2023/02/07 by Deng, Changsong, Schilling, Rene L., Xu, Lihu · 4 citations
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2302.03372
We are interested in the following two ℝd-valued stochastic differential equations (SDEs): d Xt=b(Xt) d t + σ d Lt, X0=x, % d Yt=b(Yt) d t + σ d Bt, Y0=y, where σ is an invertible d× d matrix, Lt is a rotationally symmetric α-stable Lévy process, and Bt is a d-dimensional standard Brownian motion (note that Bt is a rotationally symmetric α-stable Lévy process with α=2). We show that for any α0 ∈ (1,2) the Wasserstein-1 distance W1 satisfies for α∈ [α0,2) W1(Xtx, Yty) ≤ C1 e-C2t|x-y| +(C)/(α0-1)(2-α)dlog(1+d), which implies, in particular, W1(μα, μ2) ≤ (C)/(α0-1)(2-α)dlog(1+d), where μα and μ2 are the ergodic measures of Xt and Yt respectively. For the special case of a d-dimensional Ornstein--Uhlenbeck system, we show that W1(μα, μ2) ≥ Cd (2-α) for all α∈(1,2); this indicates that the convergence rate with respect to α in the second bound is optimal. The term dlog(1+d) appearing in this bound seems to be optimal for the dimension d as well.