boundary conditions are Dirichlet boundary conditions, defined by where ∂ Ω is the boundary of the domain, and Neumann boundary conditions, defined by where n denotes the normal to the boundary. The FKF can be modified to include such boundary conditions by modifying the law of the complementary SDE (Pardoux and R˘ a¸scanu, 2014; Skorokhod, 1962). For example, consider a system on the domain Ω with Dirichlet boundary conditions. We now have a PDE of the form with initial and boundary conditions The expectation in the FKF is now with respect to the law of an absorbed Brownian motion. Define a stopping time, γ x,t Ω = inf { s ≥ 0 | X x s ∈ R $^{d}$\ Ω , } 8 u ( x x, 0) = f ( x ) , x ∈ Ω , u ( x x, t ) = g ( x ) , x ∈ ∂ Ω , t > 0 . ∂u ∂t = d ∑ i,j =1 1 2 a$_{ij}$ ( x x, t ) ∂ $^{2}$u ∂x$_{i}$∂x$_{j}$ + d ∑ i =1 b$_{i}$ ( x x, t ) ∂u ∂x$_{i}$ − c ( x x, t ) u, x ∈ Ω , t > 0 , ∂u ( x x, t ) ∂n n = h ( x x, t ) x ∈ ∂ Ω , u ( x x, t ) = g ( x x, t ) x ∈ ∂ Ω ,