Itô’s calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the $G$-expectation theory.     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

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.     

Stochastic population dynamics driven by Lévy noise

This paper considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model.     

Pathwise Estimation of Stochastic Differential Equations with Unbounded Delay and Its Application to Stochastic Pantograph Equations

The existence and uniqueness of the global solution of stochastic differential equations with discrete variable delay is investigated in this paper, and the pathwise estimation is also done by using Lyapunov function method and exponential martingale inequality. The results can be used not only in the case of bounded delay but also in the case of unbounded delay. As the applications, this paper considers the pathwise estimation of solutions of stochastic pantograph equations.     

Fundamental solutions of stochastic differential equations with drift

The paper deals with the pathwise uniqueness of solutions to one-dimensional time homogeneous stochastic differential equations with a diffusion coefficient σ satisfying the local time condition and measurable drift term b. We show that if the functions σ and b satisfy a non-degeneracy condition and fundamental solution to considered equation is unique in law, then pathwise uniqueness of solutions holds. Our result is in some sense negative, more precisely we give an example of an equation with Holder continuous diffusion coefficient and nondegenerate drift for which a fundamental solution is not unique in law and pathwise uniqueness of solutions does not hold.     

Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

In this article we study a system of nonlinear non-parabolic stochastic evolution equations driven by Lévy noise type. This system describes the motion of second grade fluids driven by random force. Global existence of a martingale solution is proved under general conditions on the noise. Since the coefficient of the noise does not satisfy a Lipschitz property, we could not prove any pathwise uniqueness result. We note that this is the first work dealing with a stochastic model for non-Newtonian fluids excited by external forces of Lévy noise type.     

Local strong solutions to the stochastic compressible Navier–Stokes system

We study the Navier–Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which is strong in both PDE and probabilistic sense. Our approach relies on rewriting the problem as a symmetric hyperbolic system augmented by partial diffusion, which is solved via a suitable approximation procedure. We use the stochastic compactness method and the Yamada–Watanabe type argument based on the Gyöngy–Krylov characterization of convergence in probability. This leads to the existence of a strong (in the PDE sense) pathwise solution. Finally, we use various stopping time arguments to establish the local existence of a unique strong solution to the original problem.     

Martingales,scale functions and stochastic life annuities: a note

In this note we derive the most general conditions under which the probability distribution of a generalized stochastic life annuity can be obtained by using the scale function methodology. Our main result is that the cumulative distribution function (CDF) of the generalized stochastic life annuity will obey the partial differential equation (PDE) satisfied by the scale function whenever the underlying process can be “Markovianized”. The scale function is the mapping which converts a Markov diffusion process into a martingale. In many cases, the resulting PDE can be easily solved to yield a closed form expression for the CDF.     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division.Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.     

Extension to infinite dimensions of a stochastic second-order model associated with shape splines

Motivated by the development of a probabilistic model for growth of biological shapes in the context of large deformations by diffeomorphisms, we present a stochastic perturbation of the Hamiltonian equations of geodesics on shape spaces. We study the finite-dimensional case of groups of points for which we prove that the strong solutions of the stochastic system exist for all time. We extend the model to the space of parameterized curves and surfaces and we develop a convenient analytical setting to prove a strong convergence result from the finite-dimensional to the infinite-dimensional case. We then present some enhancements of the model.     

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.     

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.     

Strong solutions for stochastic partial differential equations of gradient type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the “distance” defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction–diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.     

Uniqueness Result for the 2D Navier–Stokes Equation with Additive Noise

A bidimensional stochastic Navier–Stokes equation with an additive noise is considered. Known results on existence of generalized solutions are reviewed, in order to state an existence result in the most complete form. For this kind of generalized solutions, pathwise uniqueness is proven.     

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.     

Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.