z is the corresponding O-U process, satisfies the corresponding shifted equation (see (1.4) below). Moreover, we obtain that the infinite volume p(Φ)2 quantum field is an invariant measure for the X¯=Y+Z, where Y is the unique solution to the shifted equation.

Random Perturbations of Viscous, Compressible Fluids: Global Existence of Weak Solutions

This article is devoted to the well-posedness of the stochastic compressible Navier--Stokes equations. We establish the global existence of an appropriate class of weak solutions emanating from large inital data, set within a bounded domain. The stochastic forcing is of multiplicative type, white in time and colored in space. Energy methods are used to merge techniques of P. L. Lions and E. Feireisl for the deterministic, compressible system with the theory of martingale solutions to the incompressible, stochastic system. Namely, we develop stochastic analogues of the weak compactness program of Lions, and use them to implement a martingale method. The existence proof involves four layers of approximating schemes. We combine the three layer scheme of Feireisl, Novotny, and Petzeltova for the deterministic, compressible system with a time splitting method used by Berthelin and Vovelle for the one dimensional stochastic compressible Euler equations.

Stochastic Hall-Magneto-hydrodynamics System in Three and Two and a Half Dimensions

We introduce the stochastic Hall-magneto-hydrodynamics (Hall-MHD) system in three and two and a half dimensions with infinite-dimensional multiplicative noise, white in time, and prove the global existence of a martingale solution via a stochastic Galerkin approximation and applications of Prokhorov’s, Skorokhod’s and martingale representation theorems, as well as the pressure term through de Rham’s theorem adapted to processes. The Hall term represents mathematically a very singular nonlinear term, unprecedented in the previous work. The results extend many others on the deterministic Hall-MHD and stochastic MHD systems and Navier–Stokes equations. In contrast to the stochastic MHD system, the path-wise uniqueness in the two and a half dimensional case is an open problem.

Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system

We study the three-dimensional stochastic nonhomogeneous magnetohydrodynamics system with random external forces that involve feedback, i.e., multiplicative noise, and are non-Lipschitz. We prove the existence of a global martingale solution via a semi-Galerkin approximation scheme with stochastic calculus and applications of Prokhorov's and Skorokhod's theorems. Furthermore, using de Rham's theorem for processes, we prove the existence of the pressure term.

Local martingale solutions to the stochastic one layer shallow water equations

We consider the single layer shallow water equations on a bounded domain M⊂R2 forced by a multiplicative white noise, and obtain the existence and uniqueness of a maximal pathwise solution for a short period of time. The proof relies on the Skorohod representation theorem, the Gyöngy–Krylov theorem, stopping time arguments, and isotropic estimates.

Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.

Incompressible Limit for Compressible Fluids with Stochastic Forcing

We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.