content_image/325049.jpg content_image/325048.jpg content_image/325047.jpg content_image/325046.jpg content_image/325044.jpg
boundary conditions are Dirichlet boundary conditions, defined by 

<img src='content_image/325049.jpg'>

where ∂ Ω is the boundary of the domain, and Neumann boundary conditions, defined by 

<img src='content_image/325048.jpg'>

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 

<img src='content_image/325047.jpg'>

with initial and boundary conditions 

<img src='content_image/325046.jpg'>

The expectation in the FKF is now with respect to the law of an absorbed Brownian motion. Define a stopping time, 

<img src='content_image/325044.jpg'>

