A Probabilistic Solution to the Stroock-Williams Equation

We consider the initial boundary value problem u t = µu x + 1 2 u xx (t > 0, x ≥ 0) u(0, x) = f (x) (x ≥ 0) u t (t, 0) = ν u x (t, 0) (t > 0) of Stroock and Williams [12] where µ, ν ∈ IR and the boundary condition is not of Feller's type when ν < 0. We show that when f belongs to C 1 b with f (∞) = 0 then the following probabilistic representation of the solution is valid u(t, x) = E x f (X t) − E x f (X t) 0 t (X) 0 e −2(ν−µ)s ds where X is a reflecting Brownian motion with drift µ and 0 (X) is the local time of X at 0. The solution can be interpreted in terms of X and its creation in 0 at rate proportional to 0 (X). Invoking the law of (X t , 0 t (X)) this also yields a closed integral formula for u expressed in terms of µ , ν and f .