Liouville theorem for elliptic systems with degenerate coefficients
I will discuss behavior of second order degenerate elliptic systems in divergence form with random ergodic coefficients. Assuming moment bounds on the coefficient field a and its inverse, similar to the ones used by Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014], we obtain C1,α large scale regularity estimate for a-harmonic functions. As a consequence we obtain Liouville theorem for subquadratic a-harmonic functions, which in particular implies uniqueness of the correctors.
Levy rough paths and stochastic flows
In this talk, we aim to introduce the notion of a Levy rough path and discuss their fundamental properties linked with p-variation and a Levy-Khintchine formula. We will also focus on an application of determining weak limit points of stochastic flows driven by random walks.
University of Leipzig
Fluctuation of bridges and concentration of measure phenomenon
In this talk, we have a look at the question of quantifying fluctuations of a bridge: we have a look at some reference models, such as gradient diffusions and continuous time random walks on graphs and obtain a concentration of measure inequality for the marginals in the first case, and refined large deviation estimates fin the latter case.
We study the parabolic Anderson model on 2d Riemmanian manifolds. This equation is formally ill-posed and has only recently been treated in flat space via the theory of regularity structures. We show how this structure can be modified to a geometric setting. Joint work with Antoine Dahlqvist and Bruce Driver.
Stochastic Homogenization of Isotropic Diffusions in Large Domains
We will discuss the quantitative stochastic homogenization of elliptic and parabolic pde describing, in particular, the smoothed exit distributions and exit times of certain isotropic diffusions from large domains.
A higher-order large-scale regularity theory for random elliptic operators
We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. Under the assumptions of stationarity and slightly quantified ergodicity of the ensemble, we derive a C2,α-”excess decay” estimate on large scales and a C2,α-Liouville principle: For a given a-harmonic function u on a ball BR, we show that its energy distance to the space of a-harmonic ”corrected quadratic polynomials” on some ball Br has the natural decay in the radius r above some minimal (random) radius r0. Our Liouville principle states that the space of a-harmonic functions growing at most quadratically has (almost surely) the same dimension as in the constant-coefficient case. The existence of a-harmonic ”corrected quadratic polynomials” - and therefore our regularity theory - relies on the existence of second-order correctors for the homogenization problem. By an iterative construction, we are able to establish existence of subquadratically growing second-order correctors.
Regularization by noise
We discuss regularization by noise phenomena for PDE. The talk will be introductionary, first recalling well-known results for transport equations, then briefly discussing more recent results for (stochastic) scalar conservation laws.
Quantitative Homogenization of Degenerate Random Environments
We study a random conductance model with degenerate conductivities and give a simple necessary and sufficient condition for the environment to be diffusive. This allows to infer quantitative estimates for the homogenization of the associated operator in divergence form.
Rough Gronwall lemma and weak solutions to RPDEs
In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.
Hoang Duc Luu
Exponential stability of stochastic differential equations driven by fBm
We present recent progress in the study of the asymptotic stability of stochastic differential (delay) equations driven by fractional Brownian motions in case the Hurst index H > 1∕2. The stochastic integrals, which is defined in the Young sense, can then be expressed by Riemann-Liouville fractional derivatives. Our main results are some criteria for the exponential stability of the system.
TU and WIAS Berlin
Enhanced Sanov theorem and robust propagation of chaos
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application, we obtain a large deviation principle for a suitable (k-layer, enhanced) empirical measure of weakly interacting diffusions, which in turn implies a propagation of chaos result in rough path spaces.
Next-order asymptotics and rigidity results for Coulomb and Riesz gasses
I will discuss next-order asymptotics as n goes to infinity, for the minimum energy configurations of n particles minimizing an inverse-power-law interaction energy. This is a question related to Random Matrix Theory, to Approximation Theory and to Statistical Physics. The first-order term in our large-n asymptotics was known since the 30's. The next-order term, obtained in collaboration with Sylvia Serfaty, is a new functional on micro-scale asymptotic configurations of the points. I will describe some more precise rigidity results (part of joint work with Simona Rota-Nodari) regarding the uniformity of such micro-scale configurations, which is a possible step towards the Abrikosov crystallization conjecture. The related study of energy-minimizing lattices (joint work with Laurent Betermin) and the quantum version of the problem (joint work with Codina Cotar) will also be mentioned.
Rough paths and random dynamical systems
We aim to study the long time behaviour of solutions to rough differential equations (in the sense of Lyons) driven by a rough paths valued stochastic process. Most equations of interest contain an unbounded drift term. Our first contribution is to extend the existing theory to equations containing these terms. In the second part of the talk, we connect rough paths theory to the theory of random dynamical systems. In particular, we show that rough differential equations naturally induce random dynamical systems provided the driving rough paths valued process has stationary increments. This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).
Tat Dat Tran
On free energy functional method in population genetics
In this talk, I will present some results in population genetics using the free energy functional for degenerate Fokker-Planck equations. Some necessary and sufficient conditions has been proven for the existence of a unique reversible measure as well as convergence rate to this one. The talk is based on a joint work with Julian Hofrichter and Juergen Jost.
April 13 - 15, 2016
Max Planck Institute for Mathematics in the Sciences
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