# Markov Property

• Aug 5th 2010, 11:10 AM
Sjaak
Markov Property
one more question,

I am rereading a lecture and i come across the following:
define escape time as $\xi_r=inf[t>0 | (||X_t-X_0||>r)]$
now my notes say that is easy to show by using markov property that $P[\xi_r \geq nt]\leq p^n$ for n=0,1,..
By the time i must have thought it was easy, now i cant remember how to do it anymore :(
• Aug 6th 2010, 03:24 AM
Sjaak
i found it again:
$P[\xi_r \geq nt]\leq P_x \cap_{k\leq n}[X_{kt} \in B_x(r)]=(P_t^d[x \in B_x(r)])^n \leq p^n$ where B_x(r) is closed ball of radius r around x