# Berlin-Leipzig workshop in analysis and stochastics

## Abstracts for the talks

**Oleg Butkovsky ***TU Berlin***Convergence of Markov processes to the invariant measure in the Wasserstein metric with applications to SPDEs**

While ergodic behavior of finite-dimensional Markov processes (e.g., SDEs) is quite well understood by now, less is known about the ergodic behavior of infinite-dimensional Markov processes (e.g., SPDEs). In the first part of the talk we explain how the classical methods (which are based on the strong Feller property) can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric. This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow (2011). In the second part of the talk we provide several specific applications to SPDEs (including 2D stochastic Navier-Stokes) and discuss the arising challenges. (Joint work with Alexey Kulik and Michael Scheutzow)

[1] O. Butkovsky (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, 526-552.

[2] O. Butkovsky, M. Scheutzow (2017). Invariant measures for stochastic functional differential equations. arXiv:1703.05120.

**Michele Coghi ***Universität Bielefeld***Mean field limit of interacting filaments for 3D Euler equations**

Solutions to the 3D Euler equations are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments. 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 solution of the 3D Euler 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.

**Joscha Diehl ***MPI MIS***Invariants of the signature**

We describe a complete set of elements of the signature that

is invariant to GL / SO / rotations. Joint work with Jeremy

Reizenstein (Warwick)

**Benjamin Fehrman ***MPI MIS***A Liouville theorem for stationary and ergodic ensembles of parabolic systems**

In this talk, which is based on joint work with Peter Bella and Alberto Chairini, I will discuss a first-order Liouville theorem for random ensembles of uniformly parabolic systems under the qualitative assumptions of stationarity and ergodicity. The method of proof effectively separates the probabilistic and deterministic aspects of the argument through the introduction of an extended corrector. The statistical properties of the environment are used to prove the sublinearity of the large-scale averages of this corrector, which subsequently provides the starting point for a Campanato iteration. The latter is used to establish, almost surely, an intrinsic large-scale Hölder-regularity estimate for caloric functions.

**Franziska Flegel ***WIAS Berlin***Spectral localization vs. homogenization in the random conductance model**

We study the asymptotic behavior of the top eigenvectors and eigenvalues of the random conductance Laplacian in a large domain of ℤ^{d} (d ≥ 2) with zero Dirichlet conditions. Let the conductances w be positive i.i.d. random variables, which fulﬁll certain regularity assumptions near zero. Then we show that the spectrum of the Laplacian displays a sharp transition between a completely localized and a completely homogenized phase. A simple moment condition distinguishes between the two phases.

In the homogenized phase we can even generalize our results to stationary and ergodic conductances with additional jumps of arbitrary length. Here, our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence. The investigation of the homogenized phase is joint work with M. Slowik and M. Heida.

**Peter Friz ***Technische Universität Berlin***Multiscale systems, homogenization, and rough paths**

We revisit the application of rough path theory to the homogenization of

fast-slow systems with mildly chaotic fast dynamics. Joint with

ILYA CHEVYREV (Oxford U), ALEXEY KOREPANOV (Warwick U),

IAN MELBOURNE (Warwick U) AND HUILIN ZHANG (Shandong U and TU Berlin)

**Christiane Görgen ***MPI MIS***Using computational algebra to classify probability trees**

Algebraic statistics explores how tools from algebra and algebraic geometry can be used to tackle problems in statistical inference. In this talk I will report on some recent advances in this field. In particular, we will see how primary decomposition of monomial ideals can be used to classify probability trees which make the same distributional assumptions.

**Vitalii Konarovskyi ***Universität Leipzig***Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line**

The talk is devoted to a model of interacting diffusion particles on the real line. We propose a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction. The construction is based on a new family of measures on the set of real increasing functions as reference measures for naturally associated Dirichlet forms. The processes are infinite dimensional versions of sticky reflecting dynamics on a simplex. We also identify the intrinsic metric leading to a Varadhan formula for the short time asymptotics with the Wasserstein metric for the associated measure valued diffusion. Joint work with Max von Renesse.

