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## Main Question or Discussion Point

Hi, can someone who is familiar with the analysis of random walks (statistical mechanics, condensed matter physics etc.) help me on solving a particular problem?

We define the following random walk, the random variable w(t) is evolved as

w(t+1)=w(t), with probability of p;

w(t+1)=w(t)+C*(1-w(t))*f, with probability 1-p.

C is a constant, f is a random variable with known distribution, and f is not correlated with w.

Following the way of analyzing a Brownian motion, what I have tried is to write down

E(w(t)|w(t-1),p), then show the E(w(t)|p) using the induction, for a given w(0).

Next, at least I need to find the Var(w(t)) , or equivalently E(w(t)^2) since I have known (E(w(t)))^2 for a given w(0).

I hope that someone can help me find the moment generating function of this random walk

(E(w(t)^n)), or alternatively just the mean and the variance of w(t). Any hint on this is badly needed.

We define the following random walk, the random variable w(t) is evolved as

w(t+1)=w(t), with probability of p;

w(t+1)=w(t)+C*(1-w(t))*f, with probability 1-p.

C is a constant, f is a random variable with known distribution, and f is not correlated with w.

Following the way of analyzing a Brownian motion, what I have tried is to write down

E(w(t)|w(t-1),p), then show the E(w(t)|p) using the induction, for a given w(0).

Next, at least I need to find the Var(w(t)) , or equivalently E(w(t)^2) since I have known (E(w(t)))^2 for a given w(0).

I hope that someone can help me find the moment generating function of this random walk

(E(w(t)^n)), or alternatively just the mean and the variance of w(t). Any hint on this is badly needed.