Fluctuations of bridges, reciprocal characteristics and concentration of measure

Abstract

Conditions on the generator of a Markov process to control the fluctuations of its bridges are found. In particular, continuous time random walks on graphs and gradient diffusions are considered. Under these conditions, a concentration of measure inequality for the marginals of the bridge of a gradient diffusion and refined large deviation expansions for the tails of a random walk on a graph are derived. In contrast with the existing literature about bridges, all the estimates we obtain hold for non asymptotic time scales. New concentration of measure inequalities for pinned Poisson random vectors are also established. The quantities expressing our conditions are the so called reciprocal characteristics associated with the Markov generator.

Couplings, gradient estimates and logarithmic Sobolev inequalitiy for Langevin bridges

“…Combining these two observations, we get back the usual gradient estimate (9). Therefore, our result is consistent with the BakryÉmery theory, when the two sets of hypothesis intersect.…”

“…In our simpler setting, by choosing an appropriate norm for the Malliavin derivative, we can get rid of potential terms. In Corollary 1.1 we deduce from this inequality some useful concentration of measure bounds for bridge measures, which were already partially obtained in [9]. We also investigate a natural notion of invariant measure for bridges, related to the concept of Gibbs measure on path space, in the spirit of [29], [12] (see [27] for a recent account), which is to look for the limit as T ↑ +∞ of the marginal law at t = 0 of P x,y −T,T .…”

On Properties of the Dirichlet Green's function for linear diffusions on a half line

“…There is a considerable literature on the study of bridges. In [5,6] Conforti et al study bridges associated to diffusions with a gradient drift, using the fact that it is the reciprocal characteristics which determine the bridge uniquely. In particular, diffusions with differing drifts may have the same bridge processes.…”