Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.

Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.

The gene regulatory network models all the biochemical reactions between the different species (proteins, mRNA, etc.) present in a cell. A stochastic approach to this network, using jump processes, has been studied during the '70s (Kurtz), and it was established that in large populations, the rescaled process (thought as a concentrations process) converges towards the solution of an EDO, which is entirely deterministic. In order to obtain a random limit, Crudu, Debussche and Radulescu proposed a multiscale model. In this talk, I will propose a relevant stochastic representation of the multiscale model, for which it is possible to obtain the uniform convergence towards a PDMP and a CLT for the fluctuations around this limit.

We investigate the topology and local geometry of various models of random cubical complexes. In the first part, we consider two types of random subcomplexes of the regular cubical grid: percolation clusters and the Eden cell growth model, analyzing their geometric and topological features. In the second part, we study the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube, identifying interesting threshold phenomena. This latter model serves as a cubical analogue of the Linial–Meshulam model for random simplicial complexes.

We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed. Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate “Brownian-Motion-Like” trajectories as well as diffusive motion (displacement proportional to the square-root of time). This model satisfies two conditions that grant, for the temporal motion of the heavy particle:

1. An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;
2. A best quadratic fit with an “average” non-positive curvature in a proper time interval.

We note that Planck’s constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates;to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient. Finally, we discuss whether cat states are present in the thermodynamic ensembles.

(Joint for with W.D. Wick)

File available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5287796