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Hypoelliptic Laplacian and Orbital Integrals (AM-177) by Jean-Michel Bismut

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Chapter Ten

The action functional and the harmonic oscillator

The purpose of this chapter is to solve explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. The action is the one introduced in [B05, eq. (0.10)]. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b image 0 and when image. Finally, certain heat kernels are computed in terms of the minimum value ...

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