Probability Colloquium

Minicourse "The Skorokhod embedding problem: old and new"

Speaker(s): 
Martin Huesmann (Universität Bonn)
Date: 
Tuesday, April 26, 2016 - 10:30am to Thursday, April 28, 2016 - 12:00pm
Location: 
TU Berlin (room MA748) and HU Berlin (room 1.115)

The Skorokhod embedding problem (SEP) is to represent a given probability measure as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years the SEP has become one of the important classical problems in probability theory with a huge variety of different solutions that have been employed in various branches of pure and applied probability. In recent years there has been a revived interest in the SEP due to its connection to model independent finance and martingale optimal transport.

A semigroup approach to nonlinear expectations

Speaker(s): 
Michael Kupper (Universität Konstanz)
Date: 
Wednesday, April 20, 2016 - 6:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. Based on a nonlinear version of the Daniell-Stone theorem we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite dimensional path space. As an llustration, we construct nonlinear Markov chains in discretetime. In the second part of the talk we construct sublinear semigroups and discuss the link to non-linear expectations and PDEs. The talk is based on joint work with Robert Denk and Max Nendel.

Limiting dynamics of the condensate in the reversible inclusion process on a finite set

Speaker(s): 
Alessandra Bianchi (University of Padova)
Date: 
Wednesday, February 3, 2016 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The inclusion process is a stochastic lattice gas where particles perform random walks subjected to mutual attraction, thus providing the natural bosonic counterpart of the well-studied exclusion process. Due to attractive interaction between particles, the inclusion process can exhibit a condensation transition where a finite fraction of all particles concentrates on a single site. In this talk we characterize the dynamics of the condensate for the reversible inclusion process on a finite set S, in the limit of total number of particles going to infinity.

Existence and Uniqueness of Mild Solutions to the Stochastic Neural Field Equation with Discontinuous Firing Rate

Speaker(s): 
Jennifer Krüger (TU Berlin)
Date: 
Wednesday, February 3, 2016 - 5:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The neural field equation models the spatiotemporal evolution of neural activity in thin layers of cortical tissue from a macroscopic point of view. Under the influence of spatially extended additive noise (WQ) it is given by
du(x, t) = (−u(x, t) + \int_-\infty^\infty w(x − y) F(u(y, t)) dy)dt + \sigma dW^Q(x, t) (1)

On the Membrane Potential in a Coupled Network of Integrate-and-Fire Neurons

Speaker(s): 
Alexandra Munteanu (TU Berlin)
Date: 
Wednesday, January 20, 2016 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The stochastic integrate-and-fire model describes the membrane potential of a neuron by a stochastic differential equation up until the time when it reaches a certain threshold V_F . At this point the neuron is said to ßpikeör fireänd the voltage is reset to a resting value V_R. In the model we consider, the neuron is coupled to a large network, receiving input from the surrounding neurons. Integrating this influence of the network into the model, we obtain an associated
mean-field equation.

Noise-induced stability

Speaker(s): 
Matti Leimbach (TU Berlin)
Date: 
Wednesday, January 20, 2016 - 5:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

It is known that there are certain ODEs which are explosive, but turn into non-explosive SDEs by adding white noise. In this case the Markov process associate to the SDE often admits an invariant probability measure. This phenomenon is called noise-induced stability. We investigate whether the additive noise can induce a stronger concept of stability, namely the existence of a random attractor. We present two examples which answer this question differently and sketch the main ideas of the proofs. Joint work with M. Scheutzow.

Stochastic geometry and stochastic analysis: Poisson U-statistics

Speaker(s): 
Matthias Reitzner (Universität Osnabrück)
Date: 
Wednesday, January 6, 2016 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Assume that X is a Poisson point process. A Poisson U-statistic with kernel f is the sum of f(x_1,...x_k) over all k-tuples of X. Poisson U-statistics play an important role in Stochastic Geometry. Many functionals of interest can be written as U-statistics. We investigate some elementary properties of Poisson U-statistics, and the limit behaviour of Poisson U-statistics, central limit theorems and concentration inequalities.

A paracontrolled solution for PAM in three dimensions

Speaker(s): 
Jörg Martin (HU Berlin)
Date: 
Wednesday, December 16, 2015 - 5:00pm
Location: 
TU Berlin, Raum MA004, Straße des 17. Juni 136, 10623 Berlin

We present a general link between paracontrolled analysis and regularity structures. Combining both techniques we present a new construction of the solution for the parabolic Anderson model on T^3.

Convergence and regularity of probability laws by using an interpolation method

Speaker(s): 
Vlad Bally (Marne-la-Vallée)
Date: 
Wednesday, November 25, 2015 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Fournier and Printems [Bernoulli, 2010] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Hölder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. By employing some Besov space techniques, Debussche and Romito [Probab. Theory Related Fields, 2014] have substantially improved the result of Fournier and Printems.

On the approximate Hölder index for trajectories of stable processes

Speaker(s): 
Vitalii I. Senin (TU Berlin)
Date: 
Wednesday, November 25, 2015 - 5:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

For almost all trajectories of the symmetric \alpha stable process (\alpha < 2) the following property is proved: for any \gamma with \alpha \gamma < 1 and any \epsilon > 0 there exists a Hölder continuous function with exponent \gamma which coincides with the trajectory up to a set of Lebesgue measure \leq \epsilon.
(In the collaboration with A. M. Kulik)

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