Some theorems on Feller processes: transience, local times and ultracontractivity

By René L. Schilling, Jian Wang. 2011

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We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Levy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on Rd with smooth variable index α(x) ∈ (0, 2) a transience criterion in terms of the exponent α(x); if d = 1 and infx∈R α(x) ∈ (1, 2), then the stable-like process has local times. 1. Background and main results In this paper we study sample path properties of Feller processes. Our main tool will be the fact that the infinitesimal generator of the associated Feller semigroup can be written as a pseudodifferential operator with negative definite symbol. A Feller process (Xt)t 0 with state space R is a strong Markov process whose associated operator semigroup (Tt)t 0, Ttu(x) = E x (u(Xt)) , u ∈ C∞(R), t 0, x ∈ R, (C∞(R ) is the space of continuous functions vanishing at infinity) enjoys the Feller property, i.e. maps C∞(R ) into itself. The semigroup (Tt)t 0 is said to be a Feller semigroup. That is, (Tt)t 0 is a one-parameter semigroup of contraction operators Tt : C∞(R ) → C∞(R) which is strongly continuous: limt→0 ‖Ttu− u‖∞ = 0 and has the sub-Markov property : 0 Ttu 1 whenever 0 u 1. The (infinitesimal) generator (A,D(A)) of the semigroup or the process is given by the strong limit Au := lim t→0 Ttu− u t on the set D(A) ⊂ C∞(R) of those u ∈ C∞(R) for which the above limit exists w.r.t. the sup-norm. We will call (A,D(A)) Feller generator for short. Before we proceed with general Feller semigroups it is instructive to have a brief look at Levy processes and convolution semigroups, which are a particular subclass of Feller processes. Our standard reference for Levy processes is the monograph by Sato [27]. A Levy process (Yt)t 0 is a stochastically continuous random process with stationary and independent increments. The characteristic function of a Levy Received by the editors August 16, 2011 and, in revised form, October 31, 2011. 2010 Mathematics Subject Classification. Primary 60J25, 60J75, 35S05.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2b45a345d50f64cb7e6e9d28e7a0e658c93e3da4

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In Finance

  1. J. Masamune
  2. Jian Wang
  3. T. Uemura

In Mathematical Physics

  1. Jian Wang
  2. J. Masamune
  3. T. Uemura
  4. M. Folz
  5. Xueping Huang
  6. Matthias Keller
  7. Radoslaw K. Wojciechowski

In Statistics and Probability

  1. T. Uemura
  2. Jian Wang
  3. J. Masamune

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