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

Abstract

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 → 1 2 f ′′ . We then consider the special case where the state space is the self-similar set {±q k : 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 timeboth 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.

Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

“…Note that the way (8) in which we define the extension guarantees that the functioñ( , ) is progressively measurable. In [2] the authors then defined the solution of the Δ-stochastic differential equation indicated by the notation…”

“…on the time scale ∈ T. Let us consider an extension of the (probably random) function ( ) as in (8). Let us define the so obtained extension function to bẽ( ).…”

“…It is also worth mentioning that our construction is different from [8] in that we are working with the case that the time parameter of the process is running on a time scale, whereas in [8] and related works (e.g., [9][10][11]) the authors are working with the case that the state space of the process is a time scale.…”

Stability of switched linear systems on non-uniform time domains

“…While these are valuable tools, there is an increasing interest in non-uniform time domainsdomains which contain non-uniformly spaced discrete points or a mixture of discrete and continuous parts [4]. Applications include adaptive control [5], real-time communications networks [6], [7], dynamic programming [8], switched systems [9], control theory [10], [11], [12], [13], [14], [15], signal analysis [16], [17], stochastic models [18], population models [19], and economics [20], [21].…”

State feedback stabilization of linear time-varying systems on time scales

“…The goal is not to simply reprove existing, well-known theories, but rather to view R and Z as special cases of a single, overarching theory and to extend the theory to dynamical and control systems on these more general domains. Doing so reveals a rich mathematical structure which has great potential for new applications in diverse areas such as adaptive control [20], real-time communications networks [21], [22], dynamic programming [34], switched systems [30], stochastic models [5], population models [37], and economics [3], [4]. The focus of this paper is the study of linear state feedback controllers [28], [35] in this generalized setting and to compare and contrast these results with the standard continuous and uniform discrete scenarios.…”