Seminars for members of the local unit in La Sapienza for Prin 2015 `Large Scale Random Structures`'



We announce below seminars organized for the members of the local PRIN unit. They are not announced in the weekly bullettin of the Department since they are thought for experts and not for a broad audience. In all the seminars both the speakers and the members of the local PRIN unit present their results on the subject, which is part of the PRIN project.

Martedì 27.08.2019, 17.00, Sala Gio Ponti

SPEAKER: J. Norris (Cambdridge University)
TITLE: Scaling limits for planar aggregation with subcritical fluctuations
ABSTRACT: We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability.

Martedì 27.08.19, 10:00, Aula B

SPEAKER: A.Turner (Lancaster University)
TITLE: One-dimensional scaling limits in a planar Laplacian random growth model
ABSTRACT: We consider a family of growth models defined using conformal maps in which the local growth rate is determined by the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits.

Lunedì 20.05.2019, Sala Gio Ponti, 9:00

SPEAKER: V. Silvestri (Cambridge University)
TITLE: Internal DLA on cylinder graphs: fluctuations and mixing
ABSTRACT: We use coupling ideas to analyse when to show that an IDLA process on a cylinder graph forgets a typical initial profile.

Mercoledì 11.04.2019, Sala Gio Ponti, 9:00

SPEAKER: C. Landim (IMPA, Rio de Janeiro, Brasil)
TITLE: Coalescing random walks on a torus
ABSTRACT: We consider the discrete d-dimensional torus of size N. We place a particle at each site of torus and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. We analyse the first time the set of particles is reduced to a singleton.

Lunedì 18.03.2019, Sala Gio Ponti, 13:00

SPEAKER: C. Landim (IMPA, Rio de Janeiro, Brasil)
TITLE: Metastability of the two-dimentional Blume-Capel model with zero chemical potential and small magnetic field on a large torus
ABSTRACT: We consider the Blume-Capel model with zero chemical potential and small magnetic field in a two-dimensional torus whose length increaseswith the inverse of the temeprature. We prove the mestastable behavior and that starting from a configuration with only negative spins, the process visits the configuration with only 0-spins on its way to the ground state which is the configuration with all spins equal to +1.

Mercoledì 13.03.2019, Sala Gio Ponti, 17:00

SPEAKER: C. Landim (IMPA, Rio de Janeiro, Brasil)
TITLE: Derivation of viscous Burgers equations from weakly asymmetric exclusion processes
ABSTRACT: We consider weakly asymmetric exclusion processes whose initial density profile is a small perturbation of a constant. We show that in the diffusive time-scale, in all dimensions, the density defect evolves as the solution of a viscous Burgers equation.

Martedì 29.01.2019, Sala Gio Ponti, 9:00

SPEAKER: C. Orrieri (Università Trento)
TITLE: Entropic optimal transport and mean field planning
ABSTRACT: The mean field planning problem (MFPP) is formulated by a continuity equation and Hamilton-Jacobi equation with a nonlinear coupling. Firstly introduced by P.-L. Lions in the context of mean field games theory, MFPPs describe strategic interactions among large numbers of players when the initial and final distributions are prescribed. The aim of the presentation is to recast the PDE system as an optimality system of a suitable entropic regularization of the dynamic optimal transportation problem. We will discuss existence of weak solutions using some ideas of minmax duality and dynamic superposition principles. (In collaboration with A. Porretta and G. Savare).

Venerdì 3.10.2019, 14:00, Sala Gio Ponti

SPEAKER: Mauro Mariani (HSE Moscow)
TITLE: Convergence of the one-dimensional Cahn-Hilliard equation
ABSTRACT: We consider the Cahn-Hilliard equation in one space dimension with scaling a small parameter \epsilon and a non-convex potential W. In the limit \espilon \to 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.

Giovedì 21.02.2019, 14:00, Aula Gio Ponti

SPEAKER: R.Chetrite (Nice)
TITLE: Nonequilibrium Markov processes conditioned on large deviations
ABSTRACT: We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, which is used with importance sampling to simulate rare events, and which gives rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.

Martedì 30.10.2018, Aula Gio Ponti, 15:00

SPEAKER: Carlo Orrieri (Università di Pavia)
TITLE: Ergodic maximum principle for stochastic systems
ABSTRACT: We present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a Backward SDE on infinite horizon from which we can write a version of the SMP.

Martedì 23.10.2018, Sala Gio Ponti, 9:00

SPEAKER: Carlo Orrieri (Università di Pavia)
TITLE: Necessary stochastic maximum principle for dissipative systems on infinite time horizon
ABSTRACT: We develop a necessary stochastic maximum principle for a finite-dimensional stochastic control problem in infinite horizon under a polynomial growth and joint monotonicity assumption on the coefficients. The second assumption generalizes the usual one in the sense that it is formulated as a joint condition for the drift and the diffusion term. The main difficulties concern the construction of the first and second order adjoint processes by solving backward equations on an unbounded time interval. The first adjoint process is characterized as a solution to a backward SDE, which is well-posed thanks to a duality argument. The second one can be defined via another duality relation written in terms of the Hamiltonian of the system and linearized state equation. Some known models verifying the joint monotonicity assumption are discussed as well. Joint work with Petr Veverka.

Martedì 18.09.2018, Sala Gio Ponti, 9:00

SPEAKER: M. Salvi (Ecole Polytechinique, Paris)
TITLE: A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
ABSTRACT: We explain our results obtained in collaboration with M. Biskup and T. Wolff on the central limit theorem for the effective conductance in a small finite box of Z^d of the resistor network build with i.i.d. random conductances.

Martedì 11.09.2018, Sala Gio Ponti, 9:00

SPEAKER: M. Salvi (Ecole Polytechinique, Paris)
TITLE: Scaling of sub-ballistic 1D Random Walks among biased Random Conductances
ABSTRACT: We consider two models of one-dimensional random walks among biased i.i.d. random$

Martedì 21.06.2017, Aula B, 17:00

SPEAKER: Seo Insuk (UC Berkeley)
TITLE: Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes.
ABSTRACT: We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.

Giovedì 4.05.2017, Sala Gio Ponti, 9:00

SPEAKER: N. Gantert (Munich University)
TITLE: Einstein relation for reversible diffusions in random environment
ABSTRACT: We consider reversible diffusions in random environment and prove the Einstein relation for this model. It says that the derivative of the effective velocity under an additional local drift equals the diffusivity of the model without drift. The Einstein relation is conjectured to hold for a variety of models but is proved insofar only in particular cases.

Martedì 18.04.2017, Aula B, 17:00

SPEAKER: A. Boritchev (University of Lyon, France)
TITLE: On the hyperbolicity of minimizers for 1D random Lagrangian systems
ABSTRACT: We prove the hyperbolicity of global minimizers for random Lagrangian systems in dimension 1.