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Compactness criteria for the resolvent of the Fokker-Planck operator. (English) Zbl 1404.35114

The author studies the compactness of the resolvent of the Fokker-Planck operator in \(\mathbb{R}^{2n}\), \[ P= y\cdot\partial_x- \partial_x V(x)\cdot\partial_y- \Delta_y+|y|^2/4-n/2, \] where \(x\in\mathbb{R}^n\) denotes the space variables and \(y\) denotes the velocity variables. Precise conditions on the potential \(V(x)\) are given, providing compactness of the resolvent. The results imporve in part preceding contributions on the subject, see in particular B. Helffer and F. Nier [Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians. Berlin: Springer (2005; Zbl 1072.35006)], where a link was detected with the compactness of the resolvent of the Witten Laplacian.

MSC:

35H10 Hypoelliptic equations
47A10 Spectrum, resolvent

Citations:

Zbl 1072.35006
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