Consider a random walk starting at $x_0 \in [0,1]$. At any discrete time step $k \in \mathbb{N}$, the new position is given by: $$x_k = \min\{\max\{x_{k-1}+C_k,0\},1\}$$ where $C_k \sim U(-a,a)$ are i.i.d. uniform random variables with support in $[-a,a]$. In other words, we are moving at random in $[0,1]$ and we have a certain probability to be ''absorbed'' at $0$ or $1$ and then eventually reflected back.
I am trying to find the stationary distribution of this process. Based on simulations, I have the feeling that the stationary distribution must be something like$$\pi(x) = \alpha \delta_0(x)+ \alpha \delta_1(x)+(1-2\alpha){1}_{[0,1]}(x) $$where $\alpha$ is the stationary probability of being at the extreme points and ${1}_{[0,1]}$ is the indicator function on $[0,1]$.
Is there any way to find $\pi(x)$ rigorously?
$\endgroup$ 51 Answer
$\begingroup$Consider the discrete version. Suppose there are $N$ states.
You can move from any state $x$ to any state from $x-m$ to $x+m$ with probability $1/(2m+1)$.
Adjustments are needed for states $1$ and $N$.
Build the transition matrix. Find its dominant eigenvector and plot it.
Play around with various $N$ and $m$.
