Random walk on small spheres method for solving drift-diffusion-reaction problems with spatially varying velocity and reaction rate

The well-known Random Walk on Spheres method (RWS) for the Laplace equation is here extended to drift-diffusion-reaction problems with spatially varying velocity and reaction rate coefficients. The RWS algorithm is based on a generalized spherical mean value relation with constant velocity and reaction rate which is an extension of the classical spherical mean value relation for the Laplace equation. Heuristically, the same algorithm can be applied when the velocity and the reaction rate coefficients are spatially varying just by choosing small spheres in the RWS process assuming the coefficients in small spheres are approximately constant. A fundamental question concerns the quantitative assessment of the error induced by this approximation. In the present work, we establish a rigorous bound for this error. The paper also presents the results of numerical experiments performed to validate the obtained error estimate.