Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

Comparison of Difficulty Levels of Examples in High School Mathematics Textbooks

“…[1,5,17]. Furthermore, yet under the assumption that the potential U is regular nonetheless the underlying noise (Z t ) t≥0 is a d-dimensional pure jump Lévy process, the exponential ergodicity under a multiplicative type quasi-Wasserstein distance of the Markov process solved by (1.1) was treated in [2] by invoking a refined basic coupling method. Actually, motivated by sampling from the heavy-tailed distribution arising in various applications including statistical machine learning [18] and statistical physics study [4], the long time behaviors of the SDE (1.2) with an α-stable Lévy motion (instead of a Brownian motion) will play more appropriate and important roles.…”

“…Whereas, as for the irreducible property, the noise term is confined to be a range of symmetric stable processes (or more general subordinated Brownian motions), where the key ingredient is to make fully use of the topologically irreducible property of Brownian motions. To evade the application of Harris' theorem (in the vast majority of occasions, it is extremely difficult to examine the strong Feller property and the irreducibility), the probabilistic coupling method might be an alternative to deal with ergodicity of (1.2) and, most importantly, to encompass a wide range of pure jump Lévy noises (see [2] for the case that the coefficients are regular). Due to the involvement of singular potential, for the moment, it is still a very formidable task to construct an appropriate coupling and a suitable metric function to explore the ergodicity under the (quasi-)Wasserstein distance.…”

Strong feller and ergodic properties of the (1+1)-affine process

“…The exponential ergodicity of the process in the total variation distance was established in Jin et al [9]. For a subcritical two-factor affine process driven by Lévy stable processes, the exponential ergodicity in the L 1 -Wasserstein distance was established in Bao and Wang [2] by a coupling approach. For general finite-dimensional affine Markov processes, Jin et al [11] proved a sufficient condition for the ergodicity in weak convergence, which covers partially the results of Pinsky [23] and Sato and Yamazato [26].…”