Characterization of the Levy Measures of Inverse Local Times of Gap Diffusion

Abstract

Let X(t) be a persistent nonsingular diffusion on an interval Q containing 0 (in the sense of [4]). If we assume a natural scale, then X is characterized by its speed measure m(dx) on Q, and finite endpoints are reflecting. There exists (P⁰-a.s.) the local time

$$ l(t) = \mathop{{\lim }}\limits_{{{h_i} \to 0 + }} \frac{1}{{m[ - {h_1},{h_2})}}\int\limits_0^t {{I_{{[ - {h_1},{h_2})}}}(X(s))ds} $$

((1.1))

which is continuous in t ≥ 0. The right-continuous inverse ℓ^(-1)(t) = inf{s: ℓ(s) > t} is an increasing finite process with homogeneous independent increments, and as such (see for example [4, §6.1]) it is characterized by its Lévy measure n(dy) on (0,∞) through the equation

$$ E\exp [ - \lambda {l^{{( - 1)}}}(t)] = \exp [ - t{m_0}\lambda - t\int\limits_0^{\infty } {(1 - {e^{{ - \lambda y}}})n(dy)],\lambda > 0} $$

where m₀ = m{0}. An interesting problem [4, p. 217] is to characterize the class of all n(dy) which can appear when Q and m(dx) vary. It is shown there that there is a unique measure µ on [0,∞) such that $n(dy) = dy\int_{{0 - }}^{\infty } {{e^{{ - y\gamma }}}\mu (d\gamma )}$, so the problem reduces to characterizing the class of µ.