Dates: 13th and 14th August 2015
This ACEMS workshop concentrated on the theory and applications of special functions with particular emphasis on how they arise in stochastic processes.
The 2 day workshop took place at The University of Melbourne, University College.
Below are some pictures of the event, as well as a list of speakers who presented, along with the topics they covered.
I will give an overview how representations of the quantum diffusion algebra for the three parameter Asymmetric simple Exclusion Process is related to the bi-orthogonality of a pair of polynomial sequences and to the Al-Salam-Chihara orthogonal polynomials.
We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two dimensional case.
Jan De Gier (Melbourne) Weakly asymmetric exclusion process, the KPZ equation and the delta Bose gas
I will discuss recent developments describing the relationship between the Kardar-Parisi-Zhang equation of 1+1 dimensional growth, a guise of the stochastic heat equation, and the asymmetric exclusion process and integrable delta Bose gase. This connection leads to an exact solution of the former through dualities relating integrable stochastic models.
Researchers have in recent years developed stochastic geometry models to better understand mobile phone networks. We will discuss how a point process known as the signal-to-interference ratio (SIR) process, based on one of these models, turns out to be directly related to a special case of the two-parameter Poisson-Dirichlet process, which has been
extensively in mathematics and has applications in various fields. The aforementioned SIR process and functions of it can be studied via its factorial moment measure, which involves a multi-dimensional integral, and give methods for estimating coverage in mobile phone networks.
extensively in mathematics and has applications in various fields. The aforementioned SIR process and functions of it can be studied via its factorial moment measure, which involves a multi-dimensional integral, and give methods for estimating coverage in mobile phone networks.
First I will describe the Hermite polynomial identities for continuous martingales, then an extension to Hermite functions, and then I will describe other extensions to the case of discontinuous martingales which involve the Appell functions. I will discuss also some applications of the polynomials/special functions to boundary crossing and optimal stopping problems.
Birth-death processes are integer-valued continuous-time Markov processes that permit upward and downward jumps of size 1. I will present a key formula from the theory of birth-death processes that expresses their transition probabilities in terms of an orthogonal polynomial system. This formula is used to derive various properties, including the distribution of extinction times and quasi-stationary distributions. I will speculate on how the formula might be extended to cover general continuous-time Markov processes.
Stein's method is a powerful tool for providing errors in the approximation of a complicated probability distribution of interest by a well-understood target distribution. For example, the method can be gainfully applied to the classical setting of approximating the distribution of a sample mean by a Gaussian. But the strength of the method is in its use in non-classical situations when approximating distributions arising from stochastic systems such as random networks. Differential and difference equations feature prominently in Stein's method and so special functions can appear. In this talk I will discuss some applications of Stein's method where special functions play a role.
I define a class of Markovian random graph processes on a fixed number of vertices, evolving continuously in time through the arrival, departure, and transposition of single edges. By partitioning the state space by number of edges, these processes can be viewed as level-dependent quasi-birth and death processes. As with birth-death processes, one may seek to express the transition probabilities of these level-dependent quasi-birth and death processes in terms of an orthogonal polynomial system. I show that this can be achieved in a very simple way when transition rates are constant per level, permitting one to derive various properties of these processes. Finally, I will discuss the challenge of extending to cases in which transition rates are not constant per level.
We are interested in queues with a nite capacity C, where, on \arrival", customers make a reservation for service at some time in the future. In particular, we are interested in the `rejection probability' that the entire capacity C has already been allocated at some point during a customer's desired service time, in which case the customer's request cannot be accommodated.
It appears to be an intractable problem to calculate this probability for a system in which capacity is genuinely taken to be nite, so we adopt the approach of calculating the probability that the number of customers in the corresponding innite-capacity queue exceeds C. This probability serves as an upper bound for the nite-server rejection probability and, for a well-dimensioned system, we hope that it is a good approximation. Our main result
is that the stationary `bookings diary' for the innite-capacity queue has the same distribution as the law of entire sample paths of a specied innite-server queue, which we call the `bookings queue'. This queue has distributions for the initial number of customers and the initial residual service times that depend on the service time distribution
of the original reservation queue, and a non-homogeneous Poisson arrival process with almost surely nitely-many customers in total that depends on the reservation distribution in the original queue.
The focus of this talk will be on the derivations that we need to carry out when we use this result to calculate the innite-server bounds to the rejection probabilities in two special cases: when the reservation distribution is a two-point mixture of deterministic distributions, and the service-time distribution is either exponential or deterministic.
Both of the these examples lead to expressions that involve special functions.
It appears to be an intractable problem to calculate this probability for a system in which capacity is genuinely taken to be nite, so we adopt the approach of calculating the probability that the number of customers in the corresponding innite-capacity queue exceeds C. This probability serves as an upper bound for the nite-server rejection probability and, for a well-dimensioned system, we hope that it is a good approximation. Our main result
is that the stationary `bookings diary' for the innite-capacity queue has the same distribution as the law of entire sample paths of a specied innite-server queue, which we call the `bookings queue'. This queue has distributions for the initial number of customers and the initial residual service times that depend on the service time distribution
of the original reservation queue, and a non-homogeneous Poisson arrival process with almost surely nitely-many customers in total that depends on the reservation distribution in the original queue.
The focus of this talk will be on the derivations that we need to carry out when we use this result to calculate the innite-server bounds to the rejection probabilities in two special cases: when the reservation distribution is a two-point mixture of deterministic distributions, and the service-time distribution is either exponential or deterministic.
Both of the these examples lead to expressions that involve special functions.
I will focus on arithmetic features of (generalised) uniform short random walk; in particular, on its hypergeometric representations, modular parameterisations and links to the objects from the title.
I will first describe how finite reflection groups can be used to define multidimensional analogues of the classical Gaussian integral. I will then discuss some recent work on a discrete analogue of the construction.
I will discuss an inhomogeneous multi-species asymmetric exclusion process, and identify its steady state configuration probabilities with a family of non-symmetric polynomials. I will then consider the normalization of these probabilities (obtained by summing over all possible configurations) and show that it is equal to a symmetric Macdonald polynomial.
Workshop Timetable:
Thu 13 Aug | Fri 14 Aug | |
09:15 | Welcome | |
09:30 | Brak | Novikov |
10:15 | Ross | De Gier |
11:00 | Morning tea | Morning tea |
11:30 | Borwein | Taylor |
12:15 | Pollett | Warnaar |
13:00 | Lunch | Lunch |
14:00 | Wheeler | Taimre |
14:45 | Zudilin | Keeler |
15:30 | Afternoon tea | Afternoon tea |
Organisers:
Jan De Gier (jdgier@unimelb.edu.au) and Phil Pollett (pkp@maths.uq.edu.au)
Location: The University of Melbourne, University College, 40 College Crescent, Parkville 3052.