Tree structured independence for exponential Brownian functionals

The product of GIG and gamma distributions is preserved under the transformation (x,y)↦((x+y)−1,x−1−(x+y)−1)(x,y)↦((x+y)−1,x−1−(x+y)−1). It is also known that this independence property may be reformulated and extended to an analogous property on trees. The purpose of this article is to show the independence property on trees, which was originally derived outside the framework of stochastic processes, in terms of a family of exponential Brownian functionals.