It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the chaotic noise is multidimensional and multiplicative. The tools from rough path theory prove useful in this difficult setting.[-]

Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permuton. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations.
The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permuton. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of the skew Brownian permuton.
The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. We finally explain how it is possible to construct these new limiting permutons directly from a Liouville quantum gravity decorated with two SLE curves. Building on the latter connection, we compute the density of the intensity measure of the Baxter permuton.[-]

We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.[-]

(Joint work with Gonçalo Jacinto and Patricia A. Filipe.) The effect of random fluctuations of internal and external environmental conditions on the growth dynamics of individual animals is not captured by the regression model typical approach. We use stochastic differential equation (SDE) versions of a general class of models that includes the classical growth curves as particular cases. Namely, we use models of the form $d Y_t=\beta\left(\alpha-Y_t\right) d t+\sigma d W_t$, with $X_t$ being the animal size at age $t$ and $Y_t=h\left(X_t\right)$ being the transformed size by a $C^1$ monotonous function $h$ specific of the appropriate underlying growth curve model. $\alpha$ is the average transfomed maturity size of the animal, $\beta>0$ is the rate of approach to it and $\sigma>0$ measures the intensity of the effect on the growth rate of $Y_t$ of environmental fluctuations. These models can be applied to the growth of wildlife animals and also to plant growth, particularly tree growth, but, due to data availability (data furnished by the Associação dos Produtores de Bovinos Mertolengos - ACBM) and economica interest, we have applied them to cattle growth.
We briefly mention the extensive work of this team on parameter simulation methods based on data from several animals, including alternatives to maximum likelihood to correct biases and improve confidence intervals when, as usually happens, there is shortage of data for animals at older ages. We also mention mixed SDE models, in which model parameters may vary randomly from animal to animal (due, for instance, to their different genetical values and other individual characteristics), including a new approximate parameter estimation method. The dependence on genetic values opens the possibility of evolutionary studies on the parameters.
In our application to mertolengo cattle growth, the issue of profit optimization in cattle raising is very important. For that, we have obtained expressions for the expected value and the standard deviation of the profit on raising an animal as a function of the selling age for quite complex and market realistic raising cost structures and selling prices. These results were used to determine the selling age that maximizes the expected profit. A user friendly and flexible computer app for the use of farmers was developed by Ruralbit based on our results.[-]

The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states.[-]

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security.[-]

Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal investment problem for a financial agent whose income is partly affected by market external risk. We then present the basics of existence and uniqueness theory for solutions to BSDE the coefficients of which satisfy global Lipschitz conditions.[-]