Seminari in Programma

e Seminari Passati

Scheduled and Past Seminars

 

May 14th 2024, 14:00 CET - Aula seminari, Dipartimenti di matematica, UNIPI

Speaker: Yassine Tahraoui (Scuola Normale Superiore)

Title: Stochastic obstacle problems: variational & non variational settings

Abstract: Obstacle problems are free boundary  type problems,  well known in the literature of applied mathematics and lead to numerous applications. My aim is to present some results  about the well-posedness and the regularity of  the solution to a "parabolic or hyperbolic"  obstacle problem in the presence of multiplicative noise,   studied in [1, 2]. After showing the well-posedness of such problems, we prove  Lewy-Stampacchia's inequalities, which gives an estimate of the reflected measure generated by the singularities caused by the obstacle near the free boundary.

[1]I. H. Biswas, Y. Tahraoui and G. Vallet: Obstacle problem for a stochastic conservation law and Lewy-Stampacchia inequality.   Journal of Mathematical Analysis and Applications, 527 (1) 127356 (2023)

[2]Y. Tahraoui and G.Vallet: Lewy-Stampacchia's inequality for a stochastic T-monotone obstacle problem.  Stochastic Partial Differential Equations: Analysis and Computations 10, 90-125 (2022).

Past Seminars

Anno Accademico 2023-2024 - Academic Year 2023-2024


Speaker: Patrick Charbonneau (Duke University)

Title: Toward a robust definition of random close packing

Abstract: The apparent simplicity of amorphous sphere packings can be misleading. Although jamming hard spheres to random close packing (rcp) has been studied for decades, an unambiguous definition of rcp, let alone a first-principle prediction remain elusive. Drawing inspiration from liquid state theory, rcp can be identified with the inherent structure of a hard sphere liquid. Identifying inherent structures of hard spheres through optimization, however, is a non-trivial problem. Motivated by a recent (meta-)analysis of existing algorithms, we consider the behavior of minimal models of jamming that can be studied using various approaches, notably with tools from stochastic geometry and the mean-field description of simple glasses. The resulting insights present a path toward formalizing rcp.


Speaker: Matteo Sfragara (Stockholm University)

Title: Chaos and concentration in spatial growth models

Abstract: A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations (superconcentration) and a chaotic behaviour of the ground state in certain Gaussian disordered systems. We study the connection between chaos and concentration in spatial growth models, like first-passage percolation (FPP) and last-passage percolation (LPP), and we prove that they exhibit a chaotic behaviour. This extends previous work on the topic, and illustrates that this is a phenomenon that can be expected more widely. The notion of ‘chaos' refers to the sensitivity of the optimal path (geodesic) when exposed to a slight perturbation. In FPP on Z^d the geodesic is the time-minimizing path from the origin to a vertex v, while in LPP on the square lattice [0,n]^2 the geodesic is the weight-maximizing up-right path from (0,0) to (n,n). This talk is based on two joint works with Daniel Ahlberg and Mia Deijfen (Stockholm University).


Speaker: Leonardo Tolomeo (University of Edinburgh)

Title: Transport of Gaussian measures under the flow of Hamiltonian PDEs: quasi-invariance and singularity.

Abstract: In this talk, we consider the Cauchy problem for the fractional NLS with cubic nonlinearity (FNLS), posed on the one-dimensional torus T, subject to initial data distributed according to a family of Gaussian measures.  We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. This is based on joint works with J. Forlano (UCLA/University of Edinburgh) and with J. Coe (University of Edinburgh).


Speaker: Lucio Galeati (EPFL)

Title: A.e. uniqueness for (stochastic) Lagrangian trajectories for Leray solutions to 3D Navier-Stokes

Abstract: We revisit a result due to Robinson and Sadowski (2009), who first showed a.e. uniqueness of Lagrangian trajectories for admissible weak solutions to $3$D Navier-Stokes, for sufficiently regular $u_0$. We give an alternative proof, based on a newly established asymmetric Lusin-Lipschitz property of Leray solutions, exploited crucially in the arguments from Caravenna-Crippa (2021) and Brué-Colombo-De Lellis (2021). This approach is more robust, requiring no assumptions on $u_0$ and being applicable also to the stochastic characteristics of the system.

Finally, if $u_0$ is regular (say $u_0\in H^{1/2}$), then we are able to exploit the diffusive behaviour of stochastic trajectories to further prove that, for any fixed $x_0\in\mathbb{R}^d$, path-by-path uniqueness for the SDE $d X_t = u(t,X_t) d t + d B_t, X|t=0 = x_0$.


Margherita Zanella (Politecnico di Milano)

Title: Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise

Abstract: We establish the uniqueness and the asymptotic stability of the invariant measure for the two-dimensional Navier-Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an “effectively elliptic” setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of [1] and [2], used by these authors for the stochastic Navier-Stokes equations with additive noise. Here, we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias-Prodi estimate in expected value, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.

The talk is based on a joint work with Benedetta Ferrario.

References:

[1] N. Glatt-Holtz, J. C. Mattingly, and G. Richards. On unique ergodicity in nonlinear stochastic partial differential equations. J. Stat. Phys., 166(3-4):618–649, 2017.

[2] A. Kulik and M. Scheutzow. Generalized couplings and convergence of transition probabilities. Probab. Theory Related Fields, 171(1-2):333–376, 2018.


Massimiliano Datres (Università di Trento)

Title: A two-scale complexity measure for stochastic neural networks 

Abstract: Over-parametrized deep learning models are achieving outstanding performances in solving several complex tasks such as image classification problems, object detection and natural language processing. Despite the risk of overfitting, these parametric models show impressive generalization after training. Hence, defining appropriate complexity measures becomes crucial for understanding and quantifying the generalization capabilities of deep learning models. In this talk, I will introduce a new notion of complexity measure, called two-scale effective dimension (2sED), which is a box-covering dimension related to a metric induced by the Fisher information matrix of the parametric model. I will then show how the 2sED can be used to derive a generalization bound. Furthermore, I present an approximation of the 2sED for Markovian models, called lower 2sED, that can be computed sequentially layer-by-layer with less computational demands. Finally, I present experimental evidence that the post-training performance of given parametric models is related both with 2sED and the lower 2sED.

Lucie Laurence (INRIA)

Title: Scaling methods for stochastic chemical reaction networks

Abstract: In this talk we investigate stochastic chemical reaction networks (CRNs) with scaling methods. This approach is used to study the stability properties of the associated Markov processes, but also to investigate the transient behavior of the sample paths. It also gives insight on the impact of specific features of these networks such as their polynomial reaction rates, leading to the coexistence of multiple timescales. Some examples of CRNs are discuss to illustrate the multiple timescales involved in the decay of the norm of the state when the CRN starts from a "large" initial state.  A detailed scaling analysis of an interesting CRN exhibiting a slow decrease of the norm of the state is presented.


