Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example

In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\cite{bertini2014},which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$, where $N$ is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter $\lambda$, starting from a known quasi-potential for $\lambda=0$. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively

Partial Differential Equations

The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric problems, and fourth order equations in conformal geometry

N-player games and mean-field games with absorption

We introduce a simple class of mean field games with absorbing boundary over a finite time horizon. In the corresponding N-player games, the evolution of players’ states is described by a system of weakly interacting Itô equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean field game, the renormalization appears in form of a conditional law. We justify our definition of solution in the usual way, that is, by showing that a solution of the mean field game induces approximate Nash equilibria for the N-player games with approximation error tending to zero as N tends to infinity. This convergence is established provided the diffusion coefficient is non-degenerate. The degenerate case is more delicate and gives rise to counter-examples

Some results in weak KPZ universality

Stochastic partial differential equations (SPDEs) are a central object of study in the field of stochastic analysis. Their study involves a number of different tools coming from probability theory, functional analysis, harmonic analysis, statistical mechanics, and dynamical systems. Conversely SPDEs are an extremely useful paradigm to study scaling limit phenomena encountered throughout many other areas of mathematics and physics. The present thesis is concerned mainly with one particular SPDE called the Kardar-Parisi-Zhang (KPZ) equation, which appears universally as a fluctuation limit of height profiles of microscopic models such as interacting particle systems, directed polymers, and corner growth models. Such limit results are deemed instances of ``weak KPZ universality," a field born from the seminal paper of Bertini and Giacomin. We extend results on weak KPZ universality in a number of different directions. In one direction, we prove a version of Bertini-Giacomin's result in a half-space by adapting their methods to this setting, thus extending a result of Corwin and Shen and completing the final step towards the proof of a conjecture about fluctuation behavior of half-space KPZ. In another direction, we also prove a result for the free energy for directed polymers in an octant converging to the KPZ equation in a half-space with a nontrivial normalization at the boundary. In a third direction, we return to the whole-space regime and extend the Bertini-Giacomin result to the case of several different initial data coupled together, proving joint convergence of ASEP with its basic coupling to KPZ driven by the same realization of its noise. Finally we prove a ``nonlinear" version of the law of the iterated logarithm for the KPZ equation in a weak-noise but strong-nonlinearity regime. Beyond their intrinsic purpose, one application of all these extensions and generalizations is to take limits of known results and identities for discrete systems and pass them to the limit to obtain nontrivial information about the KPZ equation itself, which is a well-known methodology launched by I. Corwin and coauthors

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio