Suppose Pt is a family of sub-Markov transition probabilities and we define by (36.12). Show that is a family of Markov transition probabilities. Show that i.e., starting at , the process stays...

Suppose P_t
is a family of sub-Markov transition probabilities and we define

by (36.12). Show that

is a family of Markov transition probabilities. Show that

i.e., starting at
, the process stays there forever.

Show that if Pt(x, A) is defined by (19.17), and P_t
f(x) = f (y) P_t(x, dy), then P_t
maps C₀
into C₀.

Show that P_t
defined by (36.13) satisfies all the parts of Assumption 36.1.

Chapter 37