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

On the relationship between stochastic turnpike and dissipativity notions