Question

Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have $$ P_x[X_t\in A] = 0 $$ for all $t\in [0,T]$ but $$ P_x[X_{t'}\in A] >0 $$ for some $t'> T$. I wonder how to find an example of such a process $X$. The simple one $$ dX_t = f(X_t)dt $$ in not interesting because it is purely deterministic.

Answer 1

This question was already asked here. Homogenous Markov chains won't work because the probability distributions of $X_t$ and $X_s$ are mutually absolutely continuous for every positive $t$ and $s$. For a homogenous diffusion with nonzero diffusion term, $P[X_{2t}\in A]\ne0$ would require that $P_y[X_t\in A]\ne0$ for $y$ in a set of positive measure with respect to the probability distribution of $X_t$ while $P_x[X_t\in A]=0$. Unless I am missing something, this is impossible.

The idea of the proof for finite Markov chains is as follows. Assume that $P_x[X_t\in A]$ is positive for a given positive $t$. The transitions of the chain can be wriiten as $$ P_y[X_t=z]=(\mathrm{e}^{tQ})(y,z), $$ where $Q$ is the infinitesimal generator of the process. Choose $u$ such that $u\ge -Q(y,y)$ for every state $y$, then $uI+Q$ has nonnegative coefficients. Now, $$ P_x[X_t\in A]=\mathrm{e}^{-ut}(\mathrm{e}^{t(uI+Q)})(x,A) $$ is a linear combination with positive coefficients of nonnegative terms $(uI+Q)^n(x,A)$, hence $(uI+Q)^n(x,y)$ is positive for at least one given $n$.

Now, for every positive time $s$, $P_x[X_s\in A]$ is also a linear combination with nonnegative coefficients of $(uI+Q)^k(x,y)$, in particular $P_x[X_s\in A]$ is at least $\mathrm{e}^{-us}(s^n/n!)(uI+Q)^n(x,A)$, hence is positive.

Answer 2

The porous medium equation provides examples. See, for instance, M. Inoue, A Markov process associated with a porous medium equation, Proc. Japan Acad. 60 (1984), 157-160.