Consider a discrete-time random walker moving on the set X = {na|n ∈ Z}, where a is the spacing between the discrete positions. Let p₊, p₀ and p− denote the probability that the walker from time t to t + δ makes a move to the right, makes no move, makes a move to the left.

Transcribed Image Text:

(a) Simulate the process and extract an effective Hurst exponent from the behaviour of (x(t)) for small values of t. (b) Establish the master equation for P(x,t) = the probability that a walker starting at x = 0, at time t = 0 is at position x at time t. (c) By numerical iteration, find P(x, t) and the moments (x(t)) and (x(t)) for, say p+ = p = 0.4, for p+ = p = 0.1 and for p+ = 0.3, p = 0.2 (d) For p+ = p-, in the limit a 0 and 80 derive a PDE for P(x,t). (e) From the behaviour of (x(t)) determine the Hurst exponent.