Vinita Mulay (University Duisburg-Essen)
Elephant Random Walk with Tampered Memory

Abstract: The Elephant Random Walk (ERW), introduced by Schütz and Trimper (2004), has gained a lot of attention in recent years. We consider a memory-based variation of the classical ERW. The aim is to understand how much memory is 'enough' to observe an ERW-like behaviour.

In our model, we partition the memory into two subsets, $D_n$ and $D_n^c$, such that the elephant behaves like the classical ERW model when a step is chosen from $D_n^c$ and behaves differently when it is chosen from $D_n$. We illustrate that in order to obtain a phase transition, the size of $D_n^c$ needs to be more than half the entire memory. This is joint work with Neeraja Sahasrabudhe (IISER Mohali) and Debleena Thacker (Durham University).

Mikhail Urusov (University Duisburg-Essen)
Representation property for 1d general diffusion semimartingales.

Abstract: A general diffusion semimartingale is a 1d continuous semimartingale that is also a regular strong Markov process. The class of general diffusion semimartingales is a natural generalization of the class of (weak) solutions to SDEs. A continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingale part. We show that the representation property holds for a general diffusion semimartingale if and only if its scale function is (locally) absolutely continuous in the interior of the state space.
Surprisingly, this means that not all general diffusion semimartingales possess the representation property, which is in contrast to the SDE case. Furthermore, it follows that the laws of general diffusion semimartingales with absolutely continuous scale functions are extreme points of their semimartingale problems. We construct a general diffusion semimartingale whose law is not an extreme point of its semimartingale problem. This contributes to the solution of the problems posed by Jacod and Yor and by Stroock and Yor on the extremality of strong Markov solutions (to martingale problems). This is a joint work with David Criens.

Anton Klimovsky (University Würzburg)
Opinion dynamics on evolving networks: a measure-valued Markov process and its scaling limit

Abstract: We study a voter-type opinion dynamics on a network whose community structure evolves endogenously through two simple mechanisms: individuals may migrate between communities (“poaching”) or split off to form singleton communities (“self-employment”), while opinions evolve by within-community resampling.
Using a two-level measure-valued representation -- communities as integer-valued measures on the opinion set and the network as a point measure on community measures -- we obtain an explicit generator and natural polynomial sampling observables.
In a diffusion scaling limit for two opinions, we derive a coupled SDE: the global opinion frequency follows a Wright–Fisher-type diffusion with stochastic volatility driven by an evolving homozygosity statistic of the community partition.