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The classical Ray-Knight theorems for the Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by the Brownian motion. We extend these results by describing the local time process jointly for all a and b , by means of the stochastic integral with respect to an appropriate white noise. Our result applies to μ -processes, and has an immediate application: a μ -process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert (2002)), whereas a Feller CSBP with immigration satisfies a stochastic differential equation (SDE) driven by a white noise (Dawson and Li (2012)); our result gives an explicit relation between these two descriptions and shows that the SDE in question is a reformulation of Tanaka’s formula.

We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

In this paper, we study the inverse local times at 0 of one-dimensional reflected diffusions on [0, ∞) and establish a comparison principle for these inverse local times. We also provide applications to Green function estimates for non-local operators.

A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the drift and variance of its Brownian component discontinuously change at its finitely many states. This reflecting diffusion process naturally arises from a state-dependent single server queue, studied by the Miyazawa (Diffusion approximation of the stationary distribution of a two-level single server queue, 2024. https://arxiv.org/abs/2312.11284 ). Our main interest is in its stationary distribution, which is important for application. We define this reflecting diffusion process as the solution of a stochastic integral equation, and show that it uniquely exists in the weak sense. This result is also proved in a different way by Atar et al. (Parallel server systems under an extended heavy traffic condition: A lower bound, 2022. https://arxiv.org/pdf/2201.07855 ). In this paper, we consider its Harris irreducibility and stability, that is, positive recurrence, and derive its stationary distribution under this stability condition. The stationary distribution has a simple analytic expression, likely extendable to a more general state-dependent SRBM. Our proofs rely on the generalized Ito formula for a convex function and local time.