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Given a Riemannian manifold, $(M,g)$, the eigenvalues $\lambda_i$ and eigenfunctions $\phi_i$ of the associated Laplace-Beltrami operator $\Delta_g$ can be used to construct the heat kernel on the manifold, as well as solve the wave and Schrodinger equations (e.g. see this question).

There are some interesting probabilistic facts about these constructs:

  • On the manifold, the infinitesimal generator of Brownian motion is $\Delta_g/2$ (e.g. see here)
  • The heat kernel (i.e. the smallest positive fundamental solution to $\partial_t u = \Delta_g u$) serves as the transition density function of Brownian motion on the manifold (e.g. see Heat Kernel on a Non-Compact Riemannian Manifold, by Grigor’yan), and can be written via a spectral decomposition as: $$ p(x,y,t) = \sum_{i=1}^\infty \exp(-t\lambda_i)\phi_i(x)\phi_i(y) $$

My question is whether there are other stochastic interpretations of the Laplace-Beltrami operator. In particular, do the eigenvalues and eigenvectors themselves have any kind of probabilistic interpretation on their own?

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