Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α-stable Processes

Dong, Zhao; Wang, Feng‐Yu; Xu, Lihu

doi:10.1007/s11118-018-9736-0

Cited by 13 publications

(6 citation statements)

References 27 publications

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“…Our main results are Theorem 2.1 and Theorem 2.2. As an application of the main results of this paper, Proposition 4.1 in Section 4 not only covers all of the results obtained in [36,37,9,13] but also requires much weaker conditions on the coefficients and the driving Lévy noise, and also covers the setting of multiplicative driving noise. In a few words, to get the irreducibility, we just impose the conditions under which the well posedness can be guaranteed and a nondegenerate condition on the intensity measure of the driving Lévy noise, i.e., Assumptions 2.…”

Section: Introductionmentioning

confidence: 88%

“…Their approach must be modified to study the irreducibility of stochastic equations with highly nonlinear terms. There are several papers doing so (see [9,13,36,37]), which we describe below.…”

Section: Introductionmentioning

confidence: 99%

“…here {γ i = 4π 2 |i| 2 } are the eigenvalues of the Laplace operator on H. By improving the methods in [36] and [29], the authors in [37] and [9] established the irreducibility of stochastic reaction-diffusion equation and stochastic Burgers equation driven by the subordinated cylindrical Wiener process with a α/2-stable subordinator, α ∈ (1, 2), respectively. However, since the main ideas of [37,9] are similar to that of [36], technical restrictions as (1.2) and…”

Section: Introductionmentioning

confidence: 99%

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Irreducibility of SPDEs driven by pure jump noise

Wang¹,

Hao²,

Zhai³

et al. 2022

Preprint

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The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. In the literature, there are very few results on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise. The existing methods on this topic are basically along the same lines as that for the Gaussian case. They heavily rely on the fact that the driving noises are additive type and more or less in the class of stable processes. The use of such methods to deal with the case of other types of additive pure jump noises appears to be unclear, let alone the case of multiplicative noises.In this paper, we develop a new, effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild, and in some sense they are necessary and sufficient. This leads to not only significantly improving all of the results in the literature, but also to new irreducibility results of a much larger class of equations driven by pure jump noise with much weaker requirements than those treatable by the known methods. As a result, we are able to apply the main results to SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrödinger equations, Euler equations etc. We emphasize that under our setting the driving noises could be compound Poisson processes, even allowed to be infinite dimensional. It is somehow surprising.

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Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Irreducibility of SPDEs driven by pure jump noise

Wang¹,

Hao²,

Zhai³

et al. 2022

Preprint

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“…The ergodicity of the stochastic Burgers equation driven by Brownian motion and Poisson process was proved in Dong (2008) in the sense that the system converges to a unique invariant measure under the weak topology, but the convergence speed is not addressed. In this paper, we prove that the system converges to the invariant measure exponentially faster under a topology stronger than total variation by constructing a Lyapunov function in the same way as in Dong et al (2019). The moderate deviation principle (MDP) for the occupation measure is also obtained.…”

Section: Introductionmentioning

confidence: 94%

Asymptotics of stochastic Burgers equation with jumps

Wang

2020

Statistics & Probability Letters

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“…Since we consider in Eq. (1.2) multiplicative noise, we cannot apply the methods developed in [21] and [9] to show irreducibility. Alternatively, we resort to showing that the system (1.2) is accessible to 0, note that accessibility to 0 is often used as a replacement of irreducibility when one proves ergodicity [10].…”

Section: Accessibility and An Associated Control Problemmentioning

confidence: 99%

Singular integrals of subordinators with applications to structural properties of SPDEs

Deng¹,

Schilling²,

Xu³

2020

Preprint

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We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates.As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution Z t , (2) a small ball probability of Z t , (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.

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Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α-stable Processes

Cited by 13 publications

References 27 publications

Irreducibility of SPDEs driven by pure jump noise

Irreducibility of SPDEs driven by pure jump noise

Asymptotics of stochastic Burgers equation with jumps

Singular integrals of subordinators with applications to structural properties of SPDEs

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