  Kinetic equations describe physical processes in a high-dimensional phase
space and are often simulated using Markov process-based Monte Carlo routines.
The quantities of interest are typically defined on the lower-dimensional
position space and estimated on a grid (histogram). In several applications,
such as the construction of diffusion Monte Carlo-like techniques and variance
prediction for particle tracing Monte Carlo methods, the cell escape
probabilities, i.e., the probabilities with which particles escape a grid cell
during one step of the Markov process, are of interest. In this paper, we
derive formulas to calculate the cell escape probabilities for common mesh
elements in one, two, and three dimensions. Deterministic calculation of cell
escape probabilities in higher dimensions becomes expensive and prone to
quadrature errors due to the involved high-dimensional integrals. We therefore
also introduce a stochastic Monte Carlo algorithm to calculate the escape
probabilities, which is more robust at the cost of a statistical error. The
code used to perform the numerical experiments and accompanying GeoGebra
tutorials are openly available at
https://gitlab.kuleuven.be/numa/public/escape-probabilities-markov-processes.