Michele Stecconi (University of Luxemburg)

Title: Sobolev-Malliavin regularity of the nodal volume

Abstract: Consider the nodal volume of a non-degenerate (in a sense to specify) Gaussian random field defined on a compact Riemannian manifold of dimension d greater or equal to 2. We prove that the law of such random variable has an absolutely continuous component, as a direct consequence of its Fréchet differentiability. Moreover, we give an esplicit formula for the derivative (the mean curvature).
The non-singularity of the law had already been established by Angst and Poly for stationary fields on the d-torus, in dimension d>2, via Malliavin calculus. In this work the two dimensional case remained open, in particular, the Malliavin differentiability of the nodal length was unknown. We prove that the nodal volume admits a L2 Malliavin derivative, for d>2 and that in the case d=2, this is false, but the Malliavin derivative still exists in L1.
A fundamental ingredient is to understand the Sobolev regularity of the function f(t) that expresses the volume of the level t of a “typical” Morse function.
(A joint work with Giovanni Peccati.)


Theresa Lange (Scuola Normale Superiore)

Title: Regularization by noise of an averaged version of the Navier-Stokes equations

Abstract: In [T16], the author constructs an averaged version of the deterministic three-dimensional Navier-Stokes equations (3D NSE) which experiences blow-up in finite time. In the last decades, various works have studied suitable perturbations of ill-posed deterministic PDEs in order to prevent or delay such behavior. A promising example is given by a particular choice of transport noise used in [FL21] in the context of the vorticity form of the 3D NSE, and in [FGL21] for more general models. In this talk, we analyze the model in [T16] in view of those two works and discuss the regularization skills of transport noise in the context of the averaged 3D NSE. This is joint work with Martina Hofmanová (U Bielefeld).


References:

[T16] T. Tao, Finite time blowup of an averaged three-dimensional Navier-Stokes equation. Journal of the American Mathematical Society 29(3), pp. 601-674 (2016)

[FL21] F. Flandoli, D. Luo, High mode transport noise improves vorticity blow-up control in 3D Navier-Stokes equations. Probability Theory and Related Fields 180, pp. 309-363 (2021)

[FGL21] F. Flandoli, L. Galeati, D. Luo, Delayed blow-up by transport noise. Communications in Partial Differential Equations 46(9), pp. 1757-1788 (2021)


Simon Michaël Schulz (Scuola Normale Superiore)

Title: Well-posedness and stationary states for a crowded active Brownian system with size-exclusion
Abstract: We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an orientation. This equation incorporates diffusion in both the spatial and angular coordinates, as well as a non-linear non-local drift term, which depends on the angle-independent density. The spatial diffusion is non-linear degenerate and also comprises diffusion of the angle-independent density, which one may interpret as cross-diffusion with infinitely many species. Our proof relies on interpreting the equation as the perturbation of a gradient flow in a Wasserstein-type space. It generalizes the boundedness-by-entropy method to this setting and makes use of a gain of integrability due to the angular diffusion. We also prove uniqueness in the particular case where the non-local drift term is null, and provide existence and uniqueness results for stationary equilibrium solutions. This is joint work with Martin Burger. 


Ernesto De Vito (University of Genoa)
Title:  Understanding Neural Networks with Reproducing Kernel Banach Spaces
Abstract: Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties.  The talk is devoted to show how the theory of reproducing kernel Banach spaces can be used to characterize the function spaces corresponding to neural networks. In particular, I will show a representer theorem for a class of reproducing kernel Banach spaces, which includes one hidden layer neural networks of possibly infinite width. Furthermore, I will prove that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure. 

The talk is based on on joint work with F. Bartolucci, L. Rosasco and S. Vigogna.


Alex Gaudilliere (CNRS)
Title:  Too many frogs cannot fall sleep
Abstract: We prove the existence of an active phase for activated random walks on the lattice in all dimensions. This interacting particle system is made of two kinds of random walkers, or frogs: active and sleeping frogs. Active frogs perform simple random walks, wake up all sleeping frogs on their trajectory and fall asleep at constant rate $\lambda$. Sleeping frogs stay where they are up to activation, when waken up by an active frog. At a large enough density, which is increasing in $\lambda$ but always less than one, such frogs on the torus form a metastable system. We prove that n active frogs in a cramped torus will typically need an exponentially long time to collectively fall asleep ---exponentially long in n. This completes the proof of existence of a non-trivial phase transition for this model designed for the study of self-organized criticality. This is a joint work with Amine Asselah and Nicolas Forien.September 19th 2023, 14:00 CET - Pisa, Dipartimento di Matematica, Aula Seminari



Giuseppe Cannizzaro (University of Warwick) 

Title: Weak coupling scaling of critical SPDEs 

Abstract: The study of stochastic PDEs has known tremendous advances in recent years and, thanks to Hairer's theory of regularity structures and Gubinelli and Perkowski's paracontrolled approach, (local) existence and uniqueness of solutions of subcritical SPDEs is by now well-understood. The goal of this talk is to move beyond the aforementioned theories and present novel tools to derive the scaling limit (in the so-called weak coupling scaling) for some stationary SPDEs at the critical dimension. Our techniques are inspired by the resolvent method developed by Landim, Olla, Yau, Varadhan, and many others, in the context of particle systems in the supercritical dimension and might be well-suited to study a much wider class of statistical mechanics models at criticality.


Giacomo Filippo Di Gesù (Università di Roma la Sapienza) 

Title: Sharp asymptotics for the Allen-Cahn equation in the limit of small noise and large volume 

Abstract: We consider the Allen-Cahn equation on a finite interval perturbed by space-time white noise. Keeping the size of the spatial domain fixed, the dynamics becomes metastable in the limit of vanishing noise. I will review some sharp metastability estimates in this regime and discuss how the invariant measure and long-time behavior is affected if one allows the system size to grow while the noise vanishes (joint work with L. Bertini and P. Buttà). 




Gigliola Staffilani (Abby Rockefeller Mauze Professor MIT)

Title: Some recent developments in wave  turbulence theory

Abstract: In this talk I will present two different approaches in the study of wave turbulence theory. The first, introduced by Bourgain, consists in analyzing the long time  behavior of high Sobolev norms for the  defocusing, cubic  NLS equation on 2D tori (periodic solutions). In this context I will emphasize  how the rationality or irrationality of the torus affects the analysis. The second approach deals with the rigorous derivation of the 3-wave kinetic equation from a weakly nonlinear multidimensional KdV type equation.


Vanessa Jacquier (University of Utrecht)

Title: Homogeneous and heterogeneous nucleation in the three-state Blume-Capel model

Abstract: We study the metastable behavior of the stochastic Blume–Capel model evolving according to the Glauber dynamics with zero boundary conditions. We will show that, due to the three–state character of the Blume–Capel model, the metastability scenario proven for periodic boundary conditions changes deeply when different boundary conditions are considered. The Hamiltonian of the Blume–Capel model depends on the magnetic field h and the chemical potential λ. We study the heuristic in the whole region λ,h > 0, where the chemical potential term equally favors minus and plus spins with respect to zeroes and the magnetic field favors pluses and disadvantages minuses with respect to the zeroes, and we compare our results with the Blume-Capel model with periodic boundary conditions. Then, we analyze in detail the region  λ>h > 0. In this region, we identify the unique metastable state -1, we compute the energy barrier from -1 to the stable state +1, and we find an estimate for the asymptotic behavior of the transition time from the metastable to the stable state as β->∞, where β is the inverse of the temperature.



