Course unit content
Characteris2c func2ons, Gauss and Lévy sta2s2cs; central limit
theorems, law of large numbers. Genera2ng Lévy flights and fractals
on a computer. Lévy sta2s2cs for long ranged interac2ng systems, the
Holtsmark problem.
2. Random walks, Polya's problem, the diffusion equa2on, classical
first passage 2mes, introduc2on to quantum first hi9ng 2mes with
applica2ons to quantum compu2ng.
3. Renewal theory for thin and fat tailed distribu2ons. Abelian and
Tauberian theorems. Sta2s2cal aging and weak ergodicity breaking.
Stochas2c model of a blinking quantum dot.
4. The con2nuous 2me random walk, long tailed wai2ng 2me
distribu2ons in the context of diffusion in disordered systems. The
trap model. Frac2onal deriva2ves. Frac2onal 2me diffusion equa2on.
5. Brownian mo2on, Einstein rela2ons, Langevin equa2on, Fokker-
Planck equa2on, Langevin equa2ons with memory, fluctua2on
dissipa2on theorem.
6. Weak ergodicity breaking for anomalous diffusion with applica2on
to single molecule tracking the cell environment.
7. Exponen2al tails of spreading packets in disordered systems, the
recent revival of an inspiring proposal of Laplace.
8. Extreme value sta2s2cs. Big jump principle.
9. Introduc2on to infinite ergodic theory using stochas2c methods
Full programme
Characteristic functions, Gauss and L evy statistics; central limit theorems, law of
large numbers. Generating L evy
ights and fractals on a computer. L evy statistics
for long ranged interacting systems, the Holtsmark problem.
2. Random walks, Polya's problem, the di usion equation, classical rst passage times,
introduction to quantum rst hitting times with applications to quantum computing.
3. Renewal theory for thin and fat tailed distributions. Abelian and Tauberian theo-
rems. Statistical aging and weak ergodicity breaking. Stochastic model of a blinking
quantum dot.
4. The continuous time random walk, long tailed waiting time distributions in the
context of di usion in disordered systems. The trap model. Fractional derivatives.
Fractional time di usion equation.
5. Brownian motion, Einstein relations, Langevin equation, Fokker-Planck equation,
Langevin equations with memory,
uctuation dissipation theorem.
6. Weak ergodicity breaking for anomalous di usion with application to single molecule
tracking the cell environment.
7. Exponential tails of spreading packets in disordered systems, the recent revival of
an inspiring proposal of Laplace.
8. Extreme value statistics. Big jump principle.
9. Introduction to in nite ergodic theory using stochastic methods.
