Rough nonlocal diffusions

We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.Comment: 55 pages. Corrected minor typos in version

Propagation of chaos for interacting particles subject to environmental noise

A system of interacting particles described by stochastic differential equations is considered. As oppopsed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (noninteracting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of inviscid type, as opposed to the case when independent noises drive the different particles.Comment: Published at http://dx.doi.org/10.1214/15-AAP1120 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mean field limit of interacting filaments and vector valued non linear PDEs

Families of $N$ interacting curves are considered, with long range, mean field type, interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as $N$ goes to infinity, to a mean field current, solution of a nonlinear, vector valued, partial differential equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero. This set-up is inspired from vortex filaments in turbulent fluids, although for technical reasons we have to restrict to smooth interaction, instead of the singular Biot-Savart kernel. All these results are based on a careful analysis of a nonlinear flow equation for 1-currents, its relation with the vector valued PDE and the continuous dependence on the initial conditions

Adaptive power method for estimating large deviations in Markov chains

We study the performance of a stochastic algorithm based on the power method that adaptively learns the large deviation functions characterizing the fluctuations of additive functionals of Markov processes, used in physics to model nonequilibrium systems. This algorithm was introduced in the context of risk-sensitive control of Markov chains and was recently adapted to diffusions evolving continuously in time. Here we provide an in-depth study of the convergence of this algorithm close to dynamical phase transitions, exploring the speed of convergence as a function of the learning rate and the effect of including transfer learning. We use as a test example the mean degree of a random walk on an Erd\"os-R\'enyi random graph, which shows a transition between high-degree trajectories of the random walk evolving in the bulk of the graph and low-degree trajectories evolving in dangling edges of the graph. The results show that the adaptive power method is efficient close to dynamical phase transitions, while having many advantages in terms of performance and complexity compared to other algorithms used to compute large deviation functions.Comment: v1: 12 pages, 7 figures; v2: typos corrected, references updated, close to published versio

Thermodynamic cost for precision of general counting observables

We analytically derive universal bounds that describe the trade-off between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.Comment: 18 pages, 6 figure

Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

We consider the 2D Euler equations on $\mathbb{R}^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise

Understanding random-walk dynamical phase coexistence through waiting times

We study the appearance of first-order dynamical phase transitions (DPTs) as `intermittent' co-existing phases in the fluctuations of random walks on graphs. We show that the diverging time scale leading to critical behaviour is the waiting time to jump from one phase to another. This time scale is crucial for observing the system's relaxation to stationarity and demonstrates ergodicity of the system at criticality. We illustrate these results through three analytical examples which provide insights into random walks exploring random graphs.Comment: 8 pages, 6 figures. v2: fixed minor typos + added reference

Convergence of the Integral Fluctuation Theorem estimator for nonequilibrium Markov systems

The Integral Fluctuation Theorem for entropy production (IFT) is among the few equalities that are known to be valid for physical systems arbitrarily driven far from equilibrium. Microscopically, it can be understood as an inherent symmetry for the fluctuating entropy production rate implying the second law of thermodynamics. Here, we examine an IFT statistical estimator based on regular sampling and discuss its limitations for nonequilibrium systems, when sampling rare events becomes pivotal. Furthermore, via a large deviation study, we discuss a method to carefully setup an experiment in the parameter region where the IFT estimator safely converges and also show how to improve the convergence region for Markov chains with finite correlation time. We corroborate our arguments with two illustrative examples.Comment: 15 pages, 7 figure