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

Abstract

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

$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes

“…(2) Recently, the coupling property of Lévy processes has been developed in [4,17,18]. The corresponding property for Ornstein-Uhlenbeck processes with jumps also has been successfully studied in [19,23]. Unlike Lévy processes and Ornstein-Uhlenbeck processes with jumps, it is impossible to write out an explicit expression for transition functions of the solution to the SDE (1.1) with general drift term b(x).…”

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

“…See e.g. [2], [21] and [22] for couplings of pure jump Lévy processes, [23], [28] and [29] for the case of Lévy-driven Ornstein-Uhlenbeck processes and [12], [31] and [25] for more general Lévy-driven SDEs with non-linear drift. See also [11] and [19] for general considerations concerning ergodicity of SDEs with jumps.…”

On the Exponential Ergodicity of Lévy-Driven Ornstein–Uhlenbeck Processes

“…Q = 0 and b = 0 in (4). According to [13,Theorem 1.7], it suffices to verify that This proves the required assertion.…”

“…Recall that the process X has successful couplings (or has the coupling property) if and only if for any x, y ∈ R d , lim t→∞ P t (x, ·) − P t (y, ·) var = 0, where P t (x, dz) is the transition kernel of the process X and · var stands for the total variation norm. The coupling property has been intensively studied for Lévy processes on R d and Ornstein-Uhlenbeck processes driven by Lévy processes on R d , see [2,11,12,13,16]. Recently, by using the lower bound conditions for the Lévy measure with respect to a nice reference probability measure, we have successfully obtained the coupling property for linear stochastic differential equations driven by non-cylindrical Lévy processes on Banach spaces, see [18,Theorem 1.2].…”