Anno Accademico 2022-2023 - Academic Year 2022-2023



Federico Pasqualotto (UC Berkeley)

Title: Singularity formation in the Boussinesq system

Abstract: In this talk, I will first review existing results on singularity formation in incompressible and inviscid fluids. I will then describe a new mechanism for singularity formation in the 2D Boussinesq system. The initial data we choose is smooth except at one point, where it has Hölder continuous first derivatives. Moreover, the singularity mechanism is connected to the classical Rayleigh–Bénard instability. Finally, I will describe how these considerations translate to a novel singularity formation scenario for the 3D incompressible Euler equations. This is joint work with Tarek Elgindi (Duke University).



Jonathan Mattingly (Duke University)

Title: Ergodicity, Positive Lyapunov Exponents, and Partial Damping for Random Switching

Abstract: I will consider some new models inspired by the PDEs/SPDE with complex dynamics such as the 2D Euler and Navier-Stokes equations. The models introduce randomness onto the system through a random splitting scheme. I will explain how the randomly split Galerkin approximations of the 2D Euler equations and other related dynamics can be shown to possess a unique invariant measure that is absolutely continuous with respect to the natural Liouville measure, despite the existence of other invariant measures corresponding to fix points of the PDEs. I will then explain how on proves that the dynamics with respect to this measure has positive Lyapunov exponents almost surely.

Lastly, I will discuss recent results which show that the system has a unique invariant measure even when damping is applied to part of the system.

This is joint work with Omar Melikechi, Andrea Agazzi, and David Herzog. 



Gianmarco Bet (University of Florence)

Title: Metastability for the Potts model with Glauber dynamics

Abstract: In this talk, I will describe some recent result on the low-temperature metastabile behavior of the ferromagnetic Potts model on a finite two-dimensional grid-graph Λ, evolving according to Glauber dynamics. More specifically, to each spin configuration is associated an energy that depends on local spin alignment, as well as on an external magnetic field that acts only on one spin value. We describe separately the case of negative, positive and, if time allows, zero external magnetic field. In the first case there are q − 1 stable configurations and a unique metastable state. In the second case there are q − 1 symmetric metastable configurations and only one global minimum. In the third scenario the system has q stable equilibria. In the negative and positive cases we study the asymptotic behavior of the first hitting time from the metastable to the stable state as the inverse temperature tends to infinity. Moreover, in both cases we identify the union of gates and prove that this union has to be crossed with high probability during the transition. Based on joint work with Anna Gallo and Francesca Nardi.


Sandra Cerrai (University of Maryland)

Title: The Smoluchowski-Kramers diffusion-approximation for a class of constrained stochastic wave equations

Abstract: We investigate the well-posedness of a class of stochastic second-order in time-damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of a d-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the L^2-norm of the solution is equal to one. We introduce a small mass mu>0 in front of the second-order derivative in time and examine the validity of a Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term.




Erwin Luesink (University of Twente)

Title: A geometric approach to deriving and numerically integrating models with transport-type noise in geophysical fluid dynamics.

Abstract: In this talk I will discuss how the framework of geometric continuum mechanics can be made stochastic by introducing the stochastic Euler-Poincare theorem. This naturally leads to noise of transport type in fluid dynamics while maintaining geometric invariants, such as the enstrophy in two-dimensional ideal fluids. In special cases, it is possible to discretise the equations of motion while preserving most of the structure, which provides strong numerical evidence for Kraichnan’s double cascade conjecture in two-dimensional incompressible fluids. I will also discuss some recent progress for the quasi-geostrophic equations.



Francesco Mattesini (MPI Leipzig, University of Munster)

Title: Annealed quantitative estimates for the 2D-discrete random matching problem

Abstract: The optimal matching problem is a classical random variational problem which may be interpreted as an optimal transport problem between two random discrete measures. Its easier instance deals with matching 2 n-clouds of i. i. d. uniformly distributed points. In recent years Caracciolo-Lucibello-Parisi-Sicuro made exact predictions on the convergence of the rescaled cost thanks to a first order linearization of the Monge-Ampère equation. This approach was later justified by Ambrosio-Stra-Trevisan and quantitative bounds for the convergence of the proxies were later shown by Ambrosio-Glaudo-Trevisan. Such techniques have been repurposed by Benedetto-Caglioti to study the case of of i. i. d. random points with non-constant densities. By subadditivity and PDE arguments Ambrosio-Goldman-Trevisan were able to justify the latter for the convergence of the rescaled cost. We show annealed quantitative upper bounds for the approximating transport map in the case of i. i. d. points and weakly correlated points with non-constant densities. We extend our results to the case of unbalanced matching, i. e. matching between point clouds of different size and to point clouds sampled from a positive recurrent Markov chain. 

Joint work with N. Clozeau (IST Austria).



Francesco Russo (ENSTA-Paris)

Title: Semimartingales with jumps, weak Dirichlet processes and path-dependent martingale problems

Abstract: In this talk we will revisit the notion of weak Dirichlet process which is the natural extension of semimartingale with jumps. If X is such a process, then it is the sum of a local martingale M and a martingale ortogonal process A in the sense that [A, N ] = 0 for every continuous local martingale N . We remark that if [A] = 0 then X is a Dirichlet process. The notion of Dirichlet process is not very suitable in the jump case since in this case A is forced to be continuous.

The talk will discuss the following points.

1. To provide a (unique) decomposition which is also new for semimartingales with jumps.

2. To discuss some new stability theorem for weak Dirichlet processes through C0,1 transformations.

3. To discuss various examples of such processes arising from path-dependent martingale problems. This includes path-dependent stochastic differential equations with involving a distributional drift and with jumps.

The talk is based on a joint paper with E. Bandini (Bologna).



Milto Hadjikyriakou (UCLan Cyprus)

Title: Some new ordering results for parallel and series systems with dependent heterogeneous exponentiated Weibull components

Abstract: In this work, ordering results are presented for parallel and series systems arising from dependent heterogeneous exponentiated Weibull components that share a common or different Archimedean copula(s). Particularly, sufficient conditions are provided under which the sample extremes are stochastically compared with respect to the usual stochastic order, the dispersive and the star-shaped order.


Silvia Morlacchi (Scuola Normale Superiore)

Title: Effect of Transport Noise on Kelvin–Helmholtz Instability

Abstract: We numerically investigate the effect of transport noise on the Kelvin- Helmholtz instability at the interface of two 2D fluids in a shear flow, to test if the transport noise acts as a stabilizing factor.

We exploit the point vortex method as a numerical discretization of the fluid equation of motion. We compare the results of the simulations in three different cases: inviscid and viscous fluid without transport noise, and inviscid fluid perturbed by transport noise. We find that when the transport noise is modeled by a large number of low-intensity point vortices, a delay in the temporal onset of the instability is present in the non viscous case, which resembles what happens in the viscous case.

