For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration and transportation inequalities. In the case of Euclidean space, there are sufficient conditions for the joint law to satisfy a logarithmic Sobolev inequality. In several cases, the constants obtained are of optimal growth with respect to the number of random variables, or are independent of this number. These results extend results known for mutually independent random variables and weakly dependent random variabels under Dobrushkin--Shlosman type conditions. The paper also contains applications to Markov processes including the ARMA process.