  Rowmotion is a certain well-studied bijective operator on the distributive
lattice $J(P)$ of order ideals of a finite poset $P$. We introduce the
rowmotion Markov chain ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to
each $x\in P$ and using these probabilities to insert randomness into the
original definition of rowmotion. More generally, we introduce a very broad
family of toggle Markov chains inspired by Striker's notion of generalized
toggling. We characterize when toggle Markov chains are irreducible, and we
show that each toggle Markov chain has a remarkably simple stationary
distribution.
  We also provide a second generalization of rowmotion Markov chains to the
context of semidistrim lattices. Given a semidistrim lattice $L$, we assign a
probability $p_j$ to each join-irreducible element $j$ of $L$ and use these
probabilities to construct a rowmotion Markov chain ${\bf M}_L$. Under the
assumption that each probability $p_j$ is strictly between $0$ and $1$, we
prove that ${\bf M}_{L}$ is irreducible. We also compute the stationary
distribution of the rowmotion Markov chain of a lattice obtained by adding a
minimal element and a maximal element to a disjoint union of two chains.
  We bound the mixing time of ${\bf M}_{L}$ for an arbitrary semidistrim
lattice $L$. In the special case when $L$ is a Boolean lattice, we use spectral
methods to obtain much stronger estimates on the mixing time, showing that
rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.
