# Spatial-Time Structured Stochastic Processes

Specific theme of research is : *From the microscale to macroscale in stochastic systems of interacting individuals in population dynamics*. We carry out a systematic analysis of non linear reaction-diffusion systems with variable geometries having the following characteristics:

[a.] nonlinearities: strong coupling of evolution equations for the (stochastic) geometries with the evolution equations of underlying fields, describing densities, concentrations, etc. of relevant quantities which interact with the phenomena subject of the analysis;

[b.] Stochastic geometries: presence of interfaces, at different Hausdorff dimensions, in patterns, clusters and networks emerging in phenomena which characterize self-organizing populations;

[c.] multiple scales: (stochastic) processes at different temporal and spatial scales.

**Description**

A central theme in current mathematical biology and social sciences is the understanding of how interactions of individuals (whether they are cells, humans or other animals) develop into population properties and dynamics, and are viceversa influenced by the state of the entire population.

The individual intrinsic behaviour influences other individuals via an underlying field, characterizing the collective state of the population, being itself influenzed by; such a field drives the feedback that induces changes on the individual behaviour. Thus interaction mechanisms may find a significant description through a functional behaviour that depends on the state of the population as a whole, possibly via mediating underlying fields, which evolves at different scales. This connection has been investigated in different contexts through rigorous upscaling procedures.

The local project aims at developing the theme of individual-to-population connection in specific areas in biology, in medicine and in social human dynamics. The local theme of research will focus on the rigorous mathematical derivation of the bridge among different scales, via suitable "laws of large numbers". The latter requires the convergence of the evolution equation for the empirical measures of the interacting individuals to reaction diffusion equations for the so called "mean-field". We expect that the resulting equations consist of non linear reaction-diffusion systems with variable geometries. On the other hand, analytical results on existence, uniqueness, and regularity of solutions of specific classes of nonlinear PDE's, studied within the national research team, will be instrumental in the rigorous derivation of deterministic mean field models, from stochastic interacting individual particles, and their analysis.

## Keywords

Interacting particles, weak convergence of measure valued stochastic processes

**Pubblications**

V. Capasso, D. Morale, "Asymptotic Behavior of a System of Stochastic Particles subject To Nonlocal Interactions " Stochastic Analysis and Applications, 27, 3, 2009 , 574 - 603

M. Burger, V. Capasso, D. Morale, "On an Aggregation Model with Long and Short Range Interactions" Nonlinear Analysis: Real World Applications, n.3, 2007, p. 939--958

Morale, D., Capasso, V. Oelschlaeger K. ``An interacting particle system modelling aggregation behavior: from individuals to populations.'' J. Mathematical Biology, Volume 50, Number 1, 2005, p. 49 - 66

**Funded Projects related to the Research Theme**

PRIN2009: From the microscale to macroscale in stochastic systems of interacting individuals in population dynamics.

PRIN2007: From stochastic modelling to statistics of space-time structured random processes in population dynamics

PRIN2003: Space-Time Stochastic Processes and Applications

## Components

Members from the Department of Mathematics | ||
---|---|---|

Vincenzo CAPASSO | info | |

Daniela MORALE | info | |

Mattia ZANELLA | ||

Collaborators | ||

Enea BONGIORNO | Università di Novara | |

Willi JAEGER | University of Heidelberg, Germania | |

Karlo OELSCHLAEGER | University of Heidelberg, Germania | |

Martin BURGER | University of Muenster, Germania |