We consider a discrete space, continuous time random walk on the integer line where the walker hops from a lattice site to a nearest neighbor site.
Suppose the walker has one step memory and it remembers whether it arrived to its location by jumping to the right or left. Here, we can define four probabilities which describe our process:
$P(rr)$ := probability that the walker jumps to the right given that its previous jump was to the right.
$P(rl)$ := probability that the walker jumps to the right given that its previous jump was to the left.
$P(lr)$ and $P(ll)$ can be analogously defined.
With this information, can we comment on the average velocity of the walker?
The answer would be proportional to the difference between probability of hopping to the right and to the left. But I'm unable to incorporate memory in this argument.
Let me know if I was unclear in my question. Any help in the right direction is appreciated 🙂
Best Answer
The four transition probabilities constitute a transition matrix between the two velocity states (going left and going right). Let $u_t$ be the probability of going left at time $t$, and $v_t$ be the probability of going right; then $$ \left(\begin{matrix}u_{t+1} \\ v_{t+1}\end{matrix}\right)=\left(\begin{matrix}P_{ll} & P_{lr} \\ P_{rl} & P_{rr}\end{matrix}\right) \left(\begin{matrix}u_{t} \\ v_{t}\end{matrix}\right)=\left(\begin{matrix}P_{ll} & 1-P_{rr} \\ 1-P_{ll} & P_{rr}\end{matrix}\right) \left(\begin{matrix}u_{t} \\ v_{t}\end{matrix}\right), $$ using the fact that $P_{ll}+P_{rl}=P_{rr}+P_{lr}=1$. The eigenvalues of the matrix are found to be $1$, with corresponding eigenvector $$u_*=\frac{1-P_{rr}}{(1-P_{rr})+(1-P_{ll})},\;v_*=\frac{1-P_{ll}}{(1-P_{rr})+(1-P_{ll})},$$ and $P_{ll}+P_{rr}-1$, with corresponding eigenvector $u=-1/2, v=1/2$. Except in the special cases where either $P_{ll}=P_{rr}=0$ (always change directions, so average velocity is zero), or $P_{ll}=P_{rr}=1$ (never change directions, so velocity is fixed at its initial value), you have $(u_t,v_t)\rightarrow(u_*, v_*)$ as $t\rightarrow\infty$. The average velocity is $$ v_*-u_*=\frac{P_{rr}-P_{ll}}{(1-P_{rr})+(1-P_{ll})}. $$