We prove the neo‐classical inequality with the optimal constant, which was conjectured by T. J. Lyons (‘Differential equations driven by rough signals’, Rev. Mat. Iberoamericana 14 (1998) 215–310). For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo‐classical inequality.
Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.
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