On Applications of Maslov Optimization Theory

Maslov optimization theory has recently emerged as a new branch of functional analysis for studying deterministic control problems and Hamilton Jacobi equations. The main purpose of this work is to use an idempotent probability calculus to study the fixed points of nonexpansive transformations on nonnecessarily finite state spaces. We will see that these fixed points can be regarded as the $(\max,+)$-version of the invariant measure of Markov semi-groups. In the second part of this work we also present the $(\max,+)$-version of Dynkin's formula in the theory of stochastic processes and we apply this formula to study the stability properties of Bellman–Maslov processes.