Almost Surely Mario

Fluctuation fields, scaling limits & hydrodynamic theory

My main laboratory consists of three conservative particle systems, the symmetric exclusion process (SEP), the symmetric inclusion process (SIP), and independent random walkers (IRW), all of which enjoy a remarkable self-duality property that makes them unusually tractable for exact calculations. Much of this work has been developed in collaboration with Frank Redig (TU Delft) and Gioia Carinci (University of Modena).

Within this setting I study two complementary scaling regimes. In the hydrodynamic regime, I derive the deterministic PDEs governing the evolution of particle density. In the fluctuation regime (the central-limit counterpart) the limiting objects are generalized stochastic processes formally solving SPDEs. A particular contribution here is the introduction of higher-order fluctuation fields, whose scaling limits require a notion of renormalized powers of distribution-valued processes.

A separate but related direction, in joint work with Nicolas Dirr (Cardiff University) and Johannes Zimmer (TU München), concerns the fixed-k scaling regime: rather than sending the number of particles to infinity, we fix k particles and study the limiting empirical measure. The limit is the atomic Dean-Kawasaki dynamics, a rigid measure-valued martingale problem whose only solutions are systems of independent diffusions.

More recently, with Michiel Renger (TU München), I have been studying a finer decomposition of the SEP dynamics: tracking not just the empirical density and net current, but also the unidirectional fluxes and collision counts (the events where a particle attempts to jump to an already-occupied site). These observables have their own hydrodynamic limits, with regime-dependent behavior: deterministic in most scaling regimes, and converging to a space-time white noise process for the net collision count.

Duality methods

Duality is a property enjoyed by certain Markov processes (including SEP, SIP, and IRW) that allows one to study a complex many-particle system through the lens of a simpler dual process. Rather than a technique applied from the outside, it is a structural feature of the system that can be exploited systematically.

In my work, duality appears in two main roles. First, as a tool within scaling limit theory: orthogonal duality polynomials (Krawtchouk, Meixner) provide the right observables for deriving Boltzmann-Gibbs principles and characterizing fluctuation fields. Second, in recent work with Frank Redig (TU Delft), we show that the OU SPDE is dual to a system of independent Brownian motions. For now this is established in the non-interacting setting, but it suggests the possibility of finding duality relations between SPDEs and interacting diffusions more generally, which could have interesting consequences for numerical methods.

Noise interpretation and stochastic calculus

The choice of stochastic integral convention (Ito, Stratonovich, Klimontovich) is not merely a technical convenience. It determines key structural features of the resulting dynamics: the geometry of the diffusion, conditions for reversibility, and behavior under coarse-graining. In joint work with Nicolas Dirr (Cardiff University), Grigorios Pavliotis (Imperial College London), and Johannes Zimmer (TU München), we study these questions systematically for Langevin dynamics with state-dependent (multiplicative) noise, combining a geometric formulation with explicit algebraic reversibility conditions and a rigorous coarse-graining analysis via Mosco convergence of Dirichlet forms.

A downstream consequence of this perspective appears in continuous-time finance: in joint work with Benjamin Vallejo (University of Colima), we show that the choice of noise interpretation shifts optimal portfolio weights in the Merton problem in a systematic and quantifiable way.