The workshop focuses on the interplay of analysis and probability theory, ranging from the theory of stochastic partial differential equations, stochastic homogenization, fluid dynamics, statistics, data science, information geometry, 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 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 tentatively start on Wednesday at 10 am and close on Friday at 4 pm. There is no registration, participation is free for academics and practitioners. Please contact Katja Heid to reserve accomodation for you until end of October 2018. Travel reimbursement is possible.

Presentations: There will be invited talks only. The talks will be around 20-30 minutes.

We investigate the behaviour of an establishing mutation which is subject to rapidly fluctuating selection under the Spatial Lambda-Fleming-Viot model and show that under a suitable scaling it converges to the SuperBrownian in a random environment. The scaling results for the behaviour of the rare allele are achieved via particle representations which belong to the family of `lookdown constructions'. To our knowledge this is the first instance of the application of the lookdown approach in which other techniques seem unavailable. Joint with Jonathan Chetwynd-Diggle.

We present a version of the multiplicative Sewing Lemma which is inspired by the work of Feyel, De La Pradelle and Mokobodzki '08. We show how this result applies to situations encountered in the treatment of Partial Differential Equations with functional analysis methods.

We show how generalized couplings can be applied to prove weak uniqueness of solutions of non-degenerate stochastic delay equations with H\"older continuous coefficients. This is joint work with Alexey Kulik (Wroclaw).

We determine all solutions of the Stratonovich SDE $dX=|X|^{\alpha}\circ dB$, $\alpha\in(-1,1)$, which are strong Markov processes spending zero time in 0. This is a joint work with G. Shevchenko (Kiev), https://arxiv.org/abs/1812.05324

Signature tensors are a useful tool in the study of paths X. When X runs among a given class of path (e.g. polynomial, piecewise linear, etc), the signature of X parametrizes an algebraic variety. The geometry of this variety reflects some of the properties of the chosen class of path. The most important example is the class of rough paths, that are widely studied in stochastic analysis. Their signature variety presents many similarities to the Veronese variety, and we'll illustrate the first nice results, as well as some open questions.

In recent years, various types of stochastic resonance phenomena have been discovered. We have found that there is a transition between self-induced stochastic resonance and inverse stochastic resonance, i a simple model, the FitzHugh-Nagumo neuron model. In fact, this transition can be induced by a simple parameter change. This may also shed some light on the emergence of stochastic resonance phenomena in more complicated models, like Hodgkin-Huxley. This is joint work with Marius Yamakou.

Pesin's formula is a relation between the entropy of a dynamical system and its positive Lyapunov exponents. This formula was first established by Pesin in the late 1970s for some deterministic dynamical systems acting on a compact Riemannian manifold. Later on the same formula was obtained in some other settings. For example, different authors have proved the formula for some random dynamical systems, or have relaxed the condition of state space compactness. Nevertheless, it has never been obtained for dynamical systems with invariant measure, which is infinite. The problem is that if invariant measure is infinite, then the notion of entropy becomes senseless. Invariant measure of isotropic Brownian flows is the Lebesgue measure on R^d, which is, clearly, infinite. Nevertheless, we define entropy for such a kind of flows using their translation invariance. For the definition we exploit ideas of Brin and Katok. Then we study the analogue of Pesin's formula for these flows using the defined entropy.

We use groupoids to describe a geometric framework which can host a generalisation of Hairer's regularity structures to manifolds. In this setup, Hairer's re-expansion map (usually denoted \Gamma) is a (direct) connection on a gauge groupoid and can therefore be viewed as a groupoid counterpart of a (local) gauge fifield. This definitions enables us to make the link between re-expansion maps (direct connections), principal connections and path connections, to understand the flatness of the direct connection in terms of that of the manifold and, finally, to easily build a polynomial regularity structure which we compare to the one given by Driver, Diehl and Dahlquist. (Join work with Sara Azzali, Alessandra Frabetti and Sylvie Paycha).

The aim of this talk is investigating the dynamical properties of stochastic delay equation(SDDE) from path-wise point of view ,analyzing the long-time behavior of these equations is quite challenging, since such equations do in general not induce a stochastic flow , However we show (SDDEs) still induce flows on spaces which depend on the current position( spaces of controlled paths) , finally a version of Multiplicative Ergodic Theorem applies and yields the existence of Lyapunov exponents.
Joint work with Dr. Sebastian Riedel

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-H\"older drift in recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to a class of everywhere discontinuous coefficients. This is a joint work with M. Gerencser.

By now there exist very powerful and general solution theories for singular SPDEs, based on regularity structures and related pathwise theories. But the evolution of the laws of these equations is still poorly understood because a priori the usual probabilistic tools break down. In my talk, I will present a probabilistic theory for a class of singular SPDEs of Burgers type. We construct a domain of controlled (and non-smooth) test functions for the infinitesimal generator and based on that we study the Kolmogorov forward and backward equations and the martingale problem. This is joint work with Massimiliano Gubinelli.

