q-deformed Brownian motion

Catherine Donati-Martin

Claudio Albanese, Stephan Lawi

We extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. As examples, we considered two versions of the q-deformed Harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules.

Charlotte Bigg

Stephan Lawi

Generating functions for integrals of stochastic processes have been known in analytically closed form for just a handful of processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In the first part of the thesis, broad extensions of these process classes are constructed. The known models are shown to fit into a classification scheme for diffusion processes for which generating functions for integrals of the process and transition probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. This scheme also extends to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. We also extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. Stochastic subordination preserves the Markov property of the process, whereas quantum subordination preserves quantum probability in the dynamics. As applications, we consider two versions of the q-deformed harmonic oscillator in both ordinary and imaginary time and show how the related q-deformed algebras can be understood as different patterns of time quantization rules. The second part of the thesis concerns the analogy between q -deformation and stochastic subordination. It is shown that the Poisson kernel of stochastic processes on chains of arbitrary length can be expressed in terms of the q-Racah polynomials, the most general q-deformed orthogonal polynomials in the discrete series of the Askey scheme. We give a new interpretation of this kernel as the transition probability density for a subordinated Markov process with only nearest neighbor hops. As an application, we give an elementary proof and extend a positivity result for a class of Poisson kernels which Gasper and Rahman established with direct methods.

R. L. Hudson

This is a personal survey of aspects of quantum probability related to the Heisenberg commutation relation for canonical pairs. Using the failure, in general, of non-negativity of the Wigner distribution for canonical pairs to motivate a more satisfactory quantum notion of joint distribution, we visit a central limit theorem for such pairs and a resulting family of quantum planar Brownian motions which deform the classical planar Brownian motion, together with a corresponding family of quantum stochastic areas.This article is part of the themed issue 'Hilbert's sixth problem'.

Horacio M. Cataldo