Abstract. We get stationary solutions of a free stochastic partial differential equation.As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra R ω embeddable. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N ≤ 0.13. IntroductionIn a fundamental series of papers, Voiculescu introduced analogs of entropy and Fisher information in the context of free probability theory. A first microstate free entropy χ(X 1 , ..., X n ) is defined as a normalized limit of the volume of sets of microstate i.e. matricial approximations (in moments) of the n-tuple of self-adjoints X i living in a (tracial) W * -probability space M. Starting from a definition of a free Fisher information [43], Voiculescu also defined a non-microstate free entropy χ * (X 1 , ..., X n ), known by the fundamental work [2] to be greater than the previous microstate entropy, and believed to be equal (at least modulo Connes' embedding conjecture). For more details, we refer the reader to the survey [45] for a list of properties as well as applications of free entropies in the theory of von Neumann algebras.Moreover in case of infinite entropy, two other invariants the microstate and nonmicrostate free entropy dimensions (respectively written δ 0 (X 1 , ..., X n ) and δ * (X 1 , ..., X n )) have been introduced to generalize results known for finite entropy. Surprisingly, Connes and Shlyakhtenko found in [10] a relation between those entropy dimensions and the first L 2 -Betti numbers they defined for finite von Neumann algebras. For instance, for (real and imaginary parts of ) generators of finitely generated groups, δ * has been proved in [27] to be equal to β-Betti numbers of groups). In [39], Dimitri Shlyakhtenko obtained lower bounds on microstate free entropy dimension (motivated by the goal of trying to prove equality with non-microstate free entropy dimension), in studying the following free stochastic differential equation :where ξ is the unique derivation densely defined (on non-commutative polynomials) such that ∂ j (X i ) = 1 i=j 1 ⊗ 1. Then the i-th conjugate variable is defined by 2000 Mathematics Subject Classification. 46L54, 60H15.

We prove that if X 1 , . . . , X n (n > 1) are self-adjoints in a W * -probability space with finite nonmicrostates free Fisher information, then the von Neumann algebra W * (X 1 , . . . , X n ) they generate doesn't have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.

Abstract:The space D ′ Γ of distributions having their wavefront sets in a closed cone Γ has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. In this paper, the topological and bornological properties of D