Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

Xu, Lihu; Zegarliński, Bogusław

doi:10.1214/ejp.v15-831

Cited by 14 publications

(13 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the study of invariant measures and the long-time behaviour of stochastic systems driven by α-stable type noises, there seem only several results (cf. [33,34,25,24,17,31]). [33,34] studied the exponential mixing of stochastic spin systems with white α-stable noises, while [25,24] obtained exponential mixing for a family of semi-linear SPDEs with Lipschitz nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

“…[33,34,25,24,17,31]). [33,34] studied the exponential mixing of stochastic spin systems with white α-stable noises, while [25,24] obtained exponential mixing for a family of semi-linear SPDEs with Lipschitz nonlinearity. [17] obtained a nice criterion for the exponential mixing of a family of SDEs forced by Lévy noises, it covers 1D SDEs driven by α-stable noises.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ergodicity of the stochastic real Ginzburg–Landau equation driven byα-stable noises

2013

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

We study the ergodicity of stochastic real Ginzburg-Landau equation driven by additive α-stable noises, showing that as α ∈ (3/2, 2), this stochastic system admits a unique invariant measure. After establishing the existence of invariant measures by the same method as in [9], we prove that the system is strong Feller and accessible to zero. These two properties imply the ergodicity by a simple but useful criterion in [16]. To establish the strong Feller property, we need to truncate the nonlinearity and apply a gradient estimate established in [26] (or see [24] for a general version for the finite dimension systems). Because the solution has discontinuous trajectories and the nonlinearity is not Lipschitz, we can not solve a control problem to get irreducibility. Alternatively, we use a replacement, i.e., the fact that the system is accessible to zero. In section 3, we establish a maximal inequality for stochastic α-stable convolution, which is crucial for studying the well-posedness, strong Feller property and the accessibility of the mild solution. We hope this inequality will also be useful for studying other SPDEs forced by α-stable noises.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ergodicity of the stochastic real Ginzburg–Landau equation driven byα-stable noises

2013

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…The finite-dimensional result established in this paper is an important contribution of independent interest. Stochastic PDEs driven by Lévy noises have been intensively studied since some time; e.g., see the papers [4,2,28,26,20,32,41], the book [27], and the references therein. Invariant measures and long-time asymptotics for stochastic systems driven by Lévy noises were studied in a number of papers.…”

Section: Introductionmentioning

confidence: 99%

Exponential ergodicity and regularity for equations with Lévy noise

Priola

Shirikyan

et al. 2012

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument [10]. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [29], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite-dimensional parabolic systems, introduced in [32], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [30] using the weak convergence induced by the Kantorovich-Wasserstein metric.

show abstract

“…[16], [49], [51] and references therein), one obtains finite speed of propagation of information which allows to show the existence of the semigroup in infinite dimensions and under additional assumptions existence of invariant measure and strong ergodicity. That is one has the following result.…”

Section: Lemmamentioning

confidence: 99%

Linear and Nonlinear Dissipative Dynamics

Zegarliński

2016

Reports on Mathematical Physics

View full text Add to dashboard Cite

Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

Abstract: Abstract. We study an infinite white α-stable systems with unbounded interactions, proving the existence by Galerkin approximation and exponential mixing property by an α-stable version of gradient bounds.

Cited by 14 publications

References 30 publications

Ergodicity of the stochastic real Ginzburg–Landau equation driven byα-stable noises

Ergodicity of the stochastic real Ginzburg–Landau equation driven byα-stable noises

Exponential ergodicity and regularity for equations with Lévy noise

Linear and Nonlinear Dissipative Dynamics

Contact Info

Product

Resources

About