Define Let (Xt, Pxa) be the diffusion on natural scale on the line whose speed measure is given by ma. Suppose x > 0. Prove that if a → ∞, then Pxa converges weakly to the law of Brownian motion...

Define

Let (X_t, P^x
_a) be the diffusion on natural scale on the line whose speed measure is given by ma. Suppose

x > 0. Prove that if a → ∞, then P^x
_a
converges weakly to the law of Brownian motion absorbed (i.e., killed) at 0, started at x. What do you think happens when a → 0?

Chapter 42