Based on joint work with Franco Flandoli and Andrea Papini.



Mauro Mariani (National Research University, Moscow)

Title: Random currents and homologies on compact manifolds

Abstract: We study the long time behavior of stochastic currents associated to diffusion processes on compact Riemannian manifolds. In the first part of the talk, sharp results about existence and tightness of stochastic currents will be discussed.

In the second part, some problems related to random homologies (homology class associated to the paths of diffusion processes) will be addressed. In particular, we give a full geometric characterization of manifolds such that the associated random homology has a gaussian asymptotic. Some simpler related problems (Gallavotti-Cohen symmetry, relation with the Riemannian metric).


Title: No blow-up by nonlinear Ito noise for Euler equations 

Abstract: We consider the 2D and 3D stochastic Euler equations. It is well-known that (under suitable assumptions on the noise) regular solutions exist locally in time. We show, by means of the Lyapunov function method and a Galerkin approximation, that the choice of a suitable non-linear multiplicative Ito noise provides a regularizing effect. Namely, we establish that with full probability the regular solutions are global in time.

The presentation is based on a joint work with Mario Maurelli and Fanhui Xu.


Vittoria Silvestri (Università Roma 1 - La Sapienza)

Title: Planar aggregation with subcritical fluctuations and the Hastings-Levitov models

Abstract: The ALE (Aggregate Loewner Evolution) models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off-lattice analogues of several fundamental discrete models, such as Eden or  Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In this talk I will discuss shape theorems and fluctuations of ALE clusters, which include Hastings-Levitov clusters as particular cases, in the subcritical regime. 

Based on joint work with James Norris and Amanda Turner.


Riccardo Montalto (Università di Milano Statale)

Title: Construction on quasi-periodic nonlinear waves in fluid mechanics

Abstract: In this talk I will discuss some recent results on Euler and Navier Stokes equations concerning the construction of quasi-periodic solutions and the problem of the invscid limit for the Navier Stokes equation. I will discuss the following results:

         These works are based on collaborations with Luca Franzoi and Nader Masmoudi.


Avi Mayorcas (TU Berlin)

Title: Blow-up criteria for an SPDE model of chemotaxis

Abstract: Chemotaxis and related phenomena have been an active area of mathematical research since statistical and PDE models were first proposed by C. Patlak (’53) and E. Keller & L. Segel (’71). They are commonly studied through mean field PDE models and a common feature of these equations is the possibility of finite time blow-up under given model parameters. Recently it was shown that advection by a sufficiently strong relaxation enhancing vector field could suppress this blow up (Kiselev & Xu ’16, Iyer, Zlatos & Xu ’20). In this talk I will discuss new results (obtained with M. Tomašević) regarding criteria for the persistence of blow-up for an SPDE model of chemotaxis with stochastic advection. The noise we cover is of a form recently shown to be almost surely relaxation enhancing (Gess & Yaroslavtsev ’21) and closely related to those studied in recent works by Galeati, Flandoli and Luo.


Marco Rehmeier (Bielefeld University): 

Title: On nonlinear Markov Processes in the sense of McKean

Abstract: We study nonlinear Markov processes in the sense of McKean and present a large new class of examples. Our notion of nonlinear Markov property is in McKean's spirit, but more general in order to include examples of such processes whose one-dimensional time marginals solve a nonlinear parabolic PDE, such as Burgers' equation, the porous media equation, or variants of the latter with transport-type drift. We show that the associated nonlinear Markov process is given by path laws of weak solutions to a corresponding distribution-dependent stochastic differential equation whose coefficients depend singularly (i.e. Nemytskii-type) on its one-dimensional time marginals. Moreover, we show that also for general nonlinear Markov processes, their path laws are uniquely determined by one-dimensional time marginals of suitable associated conditional path laws. Furthermore, we characterize the extremality of the curves of the one-dimensional time marginals of our nonlinear Markov Processes in the class of all solutions to the associated linearized PDE and, this way, obtain new interesting results also for the classical linear case. This is joint work with Michael Röckner.

Michele Aleandri (Università di Roma La Sapienza): Opinion dynamics with Lotka-Volterra type interactions 

We investigate a class of models for opinion dynamics in a population with two interacting families of individuals. Each family has an intrinsic mean field “Voter-like” dynamics which is influenced by interaction with the other family. The interaction terms describe a  cooperative/conformist or competitive/nonconformist attitude of one family with respect to the other. We prove chaos propagation, i.e., we show that on any time interval [0; T], as the size of the system goes to infinity, each individual behaves independently of the others with transition rates driven by a macroscopic equation. We focus in particular on models with Lotka-Volterra type interactions, i.e., models with cooperative vs. competitive families. For these models, although the microscopic system is driven a.s. to consensus within each family, a periodic behaviour arises in the macroscopic scale. In order to describe fluctuations between the limiting periodic orbits, we identify a slow variable in the microscopic system and, through an averaging principle, we find a diffusion which describes the macroscopic dynamics of such variable on a larger time scale.


Lihan Wang (Carnigie-Mellon University, Max Planck Institute Leipzig)

Title: Accelerate Sampling Using Birth-Death Dynamics

Abstract: In this talk, I will discuss the birth-death dynamics for sampling multimodal probability distributions, which is the spherical Hellinger gradient flow of relative entropy. The advantage of the birth-death dynamics is that, unlike any local dynamics such as Langevin dynamics, it allows global movement of mass directly from one mode to another, without the difficulty of going through low probability regions. We prove that the birth death dynamics converges to the unique invariant measure with a uniform rate under some mild conditions, showing its potential of overcoming metastability. We will also show that on torus, the kernelized dynamics, which is used for numerical simulation, Gamma-converges to the idealized dynamics as the kernel bandwidth shrinks to zero. Joint work with Yulong Lu (UMass Amherst) and Dejan Slepcev (CMU).


Elia Bisi (TU Wien): 

Title: Matrix Whittaker processes

Abstract:  Our journey starts from interacting random walks with push-and-block dynamics. We then consider their positive temperature analogues, touching upon polymer partition functions. Finally, we arrive at matrix Whittaker processes, which are integrable models of interacting Markov dynamics on matrix spaces. Our main tools are intertwining relations and the theory of Markov functions, which we will review. This talk is based on a joint work with Jonas Arista and Neil O’Connell: https://arxiv.org/abs/2203.14868.

Michele Coghi (Università di Trento): 

Title: Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Abstract: Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.


Dejun Luo (Chinese Academy of Science) : 

Title: Convergence rates and CLT for stochastic inviscid Leray-model with transport noise

Abstract: The stochastic inviscid Leray- model perturbed by multiplicative transport noise is considered on the torus. Under a suitable scaling of the noise, it is shown that the weak solutions converge, in some negative Sobolev spaces, to the unique solution of the deterministic viscous Leray- model. Interpreting such limit result as a law of large numbers, we also study the underlying central limit theorem and provide an explicit convergence rate. This talk is based a joint work with PhD Bin Tang.



