A transformation formula relating resolvents of Berezin-Toeplitz   operators by an invariance property of Brownian motion
  Using a stochastic representation provided by Wiener-regularized path integrals for the semigroups generated by certain Berezin-Toeplitz operators, a transformation formula for their resolvents is derived. The key property used in the transformation of the stochastic representation is that, up to a time change, Brownian motion is invariant under harmonic morphisms. This result for Berezin-Toeplitz operators is obtained in analogy with a well-known technique generating relations among Schr\"odinger operators that was recently generalized to Riemannian manifolds [Wittich, J. Math. Phys. {\bf 41} (2000), 244]. 
Bernhard G. Bodmann