Probability Seminar Essen

Probability Seminar Essen

Covers a wide range of topics in Probability and its applications.

Current Schedule for the Summer Semester 2017

Previous Semesters

April 25 Sandra Kliem (Universität Duisburg-Essen)
May 2 NN (NN)
May 9 NN (NN)
May 16 NN (NN)
May 23 Zakhar Kabluchko (Westfälische Wilhelms-Universität Münster)
May 30 Sander Dommers (Ruhr-Universität Bochum)
June 13 Airam Blancas Benítez (Goethe-Universität Frankfurt am Main)
June 20 Gabriel Hernán Berzunza Ojeda (Georg-August-Universität Göttingen)
June 27 Stein Andreas Bethuelsen (Technische Universität München)
July 4 NN (NN)
July 11 Manfred Opper (Technische Universität Berlin)
July 18 NN (NN)
July 25 Stephan Gufler (Goethe-Universität Frankfurt am Main)

In red are non-standard times/locations

Winter Semester 2016/17

April 25 Sandra Kliem (Universität Duisburg-Essen)

Travelling wave solutions to the KPP equation with branching noise

The one-dimensional KPP-equation driven by space-time white noise, $$ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 $$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. In a next step, I explain how to obtain a travelling wave solution to this SPDE. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions.