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Events
Berlin-Leipzig workshop in analysis and stochastics

MPI für Mathematik in den Naturwissenschaften Leipzig

G3 10 (Lecture hall)

conference

13.04.16 15.04.16

Berlin-Leipzig workshop in analysis and stochastics

The workshop focuses on the interplay of analysis and probability theory, ranging from the theory of stochastic partial differential equations, stochastic homogenization, random processes in random environments to many body problems and statistical mechanics. We aim to bring together the expertise of the analysis-stochastics groups from Berlin, and its newly founded DFG - research unit "Rough paths, stochastic partial differential equations and related topics" and the respective groups from the MPI MIS Leipzig and the University of Leipzig.

The conference will start on Wednesday at 1 pm and close on Friday at 1 pm. There is no registration, participation is free for academics and practitioners. Please contact Katja Heid to reserve accommodation for you. Travel reimbursement is possible.

Topics include

Nonlinear stochastic partial differential equations
Regularity structures
Fluid dynamics
Stochastic homogenization
Statistical mechanics
Random processes on graphs
Phase transitions, critical phenomena
Geometric PDE, interface motion
Numerical for stochastic and rough differential equations

Speakers

Peter Bella

MPI Leipzig

Rémi Catellier

University of Rennes

Ilya Chevyrev

TU Berlin

Giovanni Conforti

University of Leipzig

Joscha Diehl

TU Berlin

Benjamin Fehrman

MPI Leipzig

Julian Fischer

MPI Leipzig

Benjamin Gess

MPI Leipzig

Francesco Ghiraldin

MPI Leipzig

Arianna Giunti

MPI Leipzig

Martina Hofmanova

TU Berlin

Vitalii Konarovskyi

University of Leipzig

Hoang Duc Luu

MPI Leipzig

Jörg Martin

HU Berlin

Mario Maurelli

TU and WIAS Berlin

Mircea Petrache

MPI Leipzig

Sebastian Riedel

TU Berlin

Tat Dat Tran

MPI Leipzig

Program

12:00 - 14:00	Joint lunch at Vapiano, Augustusplatz 11
14:00 - 14:30	Rémi Catellier (University of Rennes) Interacting particle system and propagation of chaos: a rough path view
14:30 - 15:00	Sebastian Riedel (TU Berlin) 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).
15:00 - 15:30	Coffee break
15:30 - 16:00	Hoang Duc Luu (MPI Leipzig) 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.
16:00 - 16:30	Mario Maurelli (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.
16:30 - 17:00	Giovanni Conforti (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.

09:00 - 09:30	Tat Dat Tran (MPI Leipzig) 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.
09:30 - 10:00	Julian Fischer (MPI Leipzig) 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 $C^{2,\alpha}$-"excess decay" estimate on large scales and a $C^{2,\alpha}$-Liouville principle: For a given $a$-harmonic function $u$ on a ball $B_R$, we show that its energy distance to the space of $a$-harmonic "corrected quadratic polynomials" on some ball $B_r$ has the natural decay in the radius $r$ above some minimal (random) radius $r_0$. 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.
10:00 - 10:30	Coffee break
10:30 - 11:00	Francesco Ghiraldin (MPI Leipzig) Scalar conservation laws with discontinuous flux
11:00 - 11:30	Martina Hofmanova (TU Berlin) 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.
11:30 - 13:30	Lunch break
13:30 - 14:00	Joscha Diehl (TU Berlin) tba 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.
14:00 - 14:30	Benjamin Gess (MPI Leipzig) 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.
14:30 - 15:00	Coffee break
15:00 - 15:30	Ilya Chevyrev (TU Berlin) 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.
15:30 - 16:00	Peter Bella (MPI Leipzig) 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 $C^{1,\alpha}$ 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.
16:00 - 16:30	Jörg Martin (HU Berlin) A regularity structure for rough volatility
18:30 -	Conference Dinner at Auerbachs Keller

09:00 - 09:30	Benjamin Fehrman (MPI Leipzig) 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.
09:30 - 10:00	Mircea Petrache (MPI Leipzig) 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.
10:00 - 10:30	Coffee break
10:30 - 11:00	Arianna Giunti (MPI Leipzig) 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.
11:00 - 11:30	Vitalii Konarovskyi (University of Leipzig) On asymptotic behavior of the system of coalescing diffusion particles
11:30 - 12:00	Coffee break
12:30 - 22:00	End of conference

Berlin-Leipzig workshop in analysis and stochastics

Speakers

Peter Bella

Rémi Catellier

Ilya Chevyrev

Giovanni Conforti

Joscha Diehl

Benjamin Fehrman

Julian Fischer

Benjamin Gess

Francesco Ghiraldin

Arianna Giunti

Martina Hofmanova

Vitalii Konarovskyi

Hoang Duc Luu

Jörg Martin

Mario Maurelli

Mircea Petrache

Sebastian Riedel

Tat Dat Tran

Program

Participants

Peter Bella

Rémi Catellier

Ilya Chevyrev

Khalil Chouk

Giovanni Conforti

Joscha Diehl

Benjamin Fehrman

Julian Fischer

Peter Friz

Peter Friz

Benjamin Gess

Benjamin Gess

Francesco Ghiraldin

Arianna Giunti

Martina Hofmanova

Vitalii Konarovskyi

Tobias Lehmann

Hoang Duc Luu

Jörg Martin

Mario Maurelli

Mircea Petrache

Sebastian Riedel

Tat Dat Tran

Scientific Organizers

Peter Friz

Benjamin Gess

Administrative Contact

Katja Heid