## Periodic Reporting for period 2 - HyLEF (Hydrodynamic Limits and Equilibrium Fluctuations: universality from stochastic systems)

Reporting period: 2018-06-01 to 2019-11-30

The issues being addressed in this project are related to the study of the asymptotic behavior of certain stochastic processes. More precisely, the project analyzes scaling limits of interacting particle systems. Mathematically speaking, these systems belong to the class of continuous time Markov processes, whose dynamics conserve one or more quantities of interest (the thermodynamical quantities) and under a space-time scaling, the time evolution of these quantities can be described by distinct macroscopic laws. These laws can be partial differential equations or stochastic partial differential equations depending on whether one is looking at the convergence to the mean or to the fluctuations around that mean. Our aim is to give a step towards on the universality of these laws from microscopic stochastic systems.

In real life situations, we come across many different episodes where we feel that it has similarities with something we have lived before in the past. Either this is related to emotions or to physical reactions, the similarities can be present. In nature, similarities between very different organisms also occur. As an example of this puzzling reality which somehow connects all of us, as an example we can recall a rigorous winter day in which ice particles fall from the sky. When seated on a car, we see a growing pattern of ice particles which is formed in the windscreen.

The ice particles fall, randomly, from the sky and when they hit the windscreen they form a growing pattern which can be seen in other, in principle, uncorrelated, situations as coffee ring effects, bacterial growth like E-coli, the wake of a flame, tumor growth…There are various different physical systems, that when they are mathematically modeled they show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality. There are different shapes for these patterns and their study is the core of this project and it is related to a very active area of research in both mathematics and physics known as universality. This mysterious relationship between very distinct physical systems is encoded in some universal laws that one has to figure out how they can be characterized and how does one change from one universal law to another. They are linked by some parameter which somehow connects very different systems. This project intends to analyze this issue which will be important for the understanding of the surrounding and mysterious world that we live in.

In real life situations, we come across many different episodes where we feel that it has similarities with something we have lived before in the past. Either this is related to emotions or to physical reactions, the similarities can be present. In nature, similarities between very different organisms also occur. As an example of this puzzling reality which somehow connects all of us, as an example we can recall a rigorous winter day in which ice particles fall from the sky. When seated on a car, we see a growing pattern of ice particles which is formed in the windscreen.

The ice particles fall, randomly, from the sky and when they hit the windscreen they form a growing pattern which can be seen in other, in principle, uncorrelated, situations as coffee ring effects, bacterial growth like E-coli, the wake of a flame, tumor growth…There are various different physical systems, that when they are mathematically modeled they show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality. There are different shapes for these patterns and their study is the core of this project and it is related to a very active area of research in both mathematics and physics known as universality. This mysterious relationship between very distinct physical systems is encoded in some universal laws that one has to figure out how they can be characterized and how does one change from one universal law to another. They are linked by some parameter which somehow connects very different systems. This project intends to analyze this issue which will be important for the understanding of the surrounding and mysterious world that we live in.

We have already achieved results on WP1 and WP2. More precisely, we analyzed the exclusion process (EP) and its density fluctuations in equilibrium. We have put the system in contact with reservoirs and we showed the convergence to the Stochastic Burgers equation (SBE) with Dirichlet boundary conditions (BC). We also analyze the convergence of the quadratic density fluctuation field and this brings us some insight on how to attack the problems of WP3. We analyzed the equilibrium fluctuations for EP with long jumps and we proved the convergence to the SBE in and also to fractional SBE. We analyzed EP with contact with reservoirs and we derived the hydrodynamic limit, where we got to parabolic equations with different types of BC depending on the range of the parameter that rules the strength of the boundaries. We dealt with both the cases where the transition probability is symmetric and with finite and infinite variance. We analyzed the non-equilibrium fluctuations for a system in contact with slow reservoirs. The limiting process is an Ornstein-Uhlenbeck (OU) with a type of linear Robin BC. We have also analyzed a model whose hydrodynamics is given by the porous medium equation with different types of BC. We analyzed the fluctuations of systems with more than one conserved quantity and we reached other universality classes for each one of the conserved quantities. We analyzed the case of harmonic potentials and by tunning the dynamics we can reach different processes connecting the different universality classes.

