SRB-Symmetric Diffusions on Hyperbolic Attractors

The paper’s Theorem 5.1 asserts that, under the (AC)-property with uniformly bounded conditional densities, the leafwise gradient energy E(µ) on a hyperbolic attractor is closable and its closure is a local, conservative Dirichlet form, with a unique non-positive self-adjoint generator L(µ) satisfying ⟨L(µ)u,φ⟩ = −E(µ)(u,φ) and ∫Λ L(µ)u dµ = 0; the bottom of the spectrum of −L(µ) is 0. The proof localizes by a partition of unity to rectangles, disintegrates µ, invokes closability of leafwise Dirichlet integrals on weighted C1-manifolds (Appendix C), and then reassembles via a superposition/patching argument (Proposition D.1), finishing with standard Dirichlet form theory for Markov, locality, and conservativity claims. These steps are explicit in the text (definition of E(µ) and D0, partition-of-unity bounds, rectangle-level disintegration, closability on weighted plaques, and reconstruction to conclude closability and existence/uniqueness of L(µ)) . The candidate solution establishes the same result by an almost identical strategy: partition-of-unity localization, disintegration and uniform equivalence of plaque measures, closability on each plaque via an integration-by-parts argument (equivalent to Appendix C), a Fubini/diagonal argument in place of the paper’s Proposition D.1, and then Dirichlet/Markov/strong locality and conservativity, followed by the first representation theorem for the generator. Minor differences are present in the technical packaging (e.g., the explicit diagonal argument vs. Proposition D.1), but the logical content and assumptions match the paper, and the model correctly derives all claimed properties, including generator existence and spectral bottom 0 (the paper cites general theory for these steps) .