We construct self‐adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs u$u$‐measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self‐adjoint extensions of well‐known classical leafwise Laplacians. We observe a quasi‐invariance property of energy densities in the u$u$‐conformal case and the existence of nonconstant functions of zero energy.

We prove the existence of symmetric diffusions and self-adjoint Laplacians on (uniformly) hyperbolic attractors, endowed with SRB-measures. The proof is based on Dirichlet form theory. We observe some features of such diffusions, for instance, a quasi-invariance property of energy densities in the u-conformal case and the existence of non-constant harmonic functions of zero energy in the ergodic case. Contents 24 Appendix A. Disintegration and Rokhlin's theorem 26 Appendix B. The geometric construction of SRB-measures 26 Appendix C. Dirichlet integrals on weighted manifolds 27 Appendix D. Superposition of closable forms 30 References 31