Since the beginning of this project, we have obtained several results beyond the state of the art: we have contributed towards the KPZ universality conjecture by showing that a huge class of models cross from Edwards Wilkinson to KPZ. The underlying models are weakly asymmetric and the transition goes by tuning the strength of the asymmetry. In the publication list, there are several publications in which we present those results for a great variety of models. Later we extended the previous result to the case of models in contact with stochastic reservoirs. For those models, we obtained OU and KPZ/SBE with Dirichlet BC. The most difficult problem we faced is the derivation of the solution to those equations, as its BC, from a microscopic dynamics. Also, the extension of the uniqueness problem to the boundary case was very intricate. We also considered models for which we were able to deduce the fractional KPZ/SBE from microscopic systems. In this case, we difficulty was to consider models with long jumps so that the usual Laplacian operator present in the KPZ/SBE is replaced by the fractional Laplacian. Regarding this issue, we plan to attack in the near future the case of systems with long jumps and in the presence of stochastic reservoirs. Our goal is then to obtain the fractional KPZ/SBE with BC. For models of continuous (and unbounded variables), we were able to analyze their equilibrium fluctuations and we obtained solutions to other types of SPDEs. The main difficulty in working with this type of models is that a part of the dynamics is completely deterministic and it is superposed with a random dynamics which conserves more than a quantity, and the conserved quantities depend on each other (their evolution is not independent of the other quantity) and live on their own time scale. This brings an additional problem to the solution of the SPDEs, since, we obtain a system of coupled equations.

We are at the moment, exploiting this problem for a Hamiltonian system with exponential potential. The dynamics conserve the energy and the volume and the transition of the energy, has been derived and goes from an OU to SBE, and is independent of the volume evolution, but the volume depends on the evolution of the energy. In the latter case, we obtain several SPDEs where terms from the energy are present. We also want to complete the scenario of the fluctuations of this model since there are regimes still open. We are also interested in the derivation of new PDEs from microscopic dynamics. These PDEs are of a fractional type and with several types of BC. Recently we have obtained several PDEs given in terms of an operator, known in the PDE literature as the regional fractional Laplacian and with BC of Dirichlet type, Robin and Neumann. We considered models with long jumps and in contact with an infinite number of stochastic reservoirs. We also plan to obtain other types of fractional PDEs from microscopic dynamics and to derive the porous media equation (fractional) with several BC out of those systems. Finally, we also plan to derive hyperbolic laws with several BC out of the hydrodynamic limit of particle systems and we also plan to derive solutions to other SPDEs (fractional) with different types of BC out of the fluctuations of particle systems.

We are at the moment, exploiting this problem for a Hamiltonian system with exponential potential. The dynamics conserve the energy and the volume and the transition of the energy, has been derived and goes from an OU to SBE, and is independent of the volume evolution, but the volume depends on the evolution of the energy. In the latter case, we obtain several SPDEs where terms from the energy are present. We also want to complete the scenario of the fluctuations of this model since there are regimes still open. We are also interested in the derivation of new PDEs from microscopic dynamics. These PDEs are of a fractional type and with several types of BC. Recently we have obtained several PDEs given in terms of an operator, known in the PDE literature as the regional fractional Laplacian and with BC of Dirichlet type, Robin and Neumann. We considered models with long jumps and in contact with an infinite number of stochastic reservoirs. We also plan to obtain other types of fractional PDEs from microscopic dynamics and to derive the porous media equation (fractional) with several BC out of those systems. Finally, we also plan to derive hyperbolic laws with several BC out of the hydrodynamic limit of particle systems and we also plan to derive solutions to other SPDEs (fractional) with different types of BC out of the fluctuations of particle systems.