Antonio Agresti (IST Austria): 

Title: The primitive equations with transport noise

Abstract: The primitive equations are one of the fundamental models for geophysical flows used to describe oceanic and atmospheric dynamics. In this talk I will discuss some recent results on the primitive equations with noise of transport type. In addition to transport noise, we also consider non-isothermal turbulent pressure. The dependence of the turbulent pressure on the temperature is a consequence of a stochastic Boussinesq approximation. For the primitive equations with transport noise and non-isothermal turbulent pressure, we provide a physical derivation and we discuss the global well-posedness for data in the critical spaces $H^1$. The latter result gives a non-trivial extension of the celebrated work by C. Cao and E.S. Titi on the deterministic model. Our approach is based on recent developments of maximal regularity techniques in the context of stochastic parabolic PDEs. If time allows, then I will also discuss further results in presence of rough noise.

Based on joint works with M. Hieber (TU Darmstadt), A. Hussein (TU Kaiserslautern) and M. Saal (TU Darmstadt).


Andrea Papini (Scuola Normale Superiore):  

Title: Turbulence Enhancement in Kinetic Coagulation Equations

Abstract: A Smoluchowski type model of coagulation in a turbulent fluid is given, heuristically expressed by means of a stochastic model of a system of second-order (microscopic) coagulating particles, then in a suitable scaling limit as a deterministic model with enhanced diffusion in the velocity component. Existence, uniqueness and regularity is proven under different initial conditions in suitable weighted spaces for the spatially-homogeneous PDE. A precise link between mean intensity of the turbulent velocity field and coagulation enhancement is obtained by numerical simulations.

Based on joint works with Franco Flandoli and Ruojun Huang.


Milo Viviani (Centro De Giorgi): 

Title: On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations

Abstract: In this talk we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution of the Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of the Poisson algebra of functions on S², that appear to be new. Finally, we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs, via Zeitlin’s model.


Angelo Lucia (Universidad Complutense de Madrid): 

Title: Thermalization of 2D Quantum Memories

Abstract: The aim of a quantum memory is to protect an encoded quantum state against errors for long periods of time. Quantum double models are a class of 2D quantum memories proposed by Kitaev, in which the protection from noise is due to the topological properties of the ground state degeneracy. Heuristic arguments have long pointed to a weakness of these models against thermal noise processes. In this talk I will present a rigorous estimate on the lifetime of these memories under a noise process modeled by Davies generators which confirms these findings. This estimate is obtained by proving that the spectral gap of the generator is lower bounded by a positive constant uniformly in the system size.


Margherita Zanella (Politecnico di Milano): 

Title: Ergodic results for the stochastic nonlinear damped Schroedinger equation

Abstract: We study the nonlinear stochastic Schrödinger equation with linear damping. We prove the existence of invariant measures in the case of two dimensional compact Riemannian manifolds without boundary and compact smooth domains of R² with either Dirichlet or Neumann boundary conditions. We prove the uniqueness of the invariant measure in Rᵈ, d=2,3 when the damping coefficient is sufficiently large. The talk is based on joint works with B. Ferrario and Z. Brzeźniak.


Markus Tempelmayr (Max Planck Inst. Leipzig): 

Title: A diagram-free approach to the stochastic estimates in regularity structures

Abstract: We present an alternative point of view on Hairer's regularity structures that is well suited, but not restricted to, quasilinear singular SPDEs. Guided by symmetries of the equation, we approach the counterterm top-down rather than bottom-up. The model, which captures the local solution behaviour, is indexed by partial derivatives w.r.t. the nonlinearity. This allows for efficient bookkeeping and automated (inductive) proofs. The main assumption on the driving noise is a spectral gap inequality, which complements well the BPHZ choice of renormalization.

This is joint work with Pablo Linares, Felix Otto, and Pavlos Tsatsoulis.



Giulia Livieri (Scuola Normale Superiore): 

Title: N-player Games and Mean Field Games of Moderate Interaction

Abstract: We study the asymptotic organization among many optimizing individuals interacting in a suitable "moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument.

Robin Stephenson (University of Sheffield): 

Title: Scaling Limits of Multi-Type Markov Branching Trees

Abstract: Consider a population where individuals have two characteristics: a size, which is a positive integer, and a type, which is a member of a finite set. This population reproduces in a Galton-Watson fashion, with one additional condition: given that an individual has size n, the sum of the sizes of its children is less than or equal to n. We call multi-type Markov branching tree the family tree of such a population.



Wolfgang König (TU Berlin and WIAS): 

Title: The free energy of (a box version of) the interacting Bose gas

Abstract: The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose–Einstein condensation phase transition is sought for. We introduce a simplified version of the model in Z^d instead of R^d  and with an organisation of the particles in deterministic boxes instead of Brownian cycles as the marks of a reference Poisson point process.


We derive an explicit and interpretable variational formula in the thermodynamic limit for the canonical ensemble for any value of the particle density. In this formula, each of the microscopic particles and the macroscopic part of the configuration are seen explicitly (if they exist); the latter receives the interpretation of the condensate. The methods comprises a two step large-deviation approach for marked Poisson point processes and an explicit distinction into microscopic and macroscopic marks. We discuss the condensate phase transition in terms of existence of minimizer. Based on joint works with Adams/Collevecchio (2011) and Collin/Jahnel (preprint 2022).

Anno Accademico 2021-2022 - Academic Year 2021-2022




Jacob Bedrossian (University of Maryland): Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing.

Abstract: In this talk we discuss our recent results obtaining strictly positive lower bounds on the top Lyapunov exponent of stochastic differential equations such as the weakly-damped Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. We propose a new method, based on the Fisher information of an associated Markov process on the sphere bundle and uniform hypoelliptic regularity estimates, which has the ability to obtain quantitative estimates on the top Lyapunov exponents of high-dimensional, weakly dissipative SDEs.



Leonardo Maini (Université du Luxembourg): Spectral central limit theorem for additive functionals of Gaussian fields

Abstract: We consider a centered, continuous, stationary, Gaussian field on the Euclidean space and a sequence of non-linear additive functionals of the field. Since the pioneering works from the 80s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and non-central limit theorems for this kind of functionals have never ceased to be refined. The common intuition is that the limit is Gaussian when we have short-memory and non-Gaussian when we have long-memory and the Hermite rank R is different from 1. Our goal is to show that this can be a misleading intuition. To do that, we introduce a spectral central limit theorem, which highlights a variety of situations where the limit is Gaussian in a long-memory context with R different from 1. Our main mathematical tools are the Malliavin-Stein method and Fourier analysis. The talk is based on a joint work with Ivan Nourdin (University of Luxembourg).



Trishen Gunaratnam (University of Geneva): Phi 4 via random currents

Abstract: The phi 4 model is a simple candidate for a nontrivial euclidean field theory. From the point of view of statistical mechanics, however, it (rigorously or otherwise) exhibits a wide range of beautiful phenomena. I will survey some upcoming joint works with collaborators towards understanding some global probabilistic properties of this model, and highlight some irresistible conjectures. I hope to at least explain a probabilistic representation of correlations via a type of random current representation.



