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Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Shankar Bhamidi, Steven N. Evans, Ron Peled, and Peter Ralph

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Motivated by Lévy’s characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be “continuous” in the sense that its sample paths don’t skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator f↦½f''.

We then consider the special case where the state space is the self-similar set {±qk: k∈ℤ}∪{0} for some q>1. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan’s “lost” notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, q-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local time – both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve q-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

Chapter information

Deborah Nolan and Terry Speed, eds., Probability and Statistics: Essays in Honor of David A. Freedman (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 42-75

First available in Project Euclid: 7 April 2008

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Primary: 60J65: Brownian motion [See also 58J65] 60J75: Jump processes
Secondary: 30B70: Continued fractions [See also 11A55, 40A15] 30D15: Special classes of entire functions and growth estimates

birth and death process Chacon-Jamison theorem excursion Feller process local time q-binomial theorem q-exponential q-series quasidiffusion time change time-scale calculus

Copyright © 2008, Institute of Mathematical Statistics


Bhamidi, Shankar; Evans, Steven N.; Peled, Ron; Ralph, Peter. Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan. Probability and Statistics: Essays in Honor of David A. Freedman, 42--75, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000383.

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