Related papers: Limit Theorems for Cylindrical Martingale Problems…

We investigate several fundamental properties of kinetic Langevin processes in $\mathbb{R}^{2d}$, defined as solutions to the following system: $$dx\_t = v\_t \, dt, \qquad dv\_t = \mathbf{B}(x\_t, v\_t) \, dt + dL\_t$$ where $(L\_t, t \ge…

This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--L\'evy noises. The equations feature a Caputo time…

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient…

In this paper, we construct martingale suitable weak solutions for $3$-dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, ``suitable weak solutions'' are…

We consider stochastic inviscid dyadic models with energy-preserving noise. It is shown that the models admit weak solutions which are unique in law. Under a certain scaling limit of the noise, the stochastic models converge weakly to a…

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in…

We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of…

We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $R^d$, where $\alpha \in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) =…

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump L\'evy processes. Our proof consists of two parts. In the first one, mostly classical, we recall a priori…

This paper deals with linear stochastic partial differential equations with variable coefficients driven by L\'{e}vy white noise. We first derive an existence theorem for integral transforms of L\'{e}vy white noise and prove the existence…

We propose a simple and original approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon pro\-blems, and allow notably some coefficients to be stochastic. Extension to…

In this paper, we establish the existence of weak solutions for distribution-dependent stochastic differential equations (DDSDEs) driven by a broad class of L\'{e}vy noises, where the drift coefficients satisfy specific integrability…

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such…

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by L\'evy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then,…

We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient…

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract…

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative L\'evy noises in a steady-state nonequilibrium case. This solution shows that L\'evy flights suppress always a quasi-periodical motion…

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space $\HH$, where $L$ is a $\RR$ valued L\'evy process, $A:H\to…

Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation $Au=0$ associated with the infinitesimal generator $A$ of a L\'evy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial…

Given a sequence $\dot{L}^{\varepsilon}$ of L\'evy noises, we derive necessary and sufficient conditions in terms of their variances $\sigma^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise…