Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

Liang, Mingjie; Wang, Jian

doi:10.1016/j.spa.2019.09.001

Cited by 25 publications

(18 citation statements)

References 35 publications

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“…Some properties related to (1.4) are given in the appendix. Finally, we note that couplings of SDEs with multiplicative Lévy noises were treated in the recent paper [14], where part of results above have been extended.…”

Section: Strong Ergodicitymentioning

confidence: 99%

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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We establish the exponential convergence with respect to the L 1 -Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equationwhere (Z t ) t≥0 is a pure jump Lévy process whose Lévy measure ν fulfillsfor some constant κ 0 > 0, and the drift term b satisfies that for anywith some positive constants K 2 , l 0 and positive measurable function Φ 1 . The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X t ) t≥0 .where Φ 1 and Φ 2 are two nonnegative measurable functions, and l 0 ≥ 0 is a constant. For example, when Φ 2 (r) = K 2 r for some positive constantThis holds if the drift term b is dissipative outside some compact set. In particular, when Φ 1 (r) = K 1 r for some constant K 1 ≥ 0, it follows from (1.3) that for anywhich, along with (1.3), yields that the SDE (1.1) has a non-explosive and pathwise unique strong solution, see [1, Chapter 6, Theorem 6.2.3] (in the standard Lipschitz case) or [11, Theorem 2] and [25, Chapter 3, Theorem 115] (in the one-sided Lipschitz case). Note that, sincewe are sometimes concerned with only measurable drift term b, non-Lipschitz condition like B(K 1 , K 2 r, l 0 ) will also be adopted in our results below. The reader can refer to [8,20,22,26,32] and references therein for recent studies on the existence and uniqueness of strong solution to (1.1) with non-regular drift term. In particular, assuming that Z is the truncated symmetric α-stable process on R d with α ∈ (0, 2), and b is bounded and β-Hölder continuous with β > 1 − α/2, it was proved in [8, Corollary 1.4(i)] that the SDE (1.1) has a unique strong solution for each x ∈ R d . Furthermore, in the one-dimensional case, if α > 1, then the SDE (1.1) also enjoys a unique strong solution for each x ∈ R, even if the drift b is only bounded and measurable (see [26, Remark 1, p. 82]). Denote by ν the Lévy measure of the pure jump Lévy process Z. We assume that there is a constant κ 0 > 0 such that(1.4)Condition (1.4) was first used in [23] to study the coupling property of Lévy processes. It is satisfied by a large class of Lévy measures. For instance, iffor some z 0 ∈ R d and some ε > 0 such that ρ 0 (z) is positive and continuous on B(z 0 , ε), then such Lévy measure ν fulfills (1.4), see [24, Proposition 1.5] for details. Actually, as shown in Proposition 6.5, the condition (1.4) implies that there is a nonnegative measurable function ρ on R d such that ν(dz) ≥ ρ(z) dz andLet (P t ) t≥0 be the transition semigroup associated with the process (X t ) t≥0 . In this paper we are interested in the asymptotics of the Wasserstein-type distances (including the L 1 -Wasserstein distance and the total variation) between probability distributions δ x P t = P t (x, ·) and δ y P t = P t (y, ·) for any x, y ∈ R d , when the drift term b is dissipative outside some compact set, i.e. b satisfies B(Φ 1 (r), K 2 r, l 0 ) for some positive measurable function Φ 1 , and some const...

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Section: Strong Ergodicitymentioning

confidence: 99%

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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“…The aim of this paper is to establish the spatial regularity of semigroups associated with the operator L given by . We will adopt the probabilistic coupling approach, which recently has been extensively studied in . To study analytic properties for Lévy type operators via probabilistic method, as one of the standing assumptions, the existence of a strong Markov process associated with Lévy type operators was assumed.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The construction below is heavily based on the refined basic coupling for stochastic differential equations driven by additive Lévy noises first introduced in . See for the recent study on stochastic differential equations driven by multiplicative Lévy noises. However, the Lévy type operator L given by enjoys some essential difference from the infinitesimal generator associated stochastic differential equations with jumps, and the main difficulty here is that the coefficient

c (x, z)

in the operator L depends on two space variables, which requires a new idea for the construction of a coupling operator.…”

Section: Coupling Operator and Coupling Process For Lévy Type Operatorsmentioning

confidence: 99%

Spatial regularity of semigroups generated by Lévy type operators

Liang

Wang

2019

Mathematische Nachrichten

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We apply the probabilistic coupling approach to establish spatial regularity of semigroups associated with Lévy type operators, by assuming that the corresponding martingale problem is well posed. In particular, we can prove the Lipschitz continuity of the associated semigroups, when the coefficients are Hölder continuous but the corresponding Lévy kernel may be singular. K E Y W O R D Scoupling, Lévy type operator, martingale problem, spatial regularity M S C ( 2 0 1 0 ) 60J25, 60J75 www.mn-journal.org

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“…For diffusions, the well-posedness of the associated martingale problem is the method of choice, see [ In the present context, all processes are given by SDEs, so it is more natural to require the existence of a strong solution to the SDE, see e.g. [20,15].…”

Section: Coupling Operators For Sdes With Additive Lévy Noisementioning

confidence: 99%

“…(13). This coupling was first introduced in [15] when studying the regularity of semigroups and the ergodicity of the solution to the SDE (26).…”

Section: 2mentioning

confidence: 99%

A unified approach to coupling SDEs driven by Lévy noise and some applications

Liang¹,

Schilling²,

Wang³

2020

Bernoulli

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We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.2010 Mathematics Subject Classification. 60J35; 60G51; 60H10; 60J25; 60J75.

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Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

Cited by 25 publications

References 35 publications

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Spatial regularity of semigroups generated by Lévy type operators

A unified approach to coupling SDEs driven by Lévy noise and some applications

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