In a couple of recent papers Pawel Duch introduced a method to solve singular SPDEs which is based on a description of the solution along a scale parameter and an associated differential equation describing how the SPDE changes along this flow. This method is inspired by the ideas of “renormalization group” and therefore can also be seen as a nice introduction to this framework. In recent work with Paolo Rinaldi we are using this method to control the singularities in the dynamical $\Phi^4_3$ model and its fractional variants towards a stochastic quantisation on the associated invariant measures. In this talk I will try to give an introduction to the various ingredients underlying this construction.

We compute the large deviation rate functional in the limit of large time for the current flowing through a finite graph where a boundary driven system of stochastic particles is evolving with a zero-range dynamics. This result has already been obtained in prevoius papers by other authors and with different approaches. Our new approach uses new techniques and illuminate various connections and different perspectives. In particular, we use a variational approach to derive the rate functional by contraction from a level 2.5 large deviations. We use an exact minimization and show that the result has similarities with a non-reversible and non-quadratic resistor network theory. In the case of a finite lattice the variational structure is a discrete version of the continuous one of the macroscopic fluctuation theory, that we recover in the scaling limit of the mesh going to zero. This is a joint result in collaboration with Rosemary Harris, obtained long ago and unpublished; it will be available and public soon.

We present a full picture of the out of equilibrium joint fluctuations for current and occupation time in the symmetric exclusion process in dimension one. The main tools developed for that are a Kipnis-Varadhan type inequality necessary to handle the occupation time and a multiple point space time correlation estimate necessary to prove the tightness for the current. Curiously, as a corollary we obtain that, in equilibrium, current and occupation time are independent for any time (but they are not independent seen as processes).

Talk based on a joint work with D. Erhard (UFBA) and T. Xu (UFBA).

In some applications including filtering theory, one encounters stochastic differential equations of the form \begin{equation}\label{eqn:1} dY = \sigma (Y)dB + f(Y)dX , t\in[0,T], Y_0=x\end{equation} of unknown $Y_t$ in $\mathbb{R}^d$, where

• $B_t$ is a multidimensional Brownian motion;
• $X_t$ is an independent source of noise, which we assume is known, and for which we can therefore fix a realization.

To solve ($\ref{eqn:1}$), it is possible to introduce a deterministic formulation, using the rough paths theory of Lyons/Gubinelli (one also fixes a realization of $B_t$). Although it has certain advantages, this method requires very regular coefficients ($C^3$), in contrast to the usual stochastic assumption that $\sigma$ is Lipshitz (or only bounded). After a brief introduction to the theory of rough paths, I will explain how one can deal with ($\ref{eqn:1}$) with minimal assumptions on the coefficients. The key idea is to introduce a hybrid formulation for ($\ref{eqn:1}$) using a "rough semi-martingale" concept. I will also mention its usefulness in the context of mean-field rough stochastic differential equations and (quenched) propagation of chaos.

This work is the result of a collaboration with Peter Friz and Khoa Lê (TU Berlin).

After an introduction on the way large deviation functions appear in non-equilibrium systems, I will try to explain how they can be calculated for general Markov processes. Based on the theory, it is easy to establish general properties of non-equilibrium systems such as the fluctuation theorem. Then the main part of these lectures will be to review the theoretical approaches, such as matrix products, the Bethe ansatz or the macroscopic fluctuation theory, allowing to obtain a series of exact expressions of large deviation functions for lattice gas models.