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

Excursion Measure Away from an Exit Boundary of One-Dimensional Diffusion Processes

“…This result unifies the framework of our generalized excursion measure in terms of θ with that of Knight [10], Kasahara-Watanabe [8] and Kotani [12] in terms of σ * .…”

“…Based on Krein's spectral theory (see, e.g., [3], [7] and [13]), Knight [10] and Kotani-Watanabe [13] have characterized the class of the Lévy measures of (η(t)) for one-dimensional generalized (or gap) diffusion processes. For a string m, the corresponding Lévy measure has a density ρ(u)…”

Subordinators: Examples and Applications

“…The measure 1 2 ν in Proposition 9.6(i) is then known as the spectral measure of the string, and the measure 2 ν in (ii) coincides with the spectral measure of the so-called dual string d F (x), where F is the right continuous inverse of the distribution function F (x) = x 0 f (x)dx. Krein's theory yields the following remarkable formulas for the Laplace exponent Φ of σ and the tail of its Lévy measure Π, which seem to have been first observed by Knight [101] (see also Kotani and Watanabe [102] and Küchler [103]). e −xξ ν(dξ) dx .…”

“…See in particular Bertoin [7], Kasahara [92], Kent [94,95], Knight [101], Kotani and Watanabe [102], Küchler [103], Küchler and Salminen [104], Tomisaki [149], Watanabe [150,151] and references therein.…”

Can time-homogeneous diffusions produce any distribution?

“…In this section we construct, following [6] (see also [11] and [12]), timehomogeneous generalised diffusion processes as time-changes of Brownian motion. The time-change is specified in terms of the so-called speed measure.…”