$Λ$-linked coupling for drifting Brownian motions

Supremum distribution of Bessel process of drifting Brownian motion

Let (B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t + μt) be a three-dimensional Brownian motion with drift μ, starting at the origin. Then X_t = ||(B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t +μt)||, its distance from the starting point, is a diffusion with many applications. We investigate the distribution of the supremum of (X_t), give an infinite-series formula for its density and an exact estimate by elementary functions.

Optimal real-time detection of a drifting Brownian coordinate

Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a non-zero drift. Given that the position of the Brownian particle is being observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the non-zero drift. We solve this problem in the Bayesian formulation, under any prior probabilities of the non-zero drift being in any of t…

A matrix Bougerol identity and the Hua-Pickrell measures

We prove a Hermitian matrix version of Bougerol's identity. Moreover, we construct the Hua-Pickrell measures on Hermitian matrices, as stochastic integrals with respect to a drifting Hermitian Brownian motion and with an integrand involving a conjugation by an independent, matrix analogue of the exponential of a complex Brownian motion with drift.

Lévy processes linked to the lower-incomplete gamma function

We start by defining a subordinator by means of the lower-incomplete gamma function. It can be considered as an approximation of the stable subordinator, easier to be handled thank to its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes time-changed by these subordinators, with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fr…

Growth of the Brownian forest

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton--Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams' decomposition for Brownian motion with drift.

退化反射布朗运动在楔形中的马丁边界

Martin boundary of a degenerate Reflected Brownian Motion in a wedge

本文研究退化反射布朗运动在楔形中的马丁边界，通过计算格林函数的拉普拉斯变换并推导其渐进行为，确定了最小马丁边界及相应的调和函数。