Enrico Malatesta (Bocconi): Phase transitions in the landscape of solutions of overparametrized neural networks.

Abstract: Current deep neural networks are nonlinear devices composed of a number of parameters that far exceed the number of data points. Understanding how these systems can fit the data almost perfectly through variants of gradient descent algorithms and achieve exceptional levels of prediction accuracy without overfitting are key conceptual challenges. In this talk I will show how common techniques used in machine learning (e.g. the choice of the activation function or the loss) deeply affect the loss landscape, tending to mild its roughness. Then we shed light on the role of overparameterization in non-convex neural networks. By analytically studying a non-convex model of random features, we identify a novel (non-equilibrium) phase transition, that we call “Local Entropy” transition, controlled by the degree of overparameterization. In non-convex models this transition is strictly different to the SAT/UNSAT threshold and it coincides with the appearance of highly entropic minima of the error loss function. Those minima, in turn, are found to be highly attractive to the learning algorithms currently used in deep learning.



Luisa Andreis (UniFi): Large deviations for coagulation processes: an approach via graphs

Abstract: Interacting particle systems where particles interact via coagulation are of great interest for their various behaviours. In particular, interesting phenomena can occur, depending on the structure of the kernel which is giving a rate to each coagulation. Among these phenomena there is the famous phase transition that goes under the name of gelation, i.e. the appearence of one (or multiple) giant particle(s). Although fluid limits are known for the rescaled version of stochastic coagulation processes (convergence to the Smoluchowski coagulation equation and its modification), very few is known about large deviations and rare events in this framework. In this talk we will explore some connections of these processes with random graphs and how to use this connection to attack the problem of studying large deviations. This also allows a comparison with the phase transition in graphs, where a giant component appears. Some remarks about the possible generalization to coagulation kernels that depend on spatial position will be given. This is based on ongoing joint works with Wolfgang Konig (WIAS and TU Berlin), Tejas Iyer (WIAS), Heide Langhammer (WIAS), Elena Magnanini (WIAS) and Robert Patterson (WIAS).


 Giulia Carigi (University of Reading): Ergodic properties for a stochastic two-layer model of geophysical fluid dynamics

Abstract: A two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. Moreover, response theory with respect to changes in the average wind forcing is established. Specifically, it is shown that the averages of a class of observables against the invariant measure are differentiable (linear response) and locally Hölder continuous (fractional response) as functions of a deterministic additive forcing. In doing so, a framework suitable to establish (linear and fractional) response for a class of nonlinear stochastic partial differential equations is provided.



Darrick Lee (EPFL): Mapping Space Signatures

Abstract: The path signature is a foundational tool in the theory of rough paths. In this talk, we introduce the mapping space signature, a generalization of the path signature to maps from higher dimensional cubical domains, which is motivated by the topological perspective of K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to a certain equivalence relation on cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser.



Jules Pitcho (ENS Lyons): Non-uniqueness of integral curves for rough divergence-free vector fields

Abstract: Since the work of Di Perna-Lions and Ambrosio, it is known that the continuity equation with divergence-free Sobolev vector field is well-posed for densities with suitable integrability. At the Lagrangian level, these works translate into a selection principle for integral curves under which uniqueness for almost every initial data is true. Nevertheless, uniqueness of integral curves can fail almost everywhere. The deterministic technique used to construct such divergence-free Sobolev vector fields and non-unique integral curves go by the name of convex integration: we will explain some of the ideas underlying this technique. We will conclude by arguing that for rougher vector fields, a genuinely stochastic behaviour of integral curves is to be expected: we should not hope for an almost everywhere selection principle for integral curves.



Anna Paola Todino (UniMiB): Alternative forms of random spherical harmonics

Abstract: In the last decade, lot of efforts have been devoted to the analysis of the high-frequency behaviour of geometric functionals (Lipschitz-Killing Curvatures) for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). The asymptotic behavior of their expected values and variances have been investigated and quantitative central limit theorems have been established in the high energy limits. In order to generalize these results, a local study was also introduced by considering subdomains of the sphere. This topic is linked to the Berry's conjecture on planar random waves and finds its motivation in cosmological applications. Another interesting issue concerns the Gaussianity hypothesis of the random field. In this direction we introduce a model of Poisson random waves on the 2-dimensional sphere and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e. frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.



Ruojun Huang (SNS): Coagulation under environmental noise

Abstract: Environmental noise in a continuum interacting particle system is a space-dependent noise acting on all particles simultaneously. We prove that a system of locally interacting diffusions carrying discrete masses, subject to an environmental noise and undergoing mass coagulation, converges to a system of Stochastic Partial Differential Equations (SPDEs) with Smoluchowski-type nonlinearity. It partially extends a result of Hammond-Rezakhanlou (2007) who considered the PDE case, with our motivation coming from trying to understand the effect of turbulence on rain formations. Based on a joint work with Franco Flandoli.



Langxuan Su (Duke University): A Large Deviation Approach to Posterior Consistency in Dynamical Systems

Abstract: We provide asymptotic results concerning (generalized) Bayesian inference for certain dynamical systems based on a large deviation approach. Given a sequence of observations, a class of parametrized model processes and a loss function, we specify the generalized posterior distribution. We state conditions on the model family and the loss function such that the posterior distribution converges. The two conditions we require are: (1) a conditional large deviation behavior for a single model process, and (2) an exponential continuity condition over the model family for the map from the parameter to the loss incurred between a model process and the observations. The proposed framework is quite general, we apply it to two very different classes of dynamical systems: continuous time hypermixing processes and Gibbs processes on shifts of finite type. We also show that the generalized posterior distribution concentrates asymptotically on those parameters that minimize the expected loss and a divergence term, hence proving posterior consistency.

 Anno Accademico 2017-2018        -        Academic Year 2017-2018




Mario Maurelli (WIAS & TU Berlin): A McKean-Vlasov SDE with reflecting boundaries.

Abstract:  McKean-Vlasov SDEs are SDEs where the coefficients depend on the law of the solution to the SDE. Their interest is in the links with nonlinear PDEs on one side (the SDE-related Fokker-Planck equation is nonlinear) and with interacting particles on the other side (the McKean-Vlasov SDE can be approximated by a system of weakly coupled SDEs). In this talk, motivated by a model of lithium-ion batteries, we introduce a McKean-Vlasov SDE constrained on a bounded domain. The novelty lies in the particular interplay between the mean field interaction and the boundary conditions. We study well-posedness and particle approximation. Our analysis combines techniques for SDEs with Neumann boundary terms (Skorokhod equation, Lions-Sznitman) and ideas from a recent approach to McKean-Vlasov SDEs (Cass-Lyons).



Milton Jara (IMPA): Entropy methods in interacting particle systems.

