4:15 pm Wednesday, March 6, 2013
Probability: Iterating Brownian motions, ad libitum by
Takis Konstantopoulos [mail] (Department of Mathematics, Uppsala University) in RLM 11.176
Let be independent one-dimensional Brownian motions parametrized by the whole real line such that for all . We consider the nth iterated Brownian motion . Such a process first appeared in a paper by Funaki as a probabilistic con- struction of the solution of higher order diffusion equations. The process received some attention by the probability community (Burdzy, Khoshnevisan, Shi, Eisenbaum, Bertoin) in relation to the pathwise properties of this non-semimartingale process. In this talk, we will review some of the fascinating properties of such as Burdzy’s result that, given a path of , one can recover the paths up to a sign. We will then show that has a limit, in a certain very weak sense, as tends to infinity, which is exchangeable and, therefore, by the de Finetti-Hewitt-Savage theorem, a conditionally independent collection of random variables. We identify the object, , we condition on as being the limit of the random occupation measures, on [0, 1], of . It turns out that has almost surely finite support and continuous (random) density. The limiting marginal distributions have some rather curious properties, but only the 1-dimensional ones are explicitly known (from which we can obtain a new characterization of the exponential distribution). We will conclude by some open problems and by providing a glimpse towards the relation of this problem to the work of Beghin and Orsingher on fractional diffusion equations. This talk is based on joint work with Nicolas Curien, Universit\'e Paris VI. Submitted by
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