Sojourn times of diffusion processes

Aspects of Brownian Motion

“…The Ciesielski-Taylor identities in law, which we shall study in this Chapter, were published in 1962, that is one year before the publication of the papers of Ray and Knight (1963; [80] and [57]) on Brownian local times; as we shall see below, this is more than a mere coincidence!…”

“…There is no easy formulation of a Ray-Knight theorem for Brownian local times taken at a fixed time t (see Perkins [68] and Jeulin [52], who have independently obtained a semimartingale decomposition of the local times in the space variable); the situation is much easier when the fixed time is replaced by an independent exponential time, as is explained briefly in paragraph 3.4, following Biane-Yor [19]; the original result is due to Ray [80], but it is presented in a very different form than Theorem 3.5 in the present chapter.…”

The most visited sites of symmetric stable processes

“…In the case α = 2, Theorem B takes the form of the additivity property of squared Bessel processes, which is an equivalent form of the usual Ray-Knight theorem for Brownian local time at τ (1); see Ray [15], Knight [10]. In the case α = 2, Theorem B takes the form of the additivity property of squared Bessel processes, which is an equivalent form of the usual Ray-Knight theorem for Brownian local time at τ (1); see Ray [15], Knight [10].…”

Distributions of Functionals of the Brownian Local Time I

“…Before going over to the proofs of Theorems 3.1-3.8, we adduce the results of Ray and Knight [11], [12] characterizing t(t, y) as a process with respect to y. These results are used to prove Theorems 3.1-3.8 and will be formulated as befits our objective.…”