# Distribution limit of a jump process

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

For a fixed $$t>0$$, the limit does not need to exist. For example, take $$J \equiv 1$$, $$t = 1$$, and the starting position $$X(0) = a$$ to be in the set of simply normal numbers in base $$2$$. Then as $$n\to \infty$$, the probability that a jump occurred by time $$t = 1$$ is going to oscillate between $$e ^{-1}$$ and $$e^{-\frac 12 t}$$. Thus, depending on whether the $$n$$-th digit of $$a$$ is $$0$$ or $$1$$, the distribution of $$X_n(1)$$ is going to be either $$e ^{-1} \delta _a + (1 - e ^{-1}) u_{[0,1]}$$ or $$e ^{-\frac 12 } \delta _a + (1- e^{-\frac 12 }) u_{[0,1]}$$ where $$u_{[0,1]}$$ is the uniform distribution on $$[0,1]$$.