This exercise shows how to approximate E by forms whose domain is all of B. Let Show that if increases to Show that if then Suppose {Pt} are the transition probabilities corresponding to a...

This exercise shows how to approximate E by forms whose domain is all of B. Let

Show that if

increases to

Show that if

then

Suppose {P_t} are the transition probabilities corresponding to a Dirichlet form E. Suppose there exist functions pt(x, y) such that for each t,

for almost every x. Prove that for almost every pair (x, y) with respect to the product measure

m × m, pt(x, y) = pt(y, x).