Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.     

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.     

Statistical causality,martingale problems and local uniqueness

The paper considers a statistical concept of causality in continuous time in the filtered probability spaces which is based on the Granger’s definition of causality. The given causality concept is then applied to the solution of the martingale problem (associated with the stochastic differential equation driven with semimartingales). More precisely, we show that the given causality concept is closely connected to the concept of extremality of measures for the solutions of the martingale problem, for the stopped martingale problem and for the local martingale problem. We also show the equivalence between some models of causality and local uniqueness (for the solutions of the martingale problem).     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

Complete convergence and complete moment convergence for martingale difference sequence

In the paper,we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale diference sequence.Especially,we get the Baum–Katz-type Theorem and Hsu–Robbins-type Theorem for martingale diference sequence.As an application,a strong law of large numbers for martingale diference sequence is obtained.     

基于指数矩鞅差序列非参数回归函数的估计

本文研究了误差项是鞅差序列,且满足某种指数矩条件的非参数回归函数的估计.利用鞅的某种指数不等式,得到了其加权核估计的强相合以及在有限闭区间内一致强相合的性质,并在某种意义上推广了[5]的结果.     

随机序列级数的强收敛性

利用鞅收敛定理讨论随机序列级数的强收敛性,得到了该序列的强极限定理,推广了鞅差级数收敛性的一个结果     

鞅的加权Φ-不等式

本文研究了鞅空间的极大算子和极小算子,利用鞅方法,得到了关于鞅的强型和弱型加权Φ-不等式的一个必要条件.     

A certain class of diffusion processes associated with nonlinear parabolic equations

Summary We introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation. Then we show the existence and uniqueness theorems for solutions to the martingale problem.Research partially supported by the Air Force Office of Scientific Research Contract No. F4962082C0009     

Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L^∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L²-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L^∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.     

Nonlinear martingale problems involving singular integrals

We study a class of nonlinear martingale problems in one dimension, that involve a singular integral of the density in the drift term, and are related to systems of particles with singular interactions. First, we prove existence and uniqueness of regular solutions of the associated nonlinear evolution equation. Then, we establish a suitable framework and conditions where the martingale problem is well posed. This extends the results of Bonami et al. (J. Funct. Anal. 165 (1999) 390) to a wide class of coefficients and initial conditions. Finally, we obtain our solution of the martingale problem as the chaotic limit of some systems of particles interacting through regular approximating kernels.     

Stochastic Integrals of Continuous Local Martingales,II

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n-dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n-dimensional time change for reducing every n-dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.     

Brownian Representations of Cylindrical Local Martingales,Martingale Problem and Strong Markov Property of Weak Solutions of SPDEs in Banach Spaces

The paper deals with three issues. First we show a sufficient condition for a cylindrical local martingale to be a stochastic integral with respect to a cylindrical Wiener process. Secondly, we state an infinite dimensional version of the martingale problem of Stroock and Varadhan, and finally we apply the results to show that a weak existence plus uniqueness in law for deterministic initial conditions for an abstract stochastic evolution equation in a Banach space implies the strong Markov property.     

Singular dissipative stochastic equations in Hilbert spaces

 Existence of solutions to martingale problems corresponding to singular dissipative stochastic equations in Hilbert spaces are proved for any initial condition. The solutions for the single starting points form a conservative diffusion process whose transition semigroup is shown to be strong Feller. Uniqueness in a generalized sense is proved also, and a number of applications is presented. Received: 14 November 2001 / Revised version: 8 April 2002 / Published online: 10 September 2002     

Backward problems for stochastic differential equations on the Sierpinski gasket

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section 2) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman–Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.     

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.     

On the convergence of partial differential equations of parabolic type with rapidly oscillating coefficients to stochastic partial differential equations

In this paper we consider the convergence of the solutions to a sequence of partial differential equations of parabolic type with rapidly oscillating coefficients to the solutions of a stochastic partial differential equation. We use the martingale method and the characteristic functional to prove that the martingale problem has a unique solution. Our emphasis is in treating strongly mixing noises.This research was partially supported by Stifting Volkswagenwerk through Forschungszentrum BiBoS Universität Bielefeld and Grant-in-Aid for General Scientific Research (C) 61540162, the Ministry of Education, Science and Culture (Japan). Presented at the International Workshop on Diffusion Approximations and Related Topics, IISA, Laxenburg, Austria, 29 June to 3 July 1987.     

Representing filtration consistent nonlinear expectations as g-expectations in general probability spaces

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous drivers, where both the martingale and the driver terms are permitted to jump, and the martingale representation is infinite dimensional. To establish this result, we show that this domination condition is sufficient to guarantee that the comparison theorem for BSDEs will hold, and we generalise the nonlinear Doob–Meyer decomposition of Peng to a general context.     

一类非线性偏微分方程解的鞅刻划

本文以超布朗运动为工具，利用鞅刻划了一类非线性偏微分方程的正解，得到了某函数为一类非线性偏微分方程正解的充分必要鞅条件．