Reflected stochastic systems – RESYST

We are interested in reflected stochastic processes involved in several systems of queueing networks. These stochastic models have been developed due to their numerous applications in operations research, risk theory, telecommunication, data sciences and also in population biology.

The issues raised by such stochastic systems are intertwined with many fields of mathematics (probability, complex analysis, combinatorics, differential Galois theory).

A typical example of such a continuous process is obliquely reflected Brownian motion in an orthant. Their discrete analogues, random walks in cones, are also very famous processes. There are a wide variety of interesting questions regarding these reflected processes which can be either recurrent or transient. Among these questions there are the study of invariant measures, Green's functions, time to reach an edge or the vertex, algebraic nature of the generating function, harmonic functions and Martin boundary associated to the process.

A now standard method to study these problems is to establish kernel functional equations involving generating functions. The project intends to decompartmentalize certain approaches by mixing three different techniques used to solve these functional equations: Boundary value problems, Tutte's invariant approach and q-difference equations.

The project's objectives are to study:
1. Persistence, extinction and quasi-stationary distribution in a cone.
2. Ruin and escape for transient process in an orthant.
3. Monte Carlo simulation for Brownian motion in cones.
4. Additive and multiplicative decoupling for Tutte's invariant approach.
5. Transition kernel and space-time functional equation.

The funding would be mostly used to fund two years of postdoctoral fellowship(s) and to organize an international conference.

The scientific coordinator's team will be made up of four complementary dynamic young researchers: Pierre Bras, Thomas Dreyfus, Andrew Elvey Price and Sofia Tarricone.