Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Abstract

Abstract:In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H:Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a D(A −1/2 )-valued Lévy process satisfying several technical assumptions. By using a density transformation theorem type for Lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein-Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain γUsing a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on H × H. While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier-Stokes, Magnetohydrodynamics equations, the 3D Leray-α model, the GOY and Sabra shell models.

Well-posedness of Stochastic 2D Hydrodynamics type Systems with Multiplicative Lévy Noises

“…Stochastic partial differential equations (SPDEs) driven by jump-type noises such as Lévytype or Poisson-type perturbations are drastically different from those driven by Wiener noises, due to the presence of jumps, concerning the well-posedness, the Burkholder-Davis-Gundy inequality, the Girsanov theorem, the time regularity, the ergodicity, irreducibility, mixing property and other long-time behaviour of the solutions. For more details, please refer to [3] [8] [19] [20] [23][27] [28] [34], and the references therein. In general, all the results and/or techniques available for the SPDEs with Gaussian noise are not always suitable for the treatment of SPDEs with Lévy noise and therefore we require new and different techniques.…”

Well-posedness and large deviations for 2-D Stochastic Navier-Stokes equations with jumps

“…In general, all the results and/or techniques available for the SPDEs with Gaussian noise are not always suitable for the treatment of SPDEs with Lévy noise and therefore we require new and different techniques. We refer to [37,Theorem III.3.24], [8,Theorem 3.10.21], [35,36,5,10,40,41,45,46,30,60] and reference therein for more details.…”

Stochastic 3D Leray-$α$ Model with Fractional Dissipation

“…Chueshov and Millet in [16] proved the well-posedness and large deviation principle of stochastic 3D Leray-α model in the case of θ 1 = θ 2 = 1 (see also [22]). The well-posedness and irreducibility of 3D Leray-α model driven by Lévy noise have been studied in [6,23]. The α-approximation of stochastic Leray-α model to the stochastic Navier-Stokes equations was established in [8,15].…”