Monte Carlo solution for the Poisson equation on the base of spherical processes with shifted centres

Abstract

We consider a class of spherical processes rapidly 
converging to the boundary (so called Decentred 
Random Walks on Spheres or spherical processes 
with shifted centres) in comparison with the 
standard walk on spheres. The aim is to compare 
costs of the corresponding Monte Carlo estimates 
for the Poisson equation. Generally, these costs 
depend on the cost of simulation of one trajectory 
and on the variance of the estimate.
It can be proved that for the Laplace equation the 
limit variance of the estimation doesn't depend on 
the kind of spherical processes. Thus we have very 
effective estimator based on the decentred random 
walk on spheres. As for the Poisson equation, it 
can be shown that the variance is bounded by a 
constant independent of the kind of spherical 
processes (in standard form or with shifted 
centres). We use simulation for a simple model 
example to investigate variance behavior in more 
details.