KEYWORD |

#### Well-posedness of multi-agent and multi-population dynamics with diffusive effects

keywords CONTINUITY EQUATION, DIFFUSION, FOKKER-PLANCK EQUATION, MEAN-FIELD LIMIT, MULTI-AGENT SYSTEMS, STOCHASTIC DYNAMICS, NOISE-INDUCED PHENOMENA

Reference persons MARCO MORANDOTTI

External reference persons Anderson Melchor Hernandez (DISMA)

Research Groups Analisi non lineare e calcolo delle variazioni

Thesis type RESEARCH

Description This thesis aims at proposing a model for multi-agent dynamics with strategies, including a stochastic term in the form of additive noise.

In the paper "Spatially Inhomogeneous Evolutionary Games" (Ambrosio et al., CPAM 2021), a space-dependent version of the replicator dynamics is studied both from the particle point of view (by considering a system of N agents) and from the mean-field point of view (by studying the continuity equation arising in the limit as N tends to infinity). In the particle model, the micro-state of the N agents is described by their spatial positions and by a probability measure on a set of strategies, which both determines the spatial evolution of each agent and evolves in time through a binary non-local interaction mechanism with other agents of the population. The evolution of such mixed strategy follows a Darwinian evolutionary principle: strategies that maximize a given payoff functional will be chosen with higher and higher frequency. The mean-field kinetic description provides an equation that governs the evolution of a probability measure on agents with strategies: the field that drives the resulting continuity equation is determined by the evolution law of the micro-states.

The aim of this thesis is to extend the model just described by introducing a diffusive term in the spatial evolution of the agents. The expected outcomes are:

-- to prove the well-posedness of the particle system with the addition of the Brownian motion;

-- to conjecture a mean-field equation for the system with Brownian motion and to prove its well-posedness;

-- to rigorously prove the mean-field limit.

It is foreseen that the results here obtained can be extended and applied to multi-agent models which include "label-switching" effects (these models are of paramount importance in social phenomena) and to the analysis of particle-based optimization algorithms.

Required skills functional analysis, topology, measure theory.

Deadline 03/06/2025
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