We study the non-negative solution $u=u(x,t)$ to the Cauchy problem for the parabolic equation $\partial_t u=\Delta u+\xi u$ on $\mathbb Z^d\times[0,\infty)$ with initial data $u(x,0)=1_0(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, most of the total mass $U(t)=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site~$Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto frac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. In the proof, we prove and use the characterization of eigenvalue order statistics for $\Delta+\xi$ in large boxes and the exponential localisation of the corresponding eigenvectors. (Joint work with Marek Biskup and Renato dos Santos).

Existence and uniqueness of stochastic path-dependent differential equations driven by martingale noise with monotone coefficients w.r.t supremum norm is obtained. For this end, a stochastic Gronwall lemma for cadlag martingales is proved.

(Joint work with Siva Athreya, Khoa Le and Leonid Mytnik)
A standard technique for showing existence and uniqueness of solutions of SDEs with irregular drift is the Zvonkin transformation. Unfortunately, for SPDEs due to the lack of Ito’s formula this method does not work.
In the talk I will present a new approach to studying SPDEs with irregular drift, which is inspired by the stochastic sewing lemma (SSL) of Le (2018). We develop a new version of SSL, which is suitable for SPDEs and show existence and uniqueness of solutions to the stochastic heat equation with the distributional drift.

In this talk I will present my recent work on a direct method of proving the stability result for Gaussian rough differential equations. Under the strongly dissipative assumption of the drift coefficient function, I will prove that the trivial solution of the system under small noise is exponentially stable.
References:Luu, Hoang Duc: Stability theory for Gaussian rough differential equations. Part I: MIS-Preprint: 1/2019Luu, Hoang Duc: Stability theory for Gaussian rough differential equations. Part II: MIS-Preprint: 2/2019

The Dean-Kawasaki equation is an SPDE, which is used most notably in non-equilibrium statistical mechanics in order to model the evolution of fluctuating density fields. Apart from the physical reasoning, it appears as a natural candidate for a “Wasserstein-Diffusion”. However, we will prove that such dynamics are degenerate, in that the equation admits either only atomic or no solutions at all. This is joint work with Vitalii Konarovskyi and Max von Renesse.

The discussion will be devoted to a family of interacting particles on the real line which has a connection with the geometry of Wasserstein space of probability measures. We will consider a physical improvement of a classical Howitt-Warren flow (or Arratia flow, in the coalescing case), where particles sticky-reflect (or coalesce) and transfer a mass that influences their motion. We will show that such a model admits a description of each particle by a family of continuous semimartingales which satisfy some natural conditions. We will also discuss some properties of the particle system, in particular, the existence of infinite particle configurations at some moments of time.

This is a continuation of our earlier study [Stochastic Processes and their Applications, 129(1), pp.102-128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation, as well as the Green's function of the random walk in random layered conductance. This is joint work with Ryoki Fukushima.

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon some classical results in the literature. We then apply this deterministic regularity result to the corrector equation in stochastic homogenization and establish subilinearity of the corrector under essentially minimal assumptions. This is joint work with Peter Bella.

The talk is concerned with a class of two-sided stochastic processes of the form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data at time zero and $A\equiv A(W)$ is a function of $W$. Elements of the related stochastic calculus are introduced. In particular, the role of finite dimensional projections arising from the L\'{e}vy-Ciesielski representation of the Brownian motion are discussed. The calculus is adjusted to the case when $A$ is a jump process. Absolute continuity of $(X,P_{\scriptscriptstyle\pmb{\nu}})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\pmb{\nu}/dx$, and on $A$ with $A_0=0$ we verify
\begin{equation*} \frac{P_{\scriptscriptstyle\pmb{\nu}}(dX_{\cdot -t})}{P_{\scriptscriptstyle an \pmb{\nu}}(dX_\cdot)}=\frac{m(X_{-t})}{m(X_0)}\cdot\prod_i\left|\nabla_{W_0} X_{-t}\right|_i \end{equation*} a.e. where the product is taken over all coordinates. Here $\sum_i\left(\nabla_{W_0}X_{-t}\right)_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the {\it temporal homogeneity} in the sense that $X\left(W_{\cdot +v}+A_v{\, {\rm I}\mskip-10mu 1}\right) =X_{\cdot+v}(W)$, $v\in {\mathbb R}$, where $A_v{\, {\rm I}\mskip-10mu 1}$ is the trajectory taking the constant value $A_v(W)$. By means of a such a density, partial integration relative to the generator of the process $X$ is established. A further application is relative compactness of sequences of processes of the form $X^n=W+A^n$, $n\in {\mathbb N}$. ReferencesJ.-U. Löbus (2017). Absolute Continuity under Time Shift of Trajectories and Stochastic Calculus, Memoirs of the American Mathematical Society, 249, No. 1185 (2017).

