  By defining tracial states on a non-commutative analogue of a path space, we
construct Markov dilations for a class of conservative completely Markov
semigroups on finite von Neumann algebras. This class includes all symmetric
semigroups. For well chosen semigroups (for instance with generator any
divergence form operator associated to a derivation valued in the coarse
correspondence) those dilations give rise to stationary solutions of certain
free SDEs previously considered by D. Shlyakhtenko. Among applications, we
prove a non-commutative Talagrand inequality for non-microstates free entropy
(relative to a subalgebra B and a completely positive map \eta:B\to B). We also
use those new deformations in conjunction with Popa's deformation/rigidity
techniques. For instance, combining our results with techniques of Popa-Ozawa
and Peterson, we prove that the von Neumann algebra of a countable discrete
group with CMAP and positive first L^2 Betti number has no Cartan subalgebras.
