Theoretical Physics Colloquium

Arcsine law in fractional Brownian motion.

by Dr. Tridib Sadhu (TIFR, Mumbai)

Tuesday, November 21, 2017 from to (Asia/Kolkata)
at AG 69
The three arcsine laws are one of the most non-intuitive pedagogical properties of the Brownian motion. These deal with the following three observables of a Brownian motion evolving in a finite time window: (1) the fraction of time it remained positive, (2) the last time it crossed the origin, (3) and the time when it reached its maximum. All three observables have the same cumulative probability distribution expressed as an arcsine function, hence the name arcsine law. I shall discuss how these three laws change for a fractional Brownian motion which is a generalization of the Brownian motion defined by Benoit Mandelbrot to describe anomalous diffusion seen in many natural examples. This is a non-Markovian Gaussian process indexed by Hurst exponent H (H=1/2 corresponds to the standard Brownian). Due to the non-Markovian nature analytical results are hard to obtain. I shall derive the three probability distributions and show that they are all different for general H.