Penalization of a Positively Recurrent Diffusion by an Exponential Function of its Local Time

Abstract

Using Krein's theory of strings, we penalize here a large class of positively recurrent diffusions by an exponential function of their local time. After a brief study of the processes so penalized, we show that on this example the principle of penalization can be iterated, and that the family of probabilities we get forms a group. We conclude by an application to Bessel processes of dimension δ ∈ ]0, 2[ which are reflected at 1. Let P x and E x denote, respectively, the probability measure and the expectation associated with X when started from x ≥ 0. We assume that X is defined on the canonical space Ω := C(R + → R + ) (where R + := [0, +∞[), and we denote by (F t , t ≥ 0) its natural filtration, with F ∞ := t≥0 F t .Let us start by giving a definition of penalization (see also Theorem 3.1):Definition 1.1. Let (Γ t , t ≥ 0) be a measurable process taking positive values and such that 0 < E x [Γ t ] < ∞ for every t > 0 and every x ∈ I. We say that the process (Γ t , t ≥ 0) satisfies the penalization principle if there exists a probability measure Q x defined on (Ω, F ∞ ) such that

Penalizing null recurrent diffusions

“…Let us also mention that this kind of results no longer holds for positively recurrent diffusions. Indeed, it is shown in [Pro10] that if (X t , t ≥ 0) is a recurrent diffusion reflected on an interval, then, under mild assumptions, the penalization principle is satisfied by the functional (Γ t = e −αL 0 t , t ≥ 0) with α ∈ R, but unlike in Theorem 5, the penalized process so obtained remains a positively recurrent diffusion.…”

Local time penalizations with various clocks for one-dimensional diffusions

“…Such a result may be generalized to other reflected diffusions on [0, 1], under the assumption that the analytic continuation of H(q) is smaller (at infinity) than a negative power of |q| on a given strip on the complex plane, see Profeta [12].…”

“…This classical example on the reflected Brownian motion was generalized to many other processes: we may refer in particular to Debs [5] for random walks, Najnudel, Roynette and Yor [10] for Markov chains and Bessel processes, Yano, Yano and Yor [20] for stable processes, or Salminen and Vallois [17] and Profeta [11,12] for linear diffusions. In most of these papers, a (sometimes implicit but rather strong) condition is made on the considered process, basically stating that a given quantity is regularly varying.…”

On h-Transforms of One-Dimensional Diffusions Stopped upon Hitting Zero

“…2 • ). The penalization problems for one-dimensional diffusions which generalize Theorem 1.2 were studied in Profeta [19], [20].…”