Dear Colleagues, We would like to invite you to the following SPASS seminar by Marco Rehmeier (U. Bielefeld) and Michele Aleandri (Sapienza), jointly organized by UniPi, SNS, UniFi and UniSi.

The seminars will take place on TUE, 28.2.2023 starting at 14:00 CET in Sala Seminari, Dipartimento di Matematica, Pisa. The seminars will also be broadcasted on Google Meet, follow the link below for the streaming.

The organizers, A. Agazzi, G. Bet, A. Caraceni, F. Grotto, G. Zanco https://sites.google.com/unipi.it/spass

Marco Rehmeier (Bielefeld University), 14:00 Title: On nonlinear Markov Processes in the sense of McKean Abstract: We study nonlinear Markov processes in the sense of McKean and present a large new class of examples. Our notion of nonlinear Markov property is in McKean's spirit, but more general in order to include examples of such processes whose one-dimensional time marginals solve a nonlinear parabolic PDE, such as Burgers' equation, the porous media equation, or variants of the latter with transport-type drift. We show that the associated nonlinear Markov process is given by path laws of weak solutions to a corresponding distribution-dependent stochastic differential equation whose coefficients depend singularly (i.e. Nemytskii-type) on its one-dimensional time marginals. Moreover, we show that also for general nonlinear Markov processes, their path laws are uniquely determined by one-dimensional time marginals of suitable associated conditional path laws. Furthermore, we characterize the extremality of the curves of the one-dimensional time marginals of our nonlinear Markov Processes in the class of all solutions to the associated linearized PDE and, this way, obtain new interesting results also for the classical linear case. This is joint work with Michael Röckner.

Michele Aleandri (Università di Roma La Sapienza), 15:00 Title: Opinion dynamics with Lotka-Volterra type interactions Abstract: We investigate a class of models for opinion dynamics in a population with two interacting families of individuals. Each family has an intrinsic mean field “Voter-like” dynamics which is influenced by interaction with the other family. The interaction terms describe a cooperative/conformist or competitive/nonconformist attitude of one family with respect to the other. We prove chaos propagation, i.e., we show that on any time interval [0; T], as the size of the system goes to infinity, each individual behaves independently of the others with transition rates driven by a macroscopic equation. We focus in particular on models with Lotka-Volterra type interactions, i.e., models with cooperative vs. competitive families. For these models, although the microscopic system is driven a.s. to consensus within each family, a periodic behaviour arises in the macroscopic scale. In order to describe fluctuations between the limiting periodic orbits, we identify a slow variable in the microscopic system and, through an averaging principle, we find a diffusion which describes the macroscopic dynamics of such variable on a larger time scale.