Abstract:  We develop a variational formula to estimate the moment generating function of additive functionals of Markov processes. Combined with a related variational formula for the evolution of the relative entropy of the law of the process with respect to suitable reference measures, this variational formula serves as a substitute for the martingale method pioneered by Kipnis-Varadhan in the case of out-of-equilibrium interacting particle systems.


Massimiliano Gubinelli (Bonn Univ.): Weak universality of singular SPDEs.

Abstract:  I will discuss the problem of controlling the large scale limit of weakly non-linear SPDEs in the regime where the non-linearity survives at the macroscopic scale giving rise to singular non-linear SPDEs.



Nikolas Perkowski (Humboldt Univ.): A weak universality result for the Parabolic Anderson Model.

Abstract:  We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.



Daniele Cappelletti (University of Wisconsin-Madison): Chemical reaction networks: deterministic and stochastic models.

Abstract:  Chemical reaction networks are mathematical models used in biochemistry, as well as in other fields. Specifically, the time evolution of a system of biochemical reactions are modelled either deterministically, by means of a system of ordinary differential equations, or stochastically, by means of a continuous time Markov chain. It is natural to wonder whether the dynamics of the two modelling regimes are linked, and whether properties of one model can shed light on the behavior of the other one. In this talk some connections will be shown, and both classical and recent results will be discussed. However, many open questions remain. For example, it is not known under what assumptions some qualitative properties of the deterministic model imply positive recurrence, non-explosiveness or absorption events for the stochastic model.



Remi Catellier (Université de Nice Sophia-Antipolis): Averaging along irregular curves and regularization of ODEs

Abstract:  Paths of some stochastic processes such as fractional Brownian Motion have some amazing regularization properties. It is well known that in order to have uniqueness in differential systems such as dy_t = b(y_t) dt, b needs to be quite regular. However, the oscillations of a stochastic process added to the system will guarantee uniqueness for really irregular b. In this talk we will show how to solve the perturbed differential system with a certain stochastic averaging operator. As an application, we show that the stochastic transport equation driven by fractional Brownian motion has a unique solution when u0∈L∞ and b is a possibly random α-Hölder continuous function for α large enough.
 This is a joint work with Massimiliano Gubinelli.


Michel Nassif (ENS Rennes): Introduction to interacting particle systems.

Abstract:  Interacting particle systems is a recently developed field in the theory of Markov processes with many applications: particle systems have been used to model phenomena ranging from traffic behaviour to spread of infection and tumour growth. We introduce this field through the study of the simple exclusion process. We will construct the generator of this process and we will give a convergence result of the spatial particle density to the solution of the heat equation. We will also discuss a variation of the simple exclusion process with proliferation.



Marta Leocata (Università di Pisa): A particle system approach to cellular aggregation model.

Abstract:  In this talk I will present a model for cellular aggregation based on a system of PDE and I will investigate a microscopic derivations. Description of local interaction is given by the notion of moderate interactions in the sense of K. Oelshchlager.



Christian Olivera (Universidade Estadual de Campinas): Regularization by noise in (2x 2) hyperbolic systems of conservation laws.

Abstract:  In this talk we study a non-strictly hyperbolic system of conservation law by stochastic perturbation. We show existence and uniqueness of the solution. We do not assume $-regularity for the initial conditions. The proofs are based on the concept of entropy solution and on the method of charactteristics (under the influence of noise). This is the first result on the regularization by noise in hyperbolic systems of conservation law.



Dejun Luo (Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing): Constantin and Iyer's representation formula for the Navier-Stokes equations on manifolds.

Abstract:  In this talk, we will present a probabilistic representation formula for the Navier-Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case. On a Riemannian manifold, there are several different choices of Laplacian operators acting on vector fields. We shall use the de Rham-Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose it as a sum of the square of Lie derivatives. This is a joint work with Shizan Fang.




Francesco Grotto (SNS): Regularization for stationary Stochastic Burgers' Equation.

Abstract:  Stochastic Burgers' Equation (SBE) in 1+1 dimensions is the formal space derivative of Kardar-Parisi-Zhang (KPZ) equation, and shares with the latter its ill-posed nature. In the stationary case, works of M. Jara and P. Goncalves, and later M. Gubinelli and N. Perkowski, have shown how to exploit the regularizing properties of the linear part of SBE to give meaning to its quadratic drift, deriving the notion of Energy Solution for stationary SBE, and consequently for KPZ equation. We will address how this procedure is carried out, and moreover how regularization yields existence of energy solutions by means of convergence of SDEs modelling surface growth, among them the ones considered by M. Hairer and J. Quastel in their weak universality result for KPZ. 



Konstantin Matetski (U. Toronto): Convergence of general weakly asymmetric exclusion processes.

Abstract:  In my ongoing work with J. Quastel we consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite jump ranges and general jump rates. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. In contrast to the celebrated result by L. Bertini and G. Giacomin (in the case of the nearest neighbour interaction) and its extension by A. Dembo and L.-C. Tsai (for jumps of sizes at most three) we do not use the Hopf-Cole transform and work with the KPZ equation using regularity structures. The price which we have to pay for this approach is a non-trivial renormalization which has not been observed before for equations with stationary noises. In my talk i will give a general review of the Hopf-Cole solution to the KPZ equation and the results of the aforementioned authors. After that I will introduce a general weakly asymmetric exclusion process and explain the difficulty with renormalization.



Michael Röckner (Universität Bielefeld): Global solutions to random 3D vorticity equations for small initial data

Abstract:  One proves the existence and uniqueness in (Lp(R3))3, 3/2< p < 2, of a global mild solution to random vorticity equations associated to stochastic 3D Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in (L3∩L3p/(4p-6))3 with respect to the time variable. Furthermore, we obtain the path-wise continuous dependence of solutions with respect to the initial data. This is joint work with Viorel Barbu.



Maurizio Pratelli (Università di Pisa): La topologia di Meyer-Zheng sullo spazio delle funzioni "cadlag".

Abstract:  Se si considera sullo spazio delle funzioni cadlag la topologia della convergenza in probabilità anziché l'usuale topologia di Skorohod si ottengono (mediante l'immersione dello spazio delle funzioni in uno spazio di probabilità) delle condizioni di tensione molto agevoli da verificare per martingale e supermartingale.  Lo scopo del seminario è introdurre brevemente questa topologia assieme ad alcuni problemi che è stato agevole risolvere con questi metodi.



Francesca Nardi (Università di Firenze): Metastability for general dynamics with rare transitions: escape time and critical configurations.

Abstract:  Metastability is an ubiquitous physical phenomenon in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions for Markov chains. For Metropolis chains associated with statistical mechanical systems, this phenomenon has been described in an elegant way through a path-wise approach in terms of the energy landscape associated to the Hamiltonian of the system. In the seminar we will first explain the main results and ideas of this approach and compare it with other existing ones. Then we will provide a similar description in the general rare transitions setup that can be applied to irreversible systems as well. Besides their theoretical content, we believe that our results are a useful tool to approach metastability for non-Metropolis systems such as Probabilistic Cellular Automata. Moreover, we will describe results pertaining to exponential hitting times which range of applicability includes irreversible systems, systems with exponentially growing volumes and systems with a general starting measure. (Joint work with Emilio N. M. Cirillo, R. Fernandez, F. Manzo, E. Scoppola and J. Sohier).



