Divide the interval $[0,1]$ in $n$ subintervals with length $\frac{1}{n}$. The $n$ subintervals are numerated from $1$ to $n$. We have a particle that, after an exponential time of parameter $1$, chooses a site $y\in[0, 1]$ according to the kernel $J:[0,1]\times[0,1]\to\mathbb R^+$, in other words, if the particle is situated in $x$, the probability to choose a site in a region $A\subset[0, 1]$ is given by $\int_AJ(x, y)dy$. Once that the particle chooses the site it jumps on it but with the following restriction: jumps from even subintervals to odd subintervals are not allowed, all the other jumps are allowed.

Let $X_n(t)$ be the position of the particle at time t. I would like to understand which is the distribution of the position of the particle when the parameter $n\to +\infty$.

Has someone any idea to compute such limit distribution?

Thank you