## Workshop on 'Mathematical modelling of random multicomponent systems'

This meeting is part of a series of meeting partially supported by an LMS Joint Research Groups in the UK - Scheme 3 grant https://www.york.ac.uk/maths/research/mathematical-finance-stochastic-analysis/mmrms/. Main aim of our research group is to develop a mathematical framework for the study of a class of stochastic systems of a large number of elements (called "particles") described by their "positions" and "marks" (representing internal structure of the corresponding particles). The particles are coupled via (random) interaction potentials. Examples of systems of such type can be found in a wide variety of sciences, including quantum physics, astrophysics, chemical physics, biology, ecology, computer science, economics, finance, etc.

Location: Mathematics Building M100

Friday 24th of November

13:00-13:40:   Horatio Boedihardjo (Reading) "A discussion of some open problems in rough paths and Malliavin calculus"

14:00-14:40:   Eugene Lytvynov (Swansea) "An infinite dimensional umbral calculus"

15:00 -17:00:  Break

17:00-18:00:   Discussion

Dinner

Saturday 25th of November

09:30-10:10:  Alexei Daletskii (York) "Stochastic differential equations in a scale of Hilbert spaces"

10:20-11:00:  Sara Merino Aceituno (Imperial College) "A new flocking model through body attitude coordination"

11:10-11:30:  Tobias Kuna (Reading) "Characterisation of quasi-invariant measures via their Radon-Nikodym derivatives"

11:30-12:30:  Discussion

Titles and Abstracts

Title: "A new flocking model through body attitude coordination"
Sara Merino Aceituno (Imperial College London)

Abstract: We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model.
The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.
(In collaboration with Pierre Degond (Imperial College London), Amic Frouvelle (Université Paris Dauphine) and Ariane Trescases (University of Cambridge))

Title: "Stochastic differential equations in a scale of Hilbert spaces".
Alexei Daletzkii (York)

Abstract:  A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions are proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process.

Title: "A discussion of some open problems in rough paths and Malliavin calculus"

Abstract: In the past two years, there has been a lot of interests in describing ordinary and partial differential equations with noise more irregular than white noise. The existence and uniqueness results for these equations have recently been by provided by the theories of rough paths and regularity structures. There are many open problems on describing their solutions. A key tool will be Malliavin's calculus. I will talk about some problems in this field that I have recently started working on.

Title: "Characterisation of quasi-invariant measures via their Radon-Nikodym derivatives"

Abstract:  Measures quasi-invariant under a group action are the proper generalisation of invariant measure in particular in infinite dimensions. Indeed, a classical result of Mackey shows that quasi-invariant measure on a homogenous space are uniquely identified by their cocycle of Radon-Nikodym derivatives for finite dimensional groups. This result is false in infinite dimensions but the quasi-invariant measure can be identified as Gibbs measures in the general sense. The special case of point processes is discussed. This a joint work with J. Goldin (Rutgers), Yu. Kondratiev (Bielefeld) and J. Silva (Madeira)

Title: "An infinite dimensional umbral calculus"
Eugene Lytvynov (Swansea)

Abstract:  The aim of this talk is to develop umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. Our theory has remarkable similarities to the classical setting of polynomials on $\mathbb R$. For example, the form of the generating function of a Sheffer sequence on $\mathcal D'$ is similar to the generating function of a Sheffer sequence on $\mathbb R$, albeit the constants appearing in the latter function are replaced in the former function by appropriate linear continuous operators. We construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional analysis and played there a fundamental role. (Joint with Dmitri Finkelshtein, Yuri Kondratiev, and Maria Joao Oliveira)

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