We shall discuss an annealed functional CLT for ballistic RWRE in $(1/2)^-$-Hoelder rough path topology. An interesting phenomenon appears: the scaling limit of the area process is not only the Stratonovich Levy area but there is an addition of a linear term called the area anomaly. Moreover, the latter is identified in terms of the walk on a regeneration interval and naturally provides an extra information on the limiting process in case the correction is non-zero. Our result holds more generally, namely for any discrete process with bounded jumps which has a regular enough regenerative structure. This is a joint work with Olga Lopusanschi.

In this talk, we would like to consider the asymptotic behavior of a Diaconis-Freedman chain in a multi-dimensional simplex. We will give out some conditions under which we have a unique invariant (stationary) probability measure. In some special cases, we also give out explicit formulas of invariant probability density. Moreover, we completely classify all behaviors of this chain in dimensional two. The method used is based on the iterated function systems. This is the joint work with Juergen Jost and Marc Peigne.

In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity, by using new stochastic characteristics and singular heat kernel estimates.

In this talk we will discuss our recent work introducing a model of preferential attachment random graphs where the asymptotic ratio between vertices and edges of the graph is governed by a non-increasing regularly varying function $f:\mathbb{N}\to [0,1]$, which we call the edge-step function. We prove general results about the associated empirical degree distribution, as well as topological results about the clique number and the diameter. Joint work with Rémy Sanchis and Rodrigo Ribeiro.

We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by Dousse et al. in the case of a two-dimensional Poisson point process. It is an infinite-range dependent version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by Hirsch, Jahnel and Cali.
We study the SINR graph model for a stationary Cox point process in two or higher dimensions. We show that under suitable moment or boundedness conditions on the path-loss function and the intensity measure, this graph has an infinite connected component if the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds if the intensity measure is asymptotically essentially connected, and also if the intensity measure is only stabilizing but the connection radius is large. A prominent example of the intensity measure is the two-dimensional Poisson--Voronoi tessellation. We show that its total edge length in a given disk has all exponential moments. We conclude that its SINR graph has an infinite cluster if the path-loss function is bounded and has a power-law decay of exponent at least 3.

We study the large scale behavior of a branching random walk in a random environment (particles move independently and branch at a state-dependent rate provided by a random potential). We introduce a density parameter and as this parameter assumes a critical value, we can prove the scaling limit of the particle system to a rough analogous of the Superbrownian motion. We can characterize this process in dimension d=1 via a nonlinear stochastic PDE. We also study some statistical properties of the process.

Kac introduced a stochastic model that is easier to analyse than the deterministic billiard model for an ideal gas. In the Kac setting I will explain what interaction clusters are and show why they satisfy a generalisation of the Smoluchowski coagulation equation. I will also present a representation as an inhomogeneous random graph and show that the interaction clusters undergo a phase transition.

Participants

Caio Alves

Universität Leipzig

Youness Boutaib

TU Berlin

Paul Breiding

MPI MIS

Oleg Butkovsky

TU Berlin

Konstantinos Dareiotis

MPI MIS

Jean-Dominique Deuschel

TU Berlin

Joscha Diehl

MPI MIS

Ana Djurdjevac

FU Berlin

Monika Eisenmann

TU Berlin

Peter Friz

Technische Universität Berlin

Francesco Galuppi

MPI MIS

Benjamin Gess

MPI MIS

Mazyar Ghani

TU Berlin

Peter Gladbach

Universität Leipzig

Antoine Hocquet

TU Berlin

Jürgen Jost

MPI MIS

Aleksander Klimek

MPI MIS

Vitalii Konarovskii

Universität Leipzig

Wolfgang König

TU Berlin

Florian Kunick

MPI MIS

Tobias Lehmann

Universität Leipzig

Jörg-Uwe Löbus

MLU Halle-Wittenberg

Hoang Duc Luu

MPI MIS

Sima Mehri

TU Berlin + BMS

Marius Neuss

MPI MIS

Tal Orenshtein

TU Berlin

Felix Otto

MPI MIS

Robert Patterson

WIAS Berlin

Ilya Pavlyukevich

Universität Jena

Nicolas Perkowski

MPI MIS

Sebastian Riedel

TU Berlin

Tommaso Cornelis Rosato

HU Berlin

Raimundo Julián Saona

Universidad de Chile

Artem Sapozhnikov

Universität Leipzig

Mathias Schäffner

Universität Leipzig

Michael Scheutzow

TU Berlin

Vitalii Senin

TU Berlin

Scott Smith

MPI MIS

András Tóbiás

TU Berlin

Tat Dat Tran

MPI MIS

Pavlos Tsatoulis

MPI MIS

Willem van Zuijlen

WIAS Berlin

Max von Renesse

Universität Leipzig

Wu Zhaoqi

MPI MIS

Scientific Organizers

Benjamin Gess

Max-Planck-Institut für Mathematik in den Naturwissenschaften