## Thursday, June 30

Thursday: Morning session
1000-1030Coffee break
1030-1115 Mathias Beiglböck (TU Wien)
The geometry of Skorokhod embedding
1120-1145 Julio Backhoff (TU Wien)
On the symmetry between the Root and Rost embeddings and their connection to certain optimal stopping problems
1150-1215 Jiří Černý (Universität Wien)
The maximum particle of branching random walk in spatially random branching environment
1215-1400 Lunch break
Thursday: Afternoon session
1400-1425Wilfried Huss (TU Graz)
Rotor-router walks on Galton-Watson trees
1430-1455 Sebastian Müller (Aix-Marseille)
Infinite excursions of router walks on regular trees
1500-1525 Wojciech Cygan (TU Graz)
Oscillating heat kernels on ultrametric spaces
1530-1600Coffee break
1600-1625 Alexander Steinicke (U Innsbruck)
Malliavin differentiation of locally Lipschitz BSDEs in the Lévy case.
1630-1655 Florian Baumgartner (U Innsbruck)
Random jump characteristics for advection equations and related subordinators
1700-1725 Michaela Szölgyenyi (WU Wien)
Numerics for multidimensional SDEs with discontinuous drift
1830 Conference dinner Restaurant L'Osteria, Mehlplatz 1, 8010 Graz.

## Friday, July 1

Friday: Morning session
0900-0925 Evelyn Buckwar (JKU Linz)
Stochastic Differential-Algebraic Equations with Noise in the Constraints
0930-0955Harald Hinterleitner (JKU Linz)
A stochastic version of a neural mass model - properties and numerics
1000-1030Coffee break
1030-1115 Rudolf Grübel (Hannover)
Combinatorial Markov Chains
1120-1155Andreas Thalhammer (JKU Linz)
Importance sampling techniques for SPDEs
1200-1225Massimiliano Tamborrino (JKU Linz)
Noise may enhance the efficacy of latency coding
Friday: Afternoon session
1430-1515 Nikolaos Fountoulakis (University of Birmingham)
Critical phenomena for bootstrap percolation on inhomogeneous random graphs
1530-1600Annika Heckel (University of Oxford)
The chromatic number of dense random graphs
1600-1630Beata Benyi (Baja, Hungary)
Combinatorics of poly-Bernoulli numbers

## Titles and Abstracts (in alphabetical order)

1. Julio Backhoff (TU Wien)
Title: On the symmetry between the Root and Rost embeddings and their connection to certain optimal stopping problems.

Abstract: Recent works by A. Cox and J. Wang have highlighted the connection between the Rost/Root solutions to the Skorokhod Embedding Problem and certain optimal stopping problems. In this talk we present a simple probabilistic argument for this connection (as opposed to the original analytical ones) which furthermore implies an interesting symmetry between the Rost and Root solutions. This is joint work with M. Beiglböck, A. Cox and M. Huesmann.

2. Florian Baumgartner (U Innsbruck)
Title: Random jump characteristics for advection equations and related subordinators.

Abstract: Modelling irregular media with random layers, where in each layer particles propagate at constant but random speed, a layer refinement and limiting procedure yields paths of subordinators. In order to apply the method of characteristics for the advection equation in this generalized setting, it is necessary to pursue the trajectories from a given space-time point back in time. For numerical computations it is of interest if and how the trajectories can be approximated in a suitable way. The talk attempts to answer both questions. Distributional assertions on the first one require the joint distribution of the first passage time of a subordinator across a level and the corresponding left and right limits of the process.

3. Mathias Beiglböck (TU Wien)
Title: The geometry of Skorokhod embedding.

Abstract: The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructedsolutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings.

4. Beata Benyi (Baja, Hungary)
Title: Combinatorics of poly-Bernoulli numbers

Abstract: The poly-Bernoulli numbers - a natural generalization of the classical Bernoulli numbers - were introduced by Kaneko during his investigations of multiple zeta values in 1997. Poly-Bernoulli numbers of negative indices are natural numbers. In this talk we present combinatorial interpretations and interesting combinatorial properties of these numbers.

5. Evelyn Buckwar (JKU Linz)
Title: Stochastic Differential-Algebraic Equations with Noise in the Constraints.

Abstract: We provide an introduction to stochastic differential-algebraic equations, which may model for example electronic circuits with transient noise. In particular we discuss the case where there is a noise source in the circuit that will appear as a noise term in the algebraic constraint of the system of equations.

6. Jiří Černý (Universität Wien)
Title: The maximum particle of branching random walk in spatially random branching environment.

