An Explicit Local Uniform Large Deviation Bound for Brownian Bridges

By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics. Let M be an m-dimensional complete riemannian manifold with distance function d, Ω(M) := C(R0 ,M) the associated space of continuous paths and p ∈M . Let expp denote the exponential map associated to p and by i(p) > 0 its radius of injectivity. Let t > 0 and Q 0,t denote the Brownian Bridge measure on M supported by the set Ω(p, q, t) of paths starting at time 0 in p and ending up at q ∈M at time t. Provided there is a unique geodesic joining p and q, the measure Q 0,t tends, as t → 0, to the point measure δγp,q,t supported by this geodesic γ, parametrized proportional to arc-length with constant velocity v(s) = d(p, q)/t. More precisely, we have the following statement. Recall ([3], 3.4, p. 74) that a subset S ⊂ M is called strongly convex, if for any two points q, q′ ∈ S there is a unique minimizing geodesic joining q and q′ whose interior is contained in S and ([3], 4.2 Proposition, p.76) that for all p ∈M there is some number r(p) > 0 such that the geodesic balls B(p, r) := {x ∈ M : d(p, x) < r} are strongly convex for all 0 < r < r(p). Theorem. Let p ∈M and choose r > 0, ε0 > 0 such that R := r + ε0 < r(p). Let C ⊂ B(p, r) be an arbitrary closed subset. Write κ(C) := max { 1, sup q∈C,σ∈G2TqM K(σ) } (1) where G2TqM denotes the set of 2-dimensional subspaces in TpM and K(σ) is the sectional curvature of M evaluated at the subspace σ ∈ G2TpM . Then, for all q ∈ C, there is a unique geodesic joining p and q. Furthermore, there is some δ > 0 such that we have for all ε > 0 with ε < ε0 and for all q ∈ C Q 0,t (B(ε, p, q, t)) ≤ 2e− 2(R(κ)−δ)ε2 t where B(ε, p, q, t) := {ω ∈ Ω(M) : ω(0) = p, ω(t) = q, sup u∈[0,t) d(ω(u), γ(u)) ≥ ε}