We present an interacting particle system
methodology for the numerical solving of the Lyapunov exponent
of Feynman–Kac semigroups and for estimating the principal
eigenvalue of Schrödinger generators. The continuous or discrete time
models studied in this work
consists of N interacting particles evolving in an environment
with soft obstacles related to a potential function V. These
models are related to genetic algorithms and Moran type particle
schemes. Their choice
is not unique. We will examine a class of models
extending the hard obstacle model
of K. Burdzy, R. Holyst and P. March
and including the Moran type scheme presented by the authors in a
previous work.
We provide precise
uniform estimates with respect to the time parameter and we
analyze
the fluctuations of continuous time particle models.