The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions

The paper proves that for random walks on discrete subgroups of semisimple Lie groups, finite entropy alone (with total irreducibility and bi-contraction) suffices to identify the Poisson boundary with the Furstenberg boundary, via a new pin-down entropy method that avoids any moment assumptions. This is stated in Theorem 1.1 and Theorem 5.8 and emphasized as removing all moment conditions, in contrast to classical strip/ray criteria which require finite first/logarithmic moments . The candidate solution incorrectly invokes Kaimanovich’s strip criterion as needing only finite entropy and attempts a flats-based strip construction; but the paper itself explicitly notes that the strip (and ray) criteria do require moment conditions (finite logarithmic and finite first, respectively) and are therefore insufficient here, motivating their pin-down approach . Both the paper and the model agree on existence/uniqueness of hitting measures on B, forward/backward limits, and the use of opposite boundary points to parametrize oriented flats, based on Guivarc’h–Raugi-type hypotheses, but only the paper supplies a correct, moment-free maximality proof via conditional entropy pin-down and a forward/backward independence reduction HB×B(αn)=HB(αn) .