Chair & organiser: Prof Dan Crisan
15:00 – 15:20
Dr Eyal Neumann
The radius of a self-repelling star polymer
We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes N Brownian motions up to time T, started at the origin and subject to exponential penalization based on the amount of time they spend close to each other, or close to themselves. The effective radius measures the typical distance from the origin. Our main result gives estimates for the effective radius where in two and three dimensions we impose the restriction that T ≤ N. One of the highlights of our results is that in two dimensions, we find that the radius is proportional to T^{3/4}, up to logarithmiccorrections. Our result may shed light on the well-known conjecture that for a single selfavoiding random walk in two dimensions, the end-to-end distance up to time T is roughly T^{3/4}. The talk is based on a joint work with Carl Mueller.
15:20 – 15:40
Prof Greg Pavliotis
Weakly interacting diffusions: Dynamical metastability, phase transitions and coalescing Brownian motions
I will give a brief presentation of current research on the qualitative behavior of weakly interacting diffusions that exhibit discontinuous phase transitions in their mean field limit. At intermediate time scales, and when finite number of particles effects are taken into account, the dynamics of such systems is characterized by clusters. In the long time limit, these clusters merge to form a single “droplet”. I will explain how the Dean-Kawasaki SPDE can be used in order to study this dynamics, I will also explain the link between the dynamics of clusters and the problem of coalescing Brownian motion (massive Arratia flow).
15:40 – 16:00
Prof Johannes Muhle-Karbe
Stochastic Liquidity as a Proxy for Nonlinear Price Impact
Optimal execution and trading algorithms rely on price impact models, like the propagator model, to quantify trading costs. Empirically, price impact is concave in trade sizes, leading to nonlinear models for which optimization problems are intractable and even qualitative properties such as price manipulation are poorly understood. However, we show that in the diffusion limit of small and frequent orders, the nonlinear model converges to a tractable linear model. In this high-frequency limit, a stochastic liquidity parameter approximates the original impact function’s nonlinearity. We illustrate the approximation’s practical performance using limit-order data. Joint work with Zexin Wang and Kevin Webster. PAPER: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4286108.
16:00 – 16:20
Dr Ajay Chandra
Global in time existence for the generalised parabolic Anderson model
While our understanding of local theory for parabolic singular SPDE is fairly mature, we still remain unable to prove global well-posedness for many equations where one would hope for it to hold. In this talk we’ll discuss progress on this for the generalised parabolic Anderson model which, like the $\Phi^4_3$ equation, is straightforward to estimate when not rough but is much more difficult in the rough setting. However, unlike the $\Phi^4_3$ equation, we will not have a strong damping term to help us with the difficulties appearing in the rough setting. This is joint work with Guilherme Feltes and Hendrik Weber.
16:20 – 16:40
Prof Philip Ernst
Quickest real-time detection of multiple Brownian drifts
16:40 – 17:00
Prof Jack Jacquier
Transportation-cost inequalities for non-linear Gaussian functionals
We study concentration properties for laws of non-linear Gaussian functionals on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour which are beyond the reach of Talagrand’s classical Transportation-Cost Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and relying on a suitable contraction principle, we prove generalised TCIs for functionals that arise in the theory of regularity structures and, in particular, in the cases of rough volatility and the two-dimensional Parabolic Anderson Model. In doing so, we also extend existing results on TCIs for diffusions driven by Gaussian processes. Link: https://arxiv.org/abs/2310.05750.