## Probability in the North East workshop

#### 24 February 2023

Organizers: Ilya Chevyrev (Edinburgh) and Fraser Daly (Heriot Watt).

**Venue: **Appleton Tower, Room 2.14, University of Edinburgh or by Zoom.

Attendence is free but registration is **required for organisational purposes**, by emailing Ilya Chevyrev by Thursday 16th February 2023.

Please indicate whether you plan to attend in person or via Zoom. If you plan to attend in person, please indicate any special dietary requirements you may have.

## Programme

Stein method is a three-step program to measure distance between a generic probability measure of that a given target probability distribution. The first step is to find a suitable (differential) Stein operator that characterises the target distribution. In this talk, we introduce the novel notion of an algebraic polynomial Stein operator and show that every polynomial Stein operator of a Gaussian polynomial target distribution in dimension one is algebraic. This opens a new door to find algorithmically Stein operators of the complex target distributions in higher Wiener chaoses. We also discuss the class PSO(N) of the polynomial Stein operators associated to the standard Gaussian distribution and how it connects Stein method to non-commutative algebra. The talk is based on a series of joint works with Dario Gasbarra (Helsinki) and Robert Gaunt (Manchester).

In this talk I will talk about the analytical properties of a class of stochastic shallow water (SRSW) models which includes SRSW models derived using the Location Uncertainty (Mémin, 2014) approach. We prove that there exists a unique maximal strong solution and a unique global weak solution, using a method which relies on a Cauchy approximating sequence. We show that although the strong solution exists in a high order Sobolev space, it suffices to prove the Cauchy property in L^2. Comparisons with other types of SRSW models will also be made, and we will discuss conditions under which the underlying strong solution can become global with positive probability.

This is joint work with Dan Crisan and Étienne Mémin and is partly based on Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty , to appear in Journal of Mathematical Fluid Mechanics.