3.6. Brownian motion with boundary conditions Let P(s, t) = 1 √

4Dt exp[−s2/(4Dt)]

be the probability distribution in the continuum limit of the one-dimensional Brownian motion.

a. Find the probability distribution Pa(s, t) when the origin is an absorbent point, i.e.

when Pa(0, t) = 0 for all t.

b. Find the probability distribution Pr(s, t) when the origin is a pure reflecting point, i.e. when it holds the condition ∂Pr

∂x (0, t) = 0 for all t.

Hint. Use P(s, t) and the linearity of the problem to set up the method of solution.