Valeria De Mattei (Università di Firenze): Mean field games with mean field and moderate interactions.

Abstract:  We consider a mean field game where N players interact both in the cost functionals and in the dynamics. The new feature is that the dynamical interaction is of the so called ``moderate'' type, intermediate between the more classical mean field game scheme is presented. Convergence to a system of PDEs, the mean field equation for our system, is discussed.

Anno Accademico 2015-2016 - Academic Year 2015-2016




A. Hocquet (TU Berlin): Finite-time singularities of the stochastic harmonic map flow on surfaces.

Abstract: A ferromagnetic material possesses a magnetization, which, out of equilibrium, satisfies the Landau-Lifshitz-Gilbert equation (LLG). Thermal fluctuations are taken into account by Gaussian space-time white noise. At least in the deterministic case, there is an important parallel between this model and the so-called Harmonic Map Flow (HMF). This was originally used by geometers (in the early sixties) as a tool to build harmonic maps between two manifolds u:M→N. The case where M is two dimensional is critical, in the sense that the natural energy barely fails to give well-posedness. We do not address here the problem of the solvability of LLG driven by space-time white noise. Instead, we consider a spatially correlated version. We show that oppositely to the deterministic case, blow-up of solutions happens no matter how we choose the initial data.



Francesco Russo: Probabilistic representation of a generalized porous media equation. The deterministic and stochastic case.



Giovanni Zanco (IST Austria): A brief introduction to rough paths theory - II



Giovanni Zanco (IST Austria): A brief introduction to rough paths theory - I

Abstract: We will introduce basic concepts and tools from rough paths theory, motivated by the need to establish a solution theory for differential equations driven by irregular signals. In particular we will discuss controlled rough paths, integration against rough paths, we will compare it to stochastic integration theories and show how classical results can be extended to the rough paths framework. We will mainly consider α-Hölder signals with α∈(1/3,1/2]; this choice allows to simplify many concepts of the theory and, although restrictive, provides enough instruments to deal with interesting problems, like nonlinear SPDEs driven by space-time white noise.



Giuseppe Cannizzaro (TU Berlin): Calcolo di Malliavin per Strutture di Regolarità: il caso di gPAM

Abstract: Le strutture di Regolarità, introdotte da M. Hairer in A theory of Regularity Structures, hanno permesso di risolvere in modo robusto una ricca classe di equazioni alle derivate parziali stocastiche (SPDEs) mal poste. In questa presentazione vogliamo mostrare come sia possibile utilizzare tecniche di calcolo di Malliavin al fine di indagare proprietà probabilistiche delle soluzioni di tali equazioni. Ci concentreremo su un esempio standard della teoria, l'equazione parabolica di Anderson generalizzata (gPAM), e vedremo come si possa dimostrare l'esistenza della densità rispetto alla misura di Lebesgue per la sua soluzione valutata ad un punto dello spazio tempo.



Rita Giuliano (Università di Pisa): Some examples of the interplay between Probability and Number Theory.

Abstract: Probability theory have been often used to study problems in number theory: for instance the so--called "probabilistic method" is a powerful tool which traces back to P. Erdős. More recently also the large deviations theory has been applied in number theoretical settings; In this talk I shall illustrate the main results of some recent papers of mine (in collaboration with C. Macci) which are in the same circle of ideas.



Massimiliano Gubinelli (Bonn Universität): Weak universality of the stationary KPZ equation.

Abstract: I will discuss a notion of solution for the KPZ equation which has been introduced by Jara and Gonçalves (2010) and later improved by Jara and myself (2013) and goes under the name of energy solutions. In a recent work in collaboration with Perkowski we have recently obtained a uniqueness results for energy solutions. Using energy solutions is possible to prove weak universality results for KPZ: namely that a wide class of one dimensional microscopic interacting particle models with weak asymmetry and in non-equilibrium stationary states have large scale fluctuations described by the KPZ equation.


Michele Coghi (SNS): Mean field limit of interacting filaments and vector valued non linear PDEs.

Abstract: 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.



M. Neklyudov (Università di Pisa): A particle system approach to cell-cell adhesion models II

Abstract: We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlager.


Alessandra Caraceni (SNS): Le mappe esternamente planari e il loro limite di scala.

Abstract: La modellizzazione di oggetti combinatorici discreti (grafi, alberi, mappe), al crescere della loro complessità, con oggetti di natura continua, è una tecnica sviluppata a partire dai lavori di Aldous, che sta godendo in anni recenti di crescente interesse grazie ai suoi legami con combinatoria, analisi stocastica, meccanica statistica e gravitazione quantistica. Il limite di scala di numerose classi di mappe è la cosiddetta ``Mappa Browniana''; sotto determinate ipotesi, tuttavia, è possibile assistere a comportamenti asintotici differenti, e alcune classi di mappe planari con una unica faccia macroscopica ammettono il CRT (continuum random tree) di Aldous, o un suo multiplo, come limite di scala. Vedremo come sia questo il caso in particolare delle mappe ``esternamente planari, una classe di mappe di naturale interesse in combinatoria, soggette alla condizioneche tutti i vertici siano adiacenti alla faccia ``esterna''. Grazie a una bigezione di Bonichon, Gavoille e Hanusse mostreremo come una variabile aleatoria uniforme sull'insieme delle mappe esternamente planari a n vertici, riscalata di un fattore 1/\sqrt{n}, converga in legge (in senso Gromov-Hausdorff) verso 7\sqrt{2}/9 volte il CRT.



Dario Trevisan (Università di Pisa): A particle system approach to cell-cell adhesion models I

Abstract: We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlager.



V. Capasso (ADAMSS, Università di Milano): Modelli matematici per l'angiogenesi tumorale.

Abstract: Nella modellazione matematica della angiogenesi tumorale, il forte accoppiamento tra i processi stocastici di diramazione-elongazione-morte di vasi, e i campi biochimici dovuti alla massa tumorale, è causa di forte complessità dal punto di vista sia analitico che computazionale. Al fine di ridurre tale complessità, si cerca di rendere completamente deterministiche le equazioni di evoluzione dei campi, sostituendo in esse i termini stocastici derivanti dalla evoluzione delle rete di vasi, con una loro approssimazione di campo medio. In tal modo i parametri cinetici dei processi (stocastici) di formazione della rete divengono deterministici. Purtroppo a causa della anastomosi, non è possibile garantire le condizioni di applicabilità tipiche della propagazione del caos, che quindi viene messa in discussione. Una possibile derivazione di equazioni di evoluzione deterministiche per i campi fa ricorso alla media su molte repliche dei processi coinvolti, secondo leggi classiche dei grandi numeri. Simulazioni numeriche incoraggiano l'adozione di questo approccio.



Valeria De Mattei (Università di Pisa): An introduction to the theory of mean field games.



Alessandra Caraceni (SNS): Introduzione alla teoria dei limiti di scala.