Feynman-Chernoff Approximations for Stochastic Evolution Equations

This talk presents an investigation of the Feynman-Chernoff method for approximating semigroups that solve evolution equations generated by stochastic dynamics. We will discuss averages of Feynman-Chernoff iterations for a class of random operator-valued functions defined by random affine transformations, which represent a noncommutative analogue of a random walk.
The central results to be presented establish that the derivative of the averaged iteration at zero generates a strongly continuous semigroup. Sufficient conditions for the convergence of these iterations to the solution semigroup of a Fokker-Planck equation are provided. Finally, we will demonstrate the application of this theory to the diffusive limit of a quantum oscillator driven by random affine phase-space transformations.