The proper renormalization of this push-forward distribution is an issue since the renormalization constant includes infinite series along these closed geodesics. The idea is to take a Gaussian distribution in the tangent space at the mean value and to consider the push-forward distribution on the manifold that wraps the distribution along the closed geodesics. To obtain more tractable formulas, the wrapped Gaussian distribution was proposed in several domains. However, the heat kernel has a nonlinear dependency in time, which makes it difficult to use in statistics as a classical Gaussian distribution. The heat kernel is the smallest positive fundamental solution to the heat equation ∂ f ∂ t − Δ f =0, where Δ f = Tr g ( Hess f ) = g i j ( Hess f ) i j = ∇ i ∇ i f is the Laplace–Beltrami operator. The natural definition from the stochastic point of view is the heat kernel p t ( x, y ), which is the transition density of the Brownian motion (see Chapter 4). Several definitions have been proposed to extend the notion of Gaussian distribution to Riemannian manifolds. Xavier Pennec, in Riemannian Geometric Statistics in Medical Image Analysis, 2020 3.4.3 Gaussian distributions on SPD matrices
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