Large Deviation Principle for Bridges of Sub-Riemannian Diffusion Processes

Abstract

We prove that bridges of subelliptic diffusions on a compact manifold, with distinct ends, satisfy a large deviation principle in the space of Hölder continuous functions, with a good rate function, when the travel time tends to 0. This leads to the identification of the deterministic first order asymptotics of the distribution of the bridge under generic conditions on the endpoints of the bridge.

Fluctuations of bridges, reciprocal characteristics and concentration of measure

“…In Theorem 2.1 we find conditions on the potential of a gradient diffusion for its bridges to have marginals with better concentration properties than those of an Ornsetin Uhlenbeck bridge. This is one of the main novelties with respect to the existing literature about bridges where, to the best of our knowledge, only Large Deviations-type estimates have been proved, and mostly in the short time regime, see among others [1], [2], [4], [5], [22], [34], [42] and [44]. The proof of this result is done by first showing an ad hoc Girsanov formula for bridges, which differs from the usual one.…”

“…(a) The distance d(z, z ′ ) between two vertices z and z ′ is the length of the shortest walk joining z with z ′ . Due to point (1)…”

Large Deviations for Rough Path Lifts of Watanabe's Pullbacks of Delta Functions

“…We remark that Bailleul proved an LDP parallel to Corollary 2.4 on compact manifolds in [3] (and in its extended version [4] with Mesnager and Norris). Their method is basically analytic (with a little bit of rough path theory) and different from ours.…”

On Sharp Large Deviations for the Bridge of a General Diffusion

“…for some δ > 0, in the sense that the difference between these two probabilities is exponentially negligible as t → 0. In order to see this, recall Large Deviation estimates recently obtained for conditioned Diffusions (see [14], [3] for the case of a compact manifold). These state that, as t → 0, the time changed conditioned Diffusion starting at x at time s satisfies a Large Deviation Principle with rate function given by…”

“…In this note we wish to investigate a minor point in this direction. It has been proved ( [3], [7]) that the (non sharp) Large Deviation asymptotics for conditioned Diffusions do not depend on the drift b of the non conditioned process X as in (2.1). It has been a general belief that this remains true also for the sharp asymptotics of the bridge of a Diffusion.…”