Lower estimates on microstates free entropy dimension

By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},…,X_{n}. In particular, we show that \delta_{0}(X_{1},…,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{}(X_{1},…,X_{n}) and V={(\partial(X_{1}),…,\partial(X_{n})):\partial\in$\mathcal{C}$} is the set of values of derivations A=$\mathbb{C}$[X_{1},… X_{n}]\to A\otimes A with the property that \partial^{}\partial(A)\subset A. We show that for q sufficiently small (depending on n) and X_{1},…,X_{n} a q-semicircular family, \delta_{0}(X_{1},…,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).