Vortrag: "Brownian Motion with Partial Resetting"

Der Vortrag wird in englischer Sprache gehalten:

Robert Brown observed the irregular movement of small particles in water. Brownian motion, named after him, has since become increasingly important. Louis Bachelier used this movement in 1900 to model stock prices, paving the way for modern financial mathematics. 
Just five years later, Albert Einstein used this Brownian motion as strong evidence for the existence of atoms, interpreting it as the result of collisions between particles and even smaller particles.
In this lecture, we consider a one-dimensional Brownian motion with partial resetting. Processes of this type play an important role in stochastic search algorithms today.
We introduce this process and then examine its long-term stochastic behaviour. It turns out that there is a unique equilibrium distribution that can be determined explicitly, and that the exact exponential convergence rate to this distribution can be calculated using coupling techniques.
Since the polynomial eigenfunctions of the generator can be explicitly calculated, Doob's h-transform yields a new stochastic process that ultimately turns out to be the original process conditioned to stay positive. Interestingly, this process changes its jump rate, and the new rate can be explicitly described in terms of the rate of the original process. This can be seen as another example of entropic repulsion, since a change in spatial motion from a Brownian to a Bessel process would be sufficient to prevent hitting the origin.

Der Vortrag und die anschließende Diskussion finden am 13.01.2026 um 16h in Raum 5.1.06 statt.