Abstract: Branching random walk and Brownian motion have been the subject of intensive research during the last decades. We consider branching random walk and investigate the effect of introducing a spatially random branching environment. We are primarily interested in the position of the maximum particle, for which we prove a CLT. Our result correspond, on an analytic level, to a CLT for the front of the solutions to a randomized Fisher-KPP equation, and also to a CLT for the parabolic Anderson model.

7. Wojciech Cygan (TU Graz)
Title: Oscillating heat kernels on ultrametric spaces.

Abstract: We compute the precise asymptotic behaviour (on the diagonal) for the heat kernel of the hierarchical Laplacian on ultrametric spaces. More precisely, let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ of $X$ we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts in $L^{2}(X,m),$ is essentially self-adjoint and has a purely point spectrum. Under certain mild assumptions its Markov semigroup $(e^{-tL}% )_{t>0}$ admits a continuous heat kernel $p(t,x,y)$ w.r.t. $m.$ We study asymptotic behaviour of the function $t\rightarrow p(t,x,x)$ when $X$ is a group and the operator $L$ is translation invariant. In this case $p(t,x,x)$ does not depend on $x$, denote it $p(t).$ The function $p(t)$ is completely monotone and never varies regularly. When $X=\mathbb{Q}_{p}$, the ring of $p$-adic numbers, and $L=\mathcal{D}^{\alpha}$, the operator of fractional derivative of order $\alpha,$ we show that $p(t)=t^{-1/\alpha}\mathcal{A}% (\log_{p}t)$, where $\mathcal{A}(\tau)$ is a continuous non-constant $\alpha$-periodic function. We study asymptotic behaviour of $\min\mathcal{A}$ and $\max\mathcal{A}$ as the parameter $p$ tends to $\infty$. When $X=S^{(\infty)}$, the infinite symmetric group, and $L$ is a hierarchical Laplacian which has similar to $\mathcal{D}^{\alpha}$ metric structure, we show that, in contrary to the previous case, the complete monotone function $p(t)$ oscillates between two functions $\phi(t)$ and $\Phi(t)$ such that $\phi=o\left( \Phi\right)$ at infinity. This is a joint work with A. Bendikov and W.Woess.

8. Nikolaos Fountoulakis (University of Birmingham)
Title: Critical phenomena for bootstrap percolation on inhomogeneous random graphs.

Abstract: The bootstrap percolation processes were introduced in the late 1970 by Chalupa, Leath and Reich in statistical physics. Besides their s ignificance in that context, nowadays these are viewed as epidemic o r dissemination processes on graphs. In this talk, I will discuss a phase transition on the evolution of a bootstrap percolation process on the class of inhomogeneous random graphs that have a kernel of rank 1. These resemble the metastability phenomena of the bootstrap process, that were first observed by Aizenman and Lebowitz on large but finite boxes of the integer lattice. This is joint work with M. Kang, K. Koch and T. Makai.

9. Rudolf Grübel (U Hannover)
Title: Combinatorial Markov Chains.

Abstract: We consider Markov chains that are adapted to a combinatorial family in the sense that, at time $n=1,2,\ldots$, the state of the chain is an object of the family that has size $n$. These structures are of importance in Discrete Mathematics (what does a typical discrete structure look like?) but also in Theoretical Computer Science where they appear as the output of sequential algorithms with random input. Not very surprisingly, a central question is the behaviour of such structures as $n\to\infty$. It turns out that Markov chain boundary theory, as initiated by Doob in a seminal paper from 1959, provides the right' limit concept. In particular the stochastic dynamics lead to a very specific augmentation of the respective combinatorial family. Boundary theory is an intricate subject with many difficult problems. For combinatorial Markov chains the theory is considerably easier than in the general case, but it may still be a demanding task to determine the boundary in specific cases and to relate these to familiar mathematical structures. In the talk I will outline the theoretical background, and we will then walk through a zoo of examples of varying complexity.

10. Annika Heckel (University of Oxford)
Title: The chromatic number of dense random graphs.

Abstract: The chromatic number of the random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant. In this talk, I will present new upper and lower bounds which match each other up to a term of size o(1) in the denominator. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value. In contrast to earlier results, the upper bound was obtained through the second moment method, and I will give details on some aspects of the proof.

11. Harald Hinterleitner (JKU Linz)
Title: A stochastic version of a neural mass model - properties and numerics.

Abstract: Neural mass models provide a useful framework for modelling mesoscopic neural dynamics and in this talk we consider the Jansen and Rit Neural Mass Model (JR-NMM). This system of ODEs has been introduced as a model in the context of electroencephalography (EEG) rhythms and evoked potentials and has been proposed as an underlying model in various application settings. Incorporating random input, we formulate a stochastic version of the JR-NMM which has the structure of a nonlinear stochastic oscillator. We introduce the stochastic analogon of the convolution-based formulation of the JR-NMM and derive several properties of the stochastic system, e.g. estimates on the expected value and the second moment of the solution, bounds on the escape probability and long-term behaviour of the solution. Finally, we briefly address the question of efficient numerical integrators based on a splitting approach which preserve the qualitative behaviour of the solution. Co-authors: Markus Ableidinger, Evelyn Buckwar.

12. Wilfried Huss (TU Graz)
Title: Rotor-router walks on Galton-Watson trees.

Abstract: A rotor-router walk is a deterministic walk on a graph, where the exits from each vertex follow a fixed cyclic order. We give a necessary and sufficient condition for recurrence of rotor-router walks on supercritical Galton-Watson trees, in the case where the first exit of each vertex is chosen at random. Joint work with Sebastian Müller and Ecaterina Sava-Huss.

13. Sebastian Müller (Aix-Marseille)
Title: Infinite excursions of router walks on regular trees.

Abstract: A router configuration on a graph contains in every vertex an infinite ordered sequence of routers, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a router walk is a deterministic process: at each step it chooses the next unused router in its current location, and uses it to jump to the neighboring vertex to which it points. Routers walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question for natural random configuration models on regular trees. Our results suggest that in this context the router model behaves like the simple random walk unless it is not “close to” the standard rotor-router model. (joint work with Tal Orenshtein)

14. Michaela Szölgyenyi (WU Wien)
Title: Numerics for multidimensional SDEs with discontinuous drift.

Abstract: When solving certain stochastic optimization problems, e.g., in mathematical finance, the optimal control policy is of threshold type, meaning that it depends on the controlled process in a discontinuous way. The stochastic differential equations (SDEs) modeling the underlying process then typically have discontinuous drift and degenerate diffusion parameter. This motivates the study of a more general class of such SDEs. We prove an existence and uniqueness result, based on certain a transformation of the state space by which the drift is `made continuous''. As a consequence the transform becomes useful for the construction of a numerical method. The resulting scheme is proven to converge with strong order $1/2$. This is the first result of that kind for such a general class of SDEs. In examples we show the necessity of the geometric conditions we pose. Finally, we present a numerical example, in which the drift of the considered SDE is discontinuous on the unit circle. Joint work with G. Leobacher (JKU Linz).

15. Alexander Steinicke (U Innsbruck)
Title: Malliavin differentiation of locally Lipschitz BSDEs in the Lévy case.

Abstract: In order to investigate whether solutions of backward stochastic differential equations (BSDEs) possess densities, one approach is to apply results of Malliavin calculus to these equations. Therefore it is necessary to check if BSDEs are Malliavin differentiable. In the case of Lipschitz driving functions and a Brownian noise, differentiability has already been shown in the 1990s. However, recent results cover BSDEs driven by a Lévy process, certain ones even with driving functions that are of quadratic growth. In our present work, using a Lévy setting, we show that for terminal conditions which are 'smooth enough' in the sense of Malliavin calculus, results on differentiability and boundedness of solutions in the case of locally Lipschitz driving functions can be deduced. We apply the results to prove existence and boundedness of a solution to BSDEs which are beyond the locally Lipschitz assumption. This is a joint work with Christel Geiss.

16. Massimiliano Tamborrino (JKU Linz)
Title: Noise may enhance the efficacy of latency coding.

Abstract: In neuroscience, stochastic processes and their hitting times are used to describe the membrane potential dynamics of single neurons and to reproduce neuronal spikes, respectively. The time to the first spike after the stimulus onset typically varies with the stimulation intensity. Experimental evidence suggests that neural systems utilize such response latency to encode information about the stimulus. Our aim is to understand what are the ultimate limits on the accuracy of stimulus decoding based on the first-spike latency in presence of background noise, modeled by spontaneous activity. Paradoxically, the optimal performance is achieved at a non-zero level of noise. Therefore, noise may enhance signal transmission even in a setting as simple as the Brownian motion. The reported decoding accuracy improvement represents a novel manifestation of the noise-aided signal enhancement. .

17. Andreas Thalhammer (JKU Linz)
Title: Importance sampling techniques for SPDEs.

Abstract: In this talk, we consider Monte Carlo-based methods for estimating $\mathbb{E}[f(X(T))]$, where $X(T)$ denotes the mild solution of a stochastic partial differential equation (SPDE) at a given time T. It is a well-known result that the resulting Monte Carlo error can be controlled by either enlarging the number of realisations or by applying appropriate variance reduction methods. Obviously, a natural bound on the number of trajectories is imposed by the computational cost of the time integration method, which limits the possibility of increasing the number of numerical trajectories for high dimensional SODEs - especially for systems arising from semi-discretised SPDEs. For this reason, we present different approaches how importance sampling can be applied to SPDEs in order to reduce the variance of the quantity of interest.