Let a > 0 and let m(dx) = dx + a δ0(dx), where δ 0 is the point mass at 0. Let (Xt, Px ) be the diffusion on the line on natural scale whose speed measure is given by m. Show that under P 0 ,   with...

Let a > 0 and let m(dx) = dx + a δ0(dx), where δ₀
is the point mass at 0. Let (Xt, Px ) be the diffusion on the line on natural scale whose speed measure is given by m. Show that under P₀,

with probability one for each t > 0. Prove that for each t > 0, Z_t
= {t : X_t
= 0} contains no intervals. Thus the zero set of the process X spends an amount of time at 0 that has positive Lebesgue measure, but the zero set contains no intervals.