# South Wales Analysis and Probability Seminar (SWAP)

## 27 September 2019 at Swansea University (poster)

 10:30-11:00 Coffee & Welcome 11:00-11:30 Chenggui Yuan (Swansea) Comparison Theorem of Stochastic Differential Equations 11:35-12:05 Kirstin Strokorb (Cardiff) Geometry of extreme values and max-stable processes 12:10-12:40 Zeev Sobol (Swansea) Time Dependent Dirichlet forms and associated Markov processes 12:40-14:15 Lunch 14:15-14:45 Ermal Feleqi (Vlorë, Albania) Ergodic Mean Field Games with Hörmander diffusions 14:50-15:20 Guang-an Zou (Henan, China) Some results on stochastic Navier-Stokes equations

## Abstracts

Chenggui Yuan (Swansea) - Comparison Theorem of Stochastic Differential Equations

In this talk, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution independent case, and the necessity of these conditions is new even in distribution independent case.

Kirstin Strokorb (Cardiff) - Geometry of extreme values and max-stable processes

Many fundamental results in probability and statistics are based on the observation that approximating limiting objects satisfy some form of stability, the most well-known being the normal distribution which arises as the limit of sums (or averages). When particular interest lies on extreme values, limit laws of threshold exceedances or maxima come into play and the respective limiting objects are stable with respect to taking exceedances or taking maxima, respectively. In the past decade max-stable processes have gained much interest in connection with the annual maxima method in extreme value theory in order to assess the uncertainty of spatial extremal scenarios. Also from a probabilistic point of view, several structural results exist for max-stable random vectors and processes.

In this talk I will give an overview over some of these findings and reconsider max-stable processes from the perspective of random sup-measures. This has been started in the 1980s, but since then not received much attention anymore. I will show that this re-newed perspective offers a unifying approach to a number of structural results with functional analytic extensions and more streamlined proofs as well as some new insights. The talk is based on joint work with Ilya Molchanov.

Zeev Sobol (Swansea) - Time Dependent Dirichlet forms and associated Markov processes

The topic attracts an attention due to needs of Perturbation theory of propagators and Non Autonomous Markov processes. However, the Functional Analytic framework has been insufficient so far. Assumptions included time independence of the form domain or regular behaviour of the coefficients in time. In the result presented we remove these restrictions. With some approximation and perturbation results, the technique applied to construction of Markov processed with singular drifts.

Ermal Feleqi (Vlorë, Albania) - Ergodic Mean Field Games with Hörmander diffusions

I will talk about joint results with Federica Dragoni (Cardiff University, UK) on existence, uniqueness and regularity of solutions for a class of systems of subelliptic or hypoelliptic PDEs arising from ergodic Mean Field Game models with Hörmander diffusions. These results are applied to the feedback synthesis of Mean Field Game solutions and Nash equilibria of a large class of $$N$$-player differential games. In the first part of the talk I will give a brief introduction to this theory of Mean Field Games. It is a series of models introduced by J.-M. Lasry and P.-L. Lions in mid 2000s and, independently, by a group of engineers, in order to study phenomena arising from the competitive interaction of large numbers of small, similar and rational agents. It has applications in Economics and Finance (distribution of wealth and salaries, optimal consumption of resources, creation of volatility in financial markets) Dynamics of Populations (the Mexican wave “la ola”, racial segregation, traffic flows). Many classical equations of Mathematics and Physics appear as particular cases of MFG equations. This theory has attracted the interest of the mathematical community for it has posed certain deep problems in the theory of PDEs, especially in infinite-dimensional spaces.

Guang-an Zou (Henan, China) - Some results on stochastic Navier-Stokes equations

In this talk, we will consider the extended stochastic Navier–Stokes equations with Caputo derivative driven by fractional Brownian motion. Firstly, the pathwise spatial and temporal regularity of the generalized Ornstein–Uhlenbeck process is proved. Then we discuss the existence, uniqueness, and Hölder regularity of mild solutions to the given problem. Next, we present the vorticity-stream function method to compute the incompressible fluid flow, a Crank-Nicolson Fourier pseudo-spectral method is proposed for solving the formulation of stream function equation. Numerical results clearly exhibit that in the presence of stochastic forcing can affect the shapes of vortex in fluid flow. This enables us to perform optimal control experiments for the development of vortex structures.

## Organisers

Jonathan Ben-Artzi (Cardiff)
Dmitri Finkelshtein (Swansea)