Arguing for the need to combine declarative and probabilistic programming, Bárány et al. (TODS 2017) recently introduced a probabilistic extension of Datalog as a “purely declarative probabilistic programming language.” We revisit this language and propose a more principled approach towards defining its semantics based on stochastic kernels and Markov processes—standard notions from probability theory. This allows us to extend the semantics to continuous probability distributions, thereby settling an open problem posed by Bárány et al. We show that our semantics is fairly robust, allowing both parallel execution and arbitrary chase orders when evaluating a program. We cast our semantics in the framework of infinite probabilistic databases (Grohe and Lindner, LMCS 2022) and show that the semantics remains meaningful even when the input of a probabilistic Datalog program is an arbitrary probabilistic database.