**Raphael Kruse ***TU Berlin***On two quadrature rules for stochastic integrals with fractional Sobolev regularity**

In this talk we discuss the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in W^{σ,p}(0,T), σ ∈ (0,2), p ∈ [2,∞). We introduce two quadratures rules: The ﬁrst is best suited for the parameter range σ ∈ (0,1) and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule applies to the case of a deterministic integrand of fractional Sobolev regularity with σ ∈ (1,2). In both cases the order of convergence is equal to σ with respect to the norm in L^{p}(Ω). As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.

**Wolfgang König ***WIAS + TU Berlin***Cluster-size distributions in a classical many-body system**

(Joint work with Sabine Jansen and Bernd Metzger)

We consider a classical many-body system in $\R^d$ with Lennard-Jones-type pair interaction, i.e., with prevention of clumping of the particles and with a preference of a certain positive distance for each pair of particles. We consider the free energy of the canonical ensemble in the thermodynamic limit and derive interesting emerging pictures when taking afterwards the coupled limit of high temperature and low density. It turns out that the particles unite in groups of one particular cardinality, the same everywhere in the system. Our methods are a comparison to an idealised model and a Gamma-convergence for the rate function of a characteristic large-deviation principle.

**Vlada Limic ***CNRS***Multiplicative coalescent from a new angle**

It will be a talk on the black board. The speaker will use symbols and figures to explain how a two-parameter process encodes its corresponding continuous-time random graph (CTRG) component sizes evolution. CTRG component sizes have the multiplicative coalescent dynamics. Near criticality scaling limits will emerge. To each extreme scaling limit of the near critical random graph corresponds a diffusion + jump process. The talk should end by listing these latter process. It would be interesting to know if someone in the audience has seen them (or something similar) in another context.

**Mario Maurelli ***TU + WIAS Berlin***Regularization by noise for scalar conservation laws**

Joint work with Benjamin Gess

We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed (in a pathwise sense) under addition of noise. Most of the results in this direction are limited to SDEs and associated linear SPDEs.

In this talk, we show a regularization by noise result for a nonlinear SPDE, namely a stochastic scalar conservation law on $\mr^d$ with a space-irregular flux:

\begin{equation*}

\partial_t v +b\cdot\nabla[v^2] +\nabla v\circ \dot{W} =0,

\end{equation*}

where $b=b(x)$ is a given deterministic, possibly irregular vector field, $W$ is a $d$-dimensional Brownian motion ($\circ$ denotes Stratonovich integration) and $v=v(t,x,\omega)$ is the scalar solution. More precisely we prove that, under suitable Sobolev assumptions on $b$ and integrability assumptions on its divergence, the equation admits a unique entropy solution. The result is false without noise.

The proof of uniqueness is based on a careful combination of arguments used in the linear case: first we show the renormalization property for the kinetic formulation of the equation, then we use second order PDE estimates and a duality argument to conclude.

If time permits, we will discuss also some open questions.

**Leonid Mytnik ***University of Mainz + Israel Institute of Technology***Dimension of the boundary of the super-Brownian motion**

**Tal Orenshtein ***HU Berlin***Critical wetting models in (1+1) dimensions **

In this talk we will discuss wetting models in (1+1) dimensions pinned to a strip. These polymer models enjoy an interplay between two forces -- local (pinning) and global (entropic repulsion) -- and in many cases a localization-delocalization phase transition holds. The latter known as the wetting transition. In particular, whenever the strip size is fixed and the pinning function is constant and homogeneous in space, phase transition results are available. The standard case, for which the strip size is zero, is completely solved and exhibits a sharp phase transition. In particular, the asymptotic behavior of the system is drastically different in the sub-critical, the super critical and the critical phases. We will focus on the asymptotic of the strip model at criticality when the strip size is shrinking to zero. This is a joint work in progress with Jean-Dominique Deuschel.

**Martin Slowik ***TU Berlin***Quenched invariance principle for random walks among time-dependent degenerate conductances**

I will present a quenched invariance principle for the dynamic random conductance model, that is a continuous time random walk on ℤ^{d} in an environment of time-dependent random conductances. The law of the conductances is assumed to be stationary and ergodic with respect to space-time shifts and satisfy some integrability conditions. In particular, I will discuss and compare diﬀerent technique that allows to derive a maximal inequality for the corrector.

