One dimensional diffusions conditioned to be non-explosive^{∗}

Ichiro Shigekawa^{†} (Kyoto University)

We consider one dimensional diffusions conditioned to be non-explosive. Suppose we are given a diffusion process {X_{t}, P_{x}} on a state space D. Let ζ be its explosion time. If P_{x}[ζ = ∞] > 0, then the measure conditioned to be non-explosive is defied by

P_{x}[ · | ζ = ∞] = P_{x}[ · ∩ ζ = ∞]/P_{x}[ζ = ∞].

If P_{x}[ζ = ∞] = 0, then the measure conditioned to be non-explosive is defined as the limit

lim_{T →∞} P_{x}[ · | ζ > T].  (1)

If the limit exists and the limit is a diffufin process, we call it a surviving diffusion. We are interested in the following problems:

1. When does the surviving diffusion?

2. Characterizasion of the surviving diffusion.

Since

E_{x}[ · | ζ > T] = E_{x}[ · \frac{1_{\{ζ>t\}}P_{X_{t}}[ζ > T − t]]}{P_{x}[ζ > T]}],

our problem is reduce to show the existence of the limit

M_{t} = lim_{T →∞} \frac{1_{\{ζ>t\}}E_{X_{t}}[ζ > T − t]}{P_{x}[ζ > T]}  (2)

and to show that (M_{t}) is a martingale. To do this, we show that there exist a λ-harmonic function φ so that

lim_{T →∞} \frac{P_{y}[ζ > T − t]}{P_{x}[ζ > T]} = \frac{φ(y)e^{−λt}}{φ(x)} .  (3)

We consider one dimensional (minimal) diffusions on an interval (l_{1}, l_{2}). The problem is divided into sevral cases according to the classification of the boundary. The the surviving diffusion is characterized as a h-transform of the original process by the λ-harmonic function φ.

∗February 21-24, 2006, “Stochastic Analysis for Markov processes and its applications” at Kansai University

†E-mail: ichiro@math.kyoto-u.ac.jp URL: http://www.math.kyoto-u.ac.jp/~ichiro