Coupling property and gradient estimates of Lévy processes via the symbol

Schilling, René L.; Sztonyk, Paweł; Wang, Jian

doi:10.3150/11-bej375

Cited by 75 publications

(95 citation statements)

References 28 publications

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“…Assuming the conditions (3.24) and (3.25), we can mimic the proof of [12,Theorem 3.2] to show that there exist t 1 , C > 0 such that for all x, y ∈ R n and t ≤ t 1 ,…”

Section: )mentioning

confidence: 99%

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Schilling

Wang

2011

J. Evol. Equ.

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Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol. Main resultsLet (X x t ) t≥0 be an n-dimensional Ornstein-Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation(1.1)Here A is a real n × n matrix, B is a real n × d matrix and Z t is a Lévy process in R d ; note that we allow Z t to take values in a proper subspace of R d . It is well known thatThe characteristic exponent or symbol of Z t , defined byenjoys the following Lévy-Khintchine representation:where Q = (q j,k ) d j,k=1 is a positive semi-definite matrix, b ∈ R d is the drift vector and ν is the Lévy measure, i.e. a σ -finite measure on R d \{0} such that the integral z =0 (1 ∧ |z| 2 ) ν(dz) < ∞. For every ε > 0, define ν ε on R d as follows: (2000): 60J25, 60J75 Mathematics Subject Classification

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“…Assuming the conditions (3.24) and (3.25), we can mimic the proof of [12,Theorem 3.2] to show that there exist t 1 , C > 0 such that for all x, y ∈ R n and t ≤ t 1 ,…”

Section: )mentioning

confidence: 99%

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Schilling

Wang

2011

J. Evol. Equ.

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“…Our next result applies to a large class of jump Lévy processes, including stable Lévy processes. It is a direct consequence of the gradient estimates obtained in [27]…”

Section: 5mentioning

confidence: 79%

Schauder Estimates for Equations Associated with Lévy Generators

Kühn

2019

Integr. Equ. Oper. Theory

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We study the local regularity of solutions f to the integro-differential equation Af = g in U associated with the infinitesimal generator A of a Lévy process (Xt) t≥0 . Under the assumption that the transition density of (Xt) t≥0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f . Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.1991 Mathematics Subject Classification. Primary 60G51; Secondary 45K05, 60J35.

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“…(a) The condition {|z−z 0 |≤ε} ρ 0 (z) −1 dz < ∞ is very weak, as it holds provided ρ 0 has a continuous point z 0 ∈ R d such that ρ 0 (z 0 ) > 0. Successful couplings have also been constructed in [23] under a slightly different condition.…”

Section: Coupling Property For O-u Processes With Jumpmentioning

confidence: 99%

“…For detailed proofs of the above results and further developments on couplings and applications of Lévy processes, one may check with recent papers [5,22,23,32,33].…”

Section: Derivative Formula and Gradient Estimatementioning

confidence: 99%

Coupling and Applications

Wang¹

2012

Stochastic Analysis and Applications to Finance

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This paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and probability distance (or transportation-cost), then introduce applications of coupling to the study of ergodicity, Liouville theorem, convergence rate, gradient estimate, and Harnack inequality for Markov processes.

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Coupling property and gradient estimates of Lévy processes via the symbol

Cited by 75 publications

References 28 publications

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Schauder Estimates for Equations Associated with Lévy Generators

Coupling and Applications

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