On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

Abstract

We present a Cameron-Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and we obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion. IntroductionRecently the stability of properties of Markov processes and their semigroups under subordination in the sense of Bochner has attracted great interest. In [6], Wang's dimension free Harnack inequality was established for a class of subordinate semigroups. Nash and Poincaré inequalities are preserved under subordination, cf. [13,5]. In our recent paper [2], we show that shift Harnack inequalities (in the sense of [16]) remain valid under subordination in the sense of Bochner. It is a natural question whether further probabilistic properties, e.g. quasi-invariance, are preserved by subordination.The Cameron-Martin theorem, which was discovered by R.H. Cameron and W.T. Martin [1] (see e.g. [3,7,8] and the references therein for further developments), plays a fundamental role in the analysis on the path space of diffusion processes. It states that the Wiener measure (i.e. the distribution of Brownian motion) is quasi-invariant under a Cameron-Martin shift. In this paper, we shall derive an analogous result for subordinate Brownian motion.Let us recall some basic notations. Throughout this paper, we setand make the convention v u = (u,v) for all 0 ≤ u < v ≤ ∞. By S = (S t ) t∈[0,T ] , where 0 < T ≤ ∞, we denote a non-trivial subordinator, i.e. an increasing Lévy process with S 0 = 0 and Laplace transform Ee −uSt = e −tφ(u) , u > 0, t ∈ [0, T ]. 1 − e −ux ν(dx), u > 0, where b ≥ 0 is the drift parameter and ν is a Lévy measure, i.e. a measure on (0, ∞) satisfying ∞ 0 (x ∧ 1) ν(dx) < ∞; we use [12] as our standard reference for Bernstein functions and subordination. If T = ∞ then S ∞ := lim t↑∞ S t = ∞ a.s. Let M = ess sup S T = sup {r > 0 : P(S T < r) < 1} . Remark 1.1. M can attain the following values: (i) If T = ∞, then S ∞ = ∞ a.s. and M = ∞. (ii) If T < ∞ and S t is deterministic, i.e. S t = ct for some constant c > 0, then M = cT < ∞. (iii) If T < ∞ and S t is non-deterministic, then M = ∞. Indeed: Since ν = 0, there exists some finite interval [u, v] ⊂ (0, ∞) such that η := ν([u, v]) ∈ (0, ∞). The jump times of jumps with size in the interval [u, v] define a Poisson process (N t ) t∈[0,T ] with intensity η. Since S T ≥ uN T , we conclude that ess sup S T = ∞. Let (W t ) t∈[0,M ] be a standard d-dimensional Brownian motion starting from zero. The Wiener measure µ, i.e. the distribution of (W t ) t∈[0,M ] , is a probability measure on the path space W M = w : [0, M] → R d : w is continuous and w(0) = 0 ,which is endowed with the topology of locally uniform convergence. We write

On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes

“…This provides us with another approach to investigate jump processes via the corresponding results for diffusion processes. See [6] for the dimension-free Harnack inequality for subordinate semigroups, [12] for subordinate functional inequalities and [4] for the quasi-invariance property under subordination. In this paper, we will establish shift Harnack inequalities, which were introduced in [16], for subordinate semigroups.…”

Subgeometric rates of convergence for Markov processes under subordination

“…Among the most interesting examples are the symmetric α-stable Lévy processes, which can be viewed as subordinate to Brownian motions. It is known that many fine properties of Markov processes (and the corresponding Markov semigroups) are preserved under subordination; see [16], [9] for Harnack and shift Harnack inequalities for subordinate semigroups, [24], [15] for Nash and Poincaré inequalities under subordination, and [8] for the quasiinvariance property of subordinate Brownian motion.…”

Subgeometric Rates of Convergence for Discrete Time Markov Chains under Discrete Time Subordination

“…In recent years, there has been an increasing interest in the stability of properties of continuous time Markov processes and their semigroups under Bochner's subordination. See [16] for the dimension-free Harnack inequality for subordinate semigroups, [8] for shift Harnack inequality for subordinate semigroups, [9] for the quasi-invariance property of Brownian motion under random time-change, and [10] for subgeometric rates of convergence for continuous time Markov processes under continuous time subordination. Subordinate functional inequalities can be found in [4,15,24].…”