Seminarium Fizyki Materii Skondensowanej

Using a very simple setup, i.e. escape of a Brownian particle from afinite interval, some selected properties of stochastic dynamics will bepresented. Next, it will be demonstrated how these properties change wheninstead of a Brownian motion one considers more general random walk model,i.e. Levy flights, or when a finite interval is replaced with a half-lineor a point like domain. Finally, properties of first escapes and firstarrivals for Levy flights will be discussed.Levy flights provide natural extension of Brownian motion. Contrary to theBrownian motion trajectories of Levy flights are discontinuous. Discontinuity of trajectories changes conditions of absorption making themnon-local because due to long jumps Levy flight process can leave a domainof motion without hitting a boundary. Analogously, the process can hit atarget without hitting the target's boundary. This in turn makes adistinction between first arrivals and first passages. Moreover, longjumps introduces leapovers because point like boundaries are more likelyto be crossed than hit. As a consequence, properties of models with Levyflights are more intriguing and less intuitive than properties of theirBrownian counterparts.