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Journal Articles Communications on Pure and Applied Mathematics Year : 2020

Robustness of Liouville Measure under a Family of Stable Diffusions

Abstract

Consider a C 8 closed connected Riemannian manifold pM, gq with negative curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation W s of the geodesic flow. Let ∆ s be the leafwise Laplacian for W s and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ, the operator Lρ :" ∆ s`ρ X generates a diffusion for W s. We show that, as ρ Ñ´8, the unique stationary probability measure for the leafwise diffusion of Lρ converge to the normalized Liouville measure on SM .
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Dates and versions

hal-03095571 , version 1 (04-01-2021)

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Cite

Francois Ledrappier, Lin Shu. Robustness of Liouville Measure under a Family of Stable Diffusions. Communications on Pure and Applied Mathematics, 2020, 73 (12), pp.2708-2736. ⟨10.1002/cpa.21935⟩. ⟨hal-03095571⟩
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