Probability Colloquium

Adaptive dynamics, diploid models, and the escape from selection

Speaker(s): 
Anton Bovier (Bonn)
Date: 
Wednesday, July 19, 2017 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Standard stochastic models of adaptive dynamics are based on haploid reproduction schemes. They are sometimes criticised for the difficulty to create genetic diversity. This can be traced to the fact that unless parameters are fine-tuned, times of fixation and extinctions are of the same order. We show that in diploid models, it fitter alleles are dominant, extinction times are dramatically increased.

Invariant Measure for Random Walks on Random Environments

Speaker(s): 
Ilya Goldsheid (London)
Date: 
Wednesday, July 5, 2017 - 6:15pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041

It is well known that a random walk in random environment on a strip in the environment viewed from the particlesetting is a Markov chain on the set of environments. In this context, the fundamental question is that of the existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments.

Large deviations and concentration of scaling limits for weakly pinned integrated random walks

Speaker(s): 
Stefan Adams (University of Warwick)
Date: 
Wednesday, June 7, 2017 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We study scaling limits and corresponding large deviation principles of integrated random walks perturbed by an attractive force towards the origin. In particular we analyse the critical situation that the rate function admits more than one minimiser leading to concentration of measure problems. The integrated random walk models are in fact interface models with Laplacian interaction, and such linear chain models with Laplacian interaction appear naturally in the physical literature in the context of semi-flexible polymers.

Continuous spin models on annealed random graphs: Modifying the modified mean-field exponents

Speaker(s): 
Christof Külske (Ruhr-Universität Bochum)
Date: 
Wednesday, May 24, 2017 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We study Gibbs distributions of continuous spins on generalized random graphs. Our main interest lies in the critical behavior of models which show a second order phase transition. We find critical exponents which differ from the mean field exponents when the weight distribution of the generalized random graph becomes too heavy-tailed. For the Ising model this has been proved to occur by Dommers, van der Hofstad, Giardina, and others very recently (CMP 2016).

On a selection problem for small noise perturbation of ODE in multidimensional case

Speaker(s): 
Andrey Pilipenko (Kiev Polytechnic Institute)
Date: 
Wednesday, April 26, 2017 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The identification problem of the limit of an ODE with non-Lipschitz drift perturbed by a zero-noise is considered in a multidimensional framework. This problem is a classical subject of stochastic analysis, however the multidimensional case was poorly investigated. We consider two cases in particular:
(i) the drift coefficient has a jump discontinuity along an hyperplane and is Lipschitz continuous in the upper and lower half-spaces;
(ii) the drift is equivalent to a (phi) r^\alpha as r tends to 0, where (r,phi) are the polar coordinates, and \alpha < 1.

Random walks on dynamic random graphs

Speaker(s): 
Frank den Hollander (Leiden)
Date: 
Wednesday, April 19, 2017 - 6:15pm
Location: 
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

The mixing time of a Markov chain is the time it needs to approach its stationary distribution. For random walks on graphs, the characterisation of the mixing time has been the subject of intensive study. One of the motivations is the fact that the mixing time gives information about the geometry of the graph. In the last few years, much attention has been devoted to the analysis of mixing times for random walks on random graphs, which poses interesting challenges.

Large deviations for random projections of $\ell^p$ balls

Speaker(s): 
Nina Gantert (TU München)
Date: 
Wednesday, February 15, 2017 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

We give large deviation results for random projections of $\ell^p$ balls. They quantify the well-know statement that two independently drawn vectors whose law is uniform on a high-dimensional sphere, are nearly orthogonal. We give both quenched large deviation principles (fixing the sequence of projection directions) and annealed large deviation principles (averaging over the sequence of projection directions). There is an analogy with random environments, and we present a variational formula relating the two rate functions.

Gradient flows, heat equation, and Brownian motion on time-dependent metric measure spaces

Speaker(s): 
Karl-Theodor Sturm
Date: 
Wednesday, February 15, 2017 - 5:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

We study the heat equation on time-dependent metric measure spaces (being a dynamic forward gradient flow for the energy) and its dual (being a dynamic backward gradient flow for the Boltzmann entropy). Monotonicity estimates for transportation distances and for squared gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space.

Particle representations for SPDEs with boundary conditions

Speaker(s): 
Dan Crisan (Imperial College London)
Date: 
Wednesday, February 1, 2017 - 5:15pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

I will present a weighted particle representation for a class of stochastic partial differential equations with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations (SDEs). The evolution of the particles is modeled by an infinite system of stochastic differential equations with reflecting boundary condition and driven by independent finite dimensional Brownian motions.

Densities for solutions of stochastic PDEs

Speaker(s): 
Marco Romito (Università di Pisa)
Date: 
Wednesday, January 18, 2017 - 6:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 004

We present a general method to prove existence and minimal regularity of the density with respect to the Lebesgue measure of solutions of stochastic differential equations with non-smooth coefficients. We give some examples of application to suitable finite dimensional functionals of solutions of stochastic PDEs.

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