Following a recent work of Bismut, we will discuss a natural family of diffusion processes with continuously differentiable paths on the tangent bundle over a compact Riemannian manifold that interpolates between Brownian motion and the geodesic flow. We will show that they converge respectivel to the geodesic flow and Riemannian Brownian motion at the two ends of the parameter interval in the strong sense in the path space. Even in the simplest case of the standard Brownian motion, this interpolation has some interesting and desirable properties. In particular, we will show that it leads a particularly clean and short proof of the classical Ito’s formula without discrete approximation. We will mostly discuss this classical case in the talk, for which prior knowledge of Brownian motion on a manifold is not necessary.

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