Reverse juggling processes.

In a recent paper, Knutson (arXiv:1601.06391) studied a Markov chain on semi-infinite matrices $b\times \mathbb N$ over $GF(q)$ leading to two models of reverse juggling. These transitions were the same as the time-reversed transitions of previously studied (forward) juggling chains by Ayyer, Bouttier, Corteel and Nunzi (Elec. J. Prob, Vol. 20, 2015) and by Ayyer, Bouttier, Corteel, Linusson and Nunzi (arXiv:1504.02688). In this paper we generalize the reverse juggling chains of Knutson for both single and multiple species. We show that there are natural ways to place generic weights on the transitions and still obtain chains where the stationary distribution have a simple form, both for finite and infinite states. In the finite single species case, we find the phenomenon of ultrafast convergence to the stationary distribution. In the finite multispecies case, the stationary distribution turns out to be a multivariate generalisation of the inversion polynomial. Lastly, we observe a new phenomenon, which we dub "partial mixing", where the stationary distribution is independent of the probabilities of picking the ball.