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.

(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.