Let (Xt, Px) be a one-dimensional Brownian motion, the local time of Brownian motion at x, and m a positive finite measure on R. Show that is an additive functional. We consider the space-time...

Let (X_t, P_x) be a one-dimensional Brownian motion,

the local time of Brownian motion at x, and m a positive finite measure on R. Show that

is an additive functional.

We consider the space-time process. Let V_t
= V₀
+ t. The process V_t
is simply the process that increases deterministically at unit speed. Thus V_t
can represent time. If

is a Markov process, show that

is also a Markov process. Is

necessarily a strong Markov process if

is a strong Markov process?

For some applications, one lets V_t
= V_{0
− t}, and one thinks of time running backwards. Space-time processes are useful when considering parabolic partial differential equations.