Mario Ayala

I. Scaling limits of interacting particle systems

This is the core of my work. My main laboratory consists of three conservative particle systems, independent random walkers, the symmetric exclusion process (SEP) and the symmetric inclusion process (SIP), which are self-dual with orthogonal duality polynomials, i.e. Charlier, Krawtchouk and Meixner polynomials respectively. Much of this work has been developed in collaboration with Frank Redig (TU Delft) and Gioia Carinci (University of Modena and Reggio Emilia).

Duality and fluctuations

The density fluctuation field is the first term of an orthogonal expansion in duality polynomials, and the higher terms are objects in their own right. Along these lines we obtain quantitative Boltzmann-Gibbs principles, and we introduce higher order fluctuation fields, whose scaling limits satisfy a recursive martingale problem corresponding to the powers of a generalized Ornstein-Uhlenbeck process. In work in preparation with Frank Redig we prove a duality relation at the level of the SPDE, in which the duality functions are the Wick powers of the field and the dual process is a system of independent Brownian motions.

Singular limits and Mosco convergence

Duality reduces a many-particle question to a few-particle one, but the few-particle process may still have a singular scaling limit, and Mosco convergence of Dirichlet forms is the right notion in those cases. With its help we show that the distance between two condensively rescaled SIP particles converges to sticky Brownian motion, which gives the variance of the density field in the condensation regime. I show that Mosco convergence tensorizes to any number of independent particles, and with Johannes Zimmer (TU München) we obtain the non-local hydrodynamics and fluctuations of SIP with long jumps. Work in progress extends the condensation result from two particles to k.

Beyond the density field

The hydrodynamic limit keeps the conserved quantity and discards everything else, and a recurring question in my recent work is what the discarded observables do under scaling. With Michiel Renger (TU München) we study collisions and fluxes in SEP, where the unidirectional fluxes are deterministic whereas the net collision count converges to a space-time white noise, and, in work in preparation, the cumulants of contacts at criticality. With Nicolas Dirr (Cardiff University) and Johannes Zimmer we show that for a fixed number k of particles, independent walkers, SEP and SIP have the same atomic limit, i.e. the unique atomic solution of the Dean-Kawasaki martingale problem.

II. Effective dynamics and coarse-graining

Most stochastic models are reduced descriptions: a Langevin equation is obtained by eliminating fast degrees of freedom, a hydrodynamic equation by averaging over particles, and a classical Markov chain by eliminating quantum coherences. The question behind this line is when such a reduced description is faithful to the dynamics it comes from.

With Nicolas Dirr, Grigorios Pavliotis (Imperial College London) and Johannes Zimmer we obtain algebraic conditions under which a diffusion with multiplicative noise is reversible, for a parameter interpolating between the Itô, Stratonovich and Klimontovich interpretations of the noise, and we show that both reversibility and the Klimontovich interpretation are preserved under coarse-graining. With Johannes Zimmer, in work in preparation, we obtain the exclusion process as the effective classical dynamics of a Bose-Hubbard model under continuous monitoring of the local particle numbers.

The same question appears in continuous-time finance. With Benjamín Vallejo Jiménez (University of Colima) we revisit the Merton consumption-investment problem when risky returns are interpreted through a general stochastic integral, and we keep two experiments apart: if the drift is transformed together with the convention, the optimal policy is invariant, whereas if the nominal drift is held fixed, the investment opportunity set changes.

III. Stochastic population models

The same mathematics applies to individual-based models of populations, and this is in fact where my interest in stochastic processes started: as an undergraduate at the Mathematical and Theoretical Biology Institute of Arizona State University I worked on the dynamics of chytridiomycosis in a harlequin frog population.

During my postdoctoral position at INRAE, with Jérôme Coville, Raphaël Forien and Samuel Soubeyrand, I worked on measure-valued models that follow each individual of a population: a model for vector-borne viruses structured by a phenotype trait, for which we obtain the law of large numbers and the fluctuations at the landscape scale, and a model for grouped dispersal, whose large population limit is a degenerate ultra-parabolic system of PDEs.

With Johannes Zimmer, in work in preparation, we model populations that cooperate through a shared environment, such as quorum sensing, treating both the cells and the medium as interacting particle systems. The result I find most interesting is that the speed of the medium leaves every mean-field equilibrium unchanged and nevertheless changes the barrier governing rare collapse, i.e. postulating the reaction-diffusion equation gives the right equilibria and the wrong rare events.