  In this paper, we introduce and study different dissipativity notions and
different turnpike properties for discrete-time stochastic nonlinear optimal
control problems. The proposed stochastic dissipativity notions extend the
classic notion of Jan C. Willems to $L^r$ random variables and to probability
measures. Our stochastic turnpike properties range from a formulation for
random variables via turnpike phenomena in probability and in probability
measures to the turnpike property for the moments. Moreover, we investigate how
different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in
the analysis. Our results are built upon stationarity concepts in distribution
and in random variables and on the formulation of the stochastic optimal
control problem as a finite-horizon Markov decision process. We investigate how
the proposed dissipativity notions connect to the various stochastic turnpike
properties and we work out the link between these two different forms of
dissipativity.

Optimization and Control