Understanding the convergence of Ulam discretizations in higher-dimensional systems

Determine rigorous convergence properties and error estimates of Ulam’s method discretizations of the Fokker–Planck (or transfer) operator semigroup by Markov matrices as the number of partition boxes N increases for general multi-dimensional dynamical systems and stochastic differential equations, extending beyond the currently established cases of one-dimensional systems and systems with smooth invariant measures.

Background

Ulam’s method approximates transfer or Fokker–Planck operators by projecting dynamics onto a finite partition of the phase space and constructing a Markov matrix that represents the action of the semigroup over a chosen transition time. This approach is widely used to estimate spectral properties and invariant measures for deterministic and stochastic systems.

While effective in practice, the theoretical understanding of how these Markov matrix approximations converge as the partition is refined (i.e., as N increases) is limited. The authors note that rigorous results are largely confined to one-dimensional settings and systems with smooth invariant measures, highlighting the need for comprehensive convergence theory in higher dimensions and more general invariant measures.