Penalizing null recurrent diffusions

Christophe Profeta

doi:10.1214/ejp.v17-2267

Abstract

We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level $a$ up to time $t$) of a large class of null recurrent diffusions. Our results rely on hypotheses on the Lévy measure of the diffusion inverse local time at 0. As a special case, we recover some of the penalization results obtained by Najnudel, Roynette and Yor in the (reflected) Brownian setting.

Highlights

1.1 A few notationWe consider a linear regular null recurrent diffusion (Xt, t ≥ 0) taking values in R+, with 0 an instantaneously reflecting boundary and +∞ a natural boundary
For every a ∈ R+, (Lat, t ≥ 0) the local time of X at a, with the normalization: Lat
We say that the process (Γt, t ≥ 0) satisfies the penalization principle if there exists a probability measure Q(xΓ) defined on (Ω, F∞) such that:

Summary

A few notation

We consider a linear regular null recurrent diffusion (Xt, t ≥ 0) taking values in R+, with 0 an instantaneously reflecting boundary and +∞ a natural boundary. Ω := C(R+ → R+) and we denote by (Ft, t ≥ 0) its natural filtration, with F∞ := Ft. We denote by s its scale function, with the normalization s(0) = 0, and by m(dx) its speed measure, which is assumed to have no atoms. It is known that (Xt, t ≥ 0) admits a transition density q(t, x, y) with respect to m, which is jointly continuous and symmetric in x and y, that is: q(t, x, y) = q(t, y, x). This allows us to define, for λ > 0, the resolvent kernel of X by:.

Motivations

If ν belongs to class R:

Organization The remainder of the paper is organized as follows:

Preliminaries

Asymptotic of an integrated convolution product

Preliminaries: a meta-theorem and some notations