The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier-Stokes equations in unbounded domains

In this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2-dimensional non-Lipschitz Navier-Stokes equation perturbed by the cylindrical Wiener process W(t) in a bounded or unbounded domain D with the smooth boundary ∂D or D=R²:

     

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.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

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.     

Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups

Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied to characterize several properties of the set of all solutions in terms of the law of the driving noise.     

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 method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

Systems of stochastic partial differential equations with reflection: Existence and uniqueness

In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate.     

Time regularity of solutions to linear equations with Lévy noise in infinite dimensions

The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H$H$, driven by a Lévy process taking values in a Hilbert space U?H$U?H$ is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy processes.     

Almost sure explosion of solutions to stochastic differential equations

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.     

The fractional stochastic heat equation on the circle: Time regularity and potential theory

We consider a system of d$d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle S¹$S^1$. We obtain sharp results on the Hölder continuity in time of the paths of the solution u=_{{u(t,x)}t∈R+,x∈S1}$u={\left\{u(t,x)\right\}}_{t\in {\mathbb{R}}_{+},x\in S^1}$. We then establish upper and lower bounds on hitting probabilities of u$u$, in terms of the Hausdorff measure and Newtonian capacity respectively.     

Heat equation with a general stochastic measure on nested fractals

A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3.     

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.     

The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

The (Ξ,A)$(\Xi ,A)$-Fleming–Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming–Viot process except that Kingman’s coalescent is replaced by the Ξ$\Xi$-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A$A$. We then investigate its reversibility. We also study both the weak and strong uniqueness of the solution to the associated stochastic partial differential equation.