Suppose we have a Brownian flow of diffeomorphisms on R^n and we wish to represent it as a stochastic process on the metric i.e - Given a point x, the metric transforms as F*(t)(G), the pullback of G. Will the process be a brownian motion on the space of metrics(positive definite symmetric matrices) at a given point? Brownian motion on a general manifold is defined as the diffusion process of the half Laplacian(Laplace-Beltrami operator). Any help will be appreciated.
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