Analytic Subordination Consequences of Free Markovianity

In [4], under some easy to remove genericity assumptions, we proved the analytic subordination of the Cauchy transforms of distributions GμX+Y and GμX for a pair of freely independent self-adjoint random variables X, Y . We used this to obtain inequalities among p-norms of densities of distributions, free entropies and Riesz energies. This was followed by P.Biane’s discovery [1] that subordination extends, roughly speaking, to the resolvents, i.e. is an operator-valued occurrence. He also showed that this implies a noncommutative Markovtransitions property for free increment processes and that there are similar subordination results for multiplicative processes. The analytic function approach in [4] and the combinatorial one in [1] did not shed much light on why analytic subordination appears in this context. In [6] we found a simple explanation of this phenomenon based on the coalgebra structure associated with the free difference quotient derivation. In the additive case this also led to a far reaching generalization of the analytic subordination result to the B -valued case, i.e. to free independence with amalgamation over an algebra B . Here, based on simple operator-valued analytic function considerations, we build further on the result in [6]. We derive more general analytic subordination results for a freely Markovian triple A,B,C . This also includes the B -valued extension of the result for multiplication of free unitary variables.