Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem

| Lagrangian stochastic models and algorithms are constructed and justi ed for solving the footprint problem, namely, the problem of calculation of the mean concentration and the ux of particles at a xed point released from a source arbitrarily situated in the space. The direct and adjoint Monte Carlo algorithms are suggested, and rigorous justi cations are given. Two di erent backward trajectory algorithms are considered: Thomson's method and a method based on probabilistic representations of the relevant initial value problem. The cost of the latter algorithm may increase with time, but it allows to treat the general situation when a set of reacting species is scattered by the ow. Thomson's approach is extended to general stochastic di erental equations which is especially usefull when it is desired to nd a solution at a xed point, and for large time instances.