# 4th year research projects

**Title**: Metastability and large deviations for system of SDE's.

**Description**: an interesting phenomenon has been observed recently for systems of differential equations inspired by hydrodynamics: the addition of noise to a system of ODE's with a single fixed point leads to the emergence of multiple (quasi) fixed points. The aim of the project is to study classes of such ODE's and possibly apply the findings to the study of metastability in turbulence.

**Key words and phrases**: metastability; stochastic differential equations; time scale separation; large deviations; Wentzel-Freidlin theory; instanton trajectory; Fredholm determinants; Szego's theorems

**Title**: Matrix valued Brownian motions

**Description**: Random matrix theory is a multi-disciplinary subject possessing an intrinsic mathematical beauty and having applications to a range of applied sciences from string theory to data science and statistics. A dynamical version of random matrix theory is matrix valued Brownian motion. For the Hermitian matrices this gives rise to such a classical stochastic process as Dyson Brownian motion. The aim of the project is to study random matrix evolution for the non-Hermitian case.

**Key words and phrases**: matrix-valued Brownian motion; Berezin calculus; supersymmetry; determinant; pfaffian; stochastic differential equations; point processes

**Title**: Markov dualities and interacting particle systems in one dimension

**Description**: more often than not, Markov interacting particle systems in one dimensions exhibit strong fluctuations which render their approximate description using differential equations ('mean field theory') useless. One of the methods of analysing such systems is based on Markov duality which allows to extract at least partial information about the system from its dual. But how do you find these dualities? The aim of the project is to learn about Markov dualities and investigate the dualities for certain classes of strongly fluctuating interacting particle systems.

**Key words and phrases**: Markov processes in continuous time, interacting particle systems, Markov duality; Hecke algebras; Yang-Baxter equation; R-matrix; determinantal point processes; pfaffian point processes

**Title**: Moment factorisation for the stochastic heat equation.

**Description**: it has been discovered recently, that the exponential moments of the solution to the stochastic partial differential heat equation satisfy classical integrable PDE's such as Kadomtsev-Petviashvili equaitons. The discovery came from the analysis of exact formulae for the exponential moments available for certain initial conditions. One of the consequences is that certain moments for the stochastic heat equations must factor into the linear combinations of products of the lower order moments. The aim of the project is to investigate the factorisation phenomenon from the point of view of stochastic analysis studying the solutions to the stochastic heat equation itself.

**Keywords and phrases**: Stochastic heat equation: stochastic processes, Ito's calculus; KPZ equation; KP integrable hierarchy; Fredholm determinants; intermittency; moment problem