**Scott Smith ***MPI MIS***Rough diffusion equations with singular forcing**

I will discuss recent efforts to place the work of Otto/Weber on quasi-linear SPDE's in an abstract framework close to the theory of regularity structures. This leads to general tools on integration, reconstruction, and multiplication which constitute partial progress towards a general theory of singular SPDE's with variable diffusion coefficients. As a first application, we use these tools to establish a priori bounds for rough diffusion equations driven by a more singular forcing than in Otto/Weber. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

**András Tóbiás ***TU Berlin***A Gibbsian model for message routing in highly dense wireless networks**

In spatial telecommunication networks, it is a prominent question how to route many messages in the same time. We propose a random mechanism for routing messages in a network, where users are situated according to a Poisson point process in a compact subset of ℝ^{d}, and each user sends one message to the single base station. Messages are transmitted either directly or via other users, with a given upper bound on the number of hops. Given the locations of users, we deﬁne a Gibbs distribution on the set of all such trajectory families, which favours trajectories with little interference (measured in terms of the signal-to-interference ratio (SIR)) and trajectory families with little congestion (measured in terms of the number of pairs of incoming messages of the users).

We derive the behaviour of this system in the limit of a high spatial density of users using large-deviation methods, compute the limiting free energy and provide a law of large numbers for the empirical measure of message trajectories. The limit of these empirical measures is given as the minimizer(s) of a characteristic variational formula. In the special case when congestion is not penalized, we analyze the minimizer and investigate the questions of the typical number of hops, the typical length of a hop and the typical shape of a trajectory in the highly dense network.

The topic of this talk is joint work with Wolfgang König.

**Willem van Zuijlen ***HU + WIAS Berlin***The principal eigenvalue of the Anderson Hamiltonian in continuous space**

In this talk I will discuss almost sure asymptotics of the principal eigenvalue of the Anderson Hamiltonian with space white noise potential with Dirichlet boundary conditions on large boxes in ℝ^{2}. This result is joint work with Khalil Chouk.

**Max von Renesse ***Universität Leipzig***Couplings, Gradient Estimates and Logarithmic Sobolev Inequality for Langevin Bridges**

with Giovanni Conforti

We establish quantitative results about the bridges of the

Langevin dynamics and the associated reciprocal processes. They include an equivalence

between gradient estimates for bridge semigroups and couplings, comparison

principles, bounds of the distance between bridges of different Langevin dynamics, and

a Logarithmic Sobolev inequality for bridge measures.

**Lukas Wresch ***Universität Bielefeld***Path-by-Path Uniqueness in Infinite Dimensions**

Let f be a bounded measurable function and W a Wiener process. The question whether the integral equation

<center class="math-display"> <img src="/fileadmin/conf_img/tex_3221c0x.png" alt=" ∫t Xt(ω) = x0 + f(Xs(ω))ds+ Wt (ω ) 0 " class="math-display"></center>

has at most one solution X_{t} for almost all ω (so-called path-by-path uniqueness) has been posed by N. Krylov and answered aﬃrmatively by A. Davie in ℝ^{d}. This result can be understood as a “regularisation by noise” eﬀect since in the absence of noise the above result fails to hold.

Let A be a positive linear operator on a separable Hilbert and W a cylindrical Wiener process. In this talk we consider the equation

<center class="math-display"> <img src="/fileadmin/conf_img/tex_3221c1x.png" alt="dXt = − AXtdt + f(t,Xt )dt+ dWt, " class="math-display"></center>

where f is a bounded Borel measurable function and W a cylindrical Wiener process. If the components of f decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this SDE giving a positive answer to N. Krylov’s original question for SDEs taking values in an abstract Hilbert spaces for small nonlinearties f. Naturally, this notion of uniqueness is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.

**Yue Wu ***TU Berlin***Randomized numerical schemes**

This research is targeted to develop suitable randomized numerical schemes

for a variety of differential equations, say, ODEs and S(P)DEs, with the

order of convergence improved while conditions imposed to coefficient

functions relaxed.

## Date and Location

**November 29 - December 01, 2017**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Peter Friz**

Technische Universität Berlin**Benjamin Gess**

MPI für Mathematik in den Naturwissenschaften

## Administrative Contact

**Katja Heid**

MPI for Mathematics in the Sciences

Contact by Email