Brownian motion on time scales, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by L\'evy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary unbounded closed subset of the line: such a process will a martingale, has the identity function as its quadratic variation process, and is ``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 Feller-Dynkin Markov process, and find its generator. We then consider the special case where the state space is the self-similar set $\{\pm q^k : k \in \Z\} \cup \{0\}$ 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 $q$-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. We find the entrance laws of the corresponding It\^o excursion measure and the Laplace exponent of the inverse local time -- both again in terms of basic hypergeometric functions -- and hence obtain explicit formulae for the resolvent of the process.