Invariant Measures for Monotone SPDEs with Multiplicative Noise Term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.     

Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence,Feller and strong Feller properties,and exponential ergodicity

This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non-confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman–Kac formula for Lévy type operators.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

On the strong Feller property for stochastic delay differential equations with singular drift

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations, see Maslowski and Seidler (2000). The argumentation is based on the well-posedness and the strong Feller property of the equations’ drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

Abstract

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

Smooth properties for semigroups of Lévy processes and their application

Explicit smooth properties for the semigroup of Lévy processes are derived in terms of its symbol. As an application, we obtain new sufficient conditions for the strong Feller property of stochastic differential equations driven by the additive Lévy process.     

Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces

We provide some sharp criteria for studying the ergodicity and asymptotic stability of general Feller semigroups on Polish metric spaces. As an application, the 2D Navier-Stokes equations with degenerate stochastic forcing will be simply revisited.     

Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.     

Stochastic flows of mappings

In this paper,the stochastic flow of mappings generated by a Feller convolution semigroup on a compact metric space is studied.This kind of flow is the generalization of superprocesses of stochastic flows and stochastic diffeomorphism induced by the strong solutions of stochastic differential equations.     

A viability theorem of stochastic semilinear evolution equations

A viability theorem of stochastic semilinear evolution equations is discussed under a dissipative condition in terms of uniqueness functions and a stochastic subtangential condition. Our strategy is to interpret a stochastic viability problem into a characterization problem of evolution operators associated with stochastic semilinear evolution equations. The main theorem is a generalization of the results due to Aubin and Da Prato in the case of stochastic differential equations in ℝ ^d .     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Equivalence of laws and null controllability for SPDEs driven by a fractional Brownian motion

We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control problem.     

Periodic solutions of hybrid jump diffusion processes

We investigate periodic solutions of regime-switching jump diffusions. We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system. Then, we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups. Finally, we establish the existence and uniqueness of periodic solutions. Concrete examples are presented to illustrate the results.     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Non-Autonomous stochastic differential equations in hubert spaces: Lyapunov Function,stability and ultimate boundedness

In this paper, we establish the Lyapunov function characterization which is a sufficient and necessary condition for mild solutions of semilinear stochastic evolution equations to be exponentially stable in mean square. We also study the Lyapunov function characterization of ultimate exponential boundedness, a concept which is closely related to the existence of invariant measures of non-stationary stochastic evolution equations     

On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

Journal of Theoretical Probability - We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical...     

Hypercontractivity of Solutions to Hamilton-Jacobi Equations

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.