Prove (40.6). Let D be a ball in and suppose u is the solution to the Cauchy problem in the domain D × [0,t1] as described in Section 40.3. Show that Suppose (Xt, Px )is a Brownian motion on [0,...

Prove (40.6).

Let D be a ball in

and suppose u is the solution to the Cauchy problem in the domain D × [0,t₁] as described in Section 40.3. Show that

Suppose (Xt, Px )is a Brownian motion on [0, b] with reflection at 0 and b. Find a series expansion for p(t, x, y), the transition densities for X.

Chapter 41