Neil O’Connell

Exponential functionals of brownian motion and class-one Whittaker functions

We consider exponential functionals of a Brownian motion with drift in R(n), defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrodinger-type partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes.

In Memoriam Marc Yor - Séminaire de Probabilités XLVII

How does one introduce randomness into a classical dynamical system in order to produce something which is related to the ‘corresponding’ quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems.

9. Moderate Deviations Scalings

9.1 Motivation 9.2 Traffic Processes 9.3 Queue Scalings 9.4 Shared Buffers 9.5 Mixed Limits

8. Long Range Dependence

8.1 What Is Long Range Dependence? 8.2 Implications for Queues 8.3 Sample Path LDP for Fractional Brownian Motion 8.4 Scaling Properties 8.5 How Does Long Range Dependence Arise? 8.6 Philosophical Difficulties with LRD Modelling

7. Many-Flows Scalings

7.1 Traffic Scaling 7.2 Topology for Sample Paths 7.3 The Sample Path LDP 7.4 Example Sample Path LDPs 7.5 Applying the Contraction Principle 7.6 Queues with Infinite Buffers 7.7 Queues with Finite Buffers 7.8 Overflow and Underflow 7.9 Paths to Overflow 7.10 Priority Queues 7.11 Departures from a Queue

6. Large-Buffer Scalings

6.1 The Space of Input Processes 6.2 Large Deviations for Partial Sums Processes 6.3 Linear Geodesics 6.4 Queues with Infinite Buffers 6.5 Queues with Finite Buffers 6.6 Queueing Delay 6.7 Departure Process 6.8 Mean Rate of Departures 6.9 Quasi-Reversibility 6.10 Scaling Properties of Networks 6.11 Statistical Inference for the Tail-Behaviour of Queues

5. Continuous Queueing Maps

5.1 Introduction 5.2 An Example: Queues with Large Buffers 5.3 The Continuous Mapping Approach 5.4 Continuous Functions 5.5 Some Convenient Notation 5.6 Queues with Infinite Buffers 5.7 Queues with Finite Buffers 5.8 Queueing Delay 5.9 Priority Queues 5.10 Processor Sharing 5.11 Departures from a Queue 5.12 Conclusion

4. Introduction to Abstract Large Deviations

4.1 Topology and Metric Spaces 4.2 Definition of LDP 4.3 The Contraction Principle 4.4 Other Useful LDP Results

3. More on the Single Server Queue

3.1 Queues with Correlated Inputs 3.2 Queues with Many Sources and Power-Law Scalings 3.3 Queues with Large Buffers and Power-Law Scalings

2. Large Deviations in Euclidean Spaces

2.1 Some Examples 2.2 Principle of the Largest Term 2.3 Large Deviations Principle 2.4 Cumulant Generating Functions 2.5 Convex Duality 2.6 Cramer’s Theorem 2.7 Sanov’s Theorem for Finite Alphabets 2.8 A Generalisation of Cramer’s Theorem