02-332 J.-P. Eckmann and M. Hairer
Spectral Properties of Hypoelliptic Operators (294K, PDF) Jul 30, 02
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Abstract. We study hypoelliptic operators with polynomially bounded coefficients that are of the form $K = \sum_{i=1}^m X_i^T X_i^{} + X_0 + f$, where the $X_j$ denote first order differential operators, $f$ is a function with at most polynomial growth, and $X_i^T$ denotes the formal adjoint of $X_i$ in $\L^2$. For any $\eps>0$ we show that an inequality of the form $ \|u\|_{\delta,\delta} \le C\(\|u\|_{0,\eps} + \|(K+iy) u\|_{0,0}\)$ holds for suitable $\delta $ and $C$ which are independent of $y\in\R$, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of H\'erau and Nier [HN02], we conclude that its spectrum lies in a cusp $\{x+iy~|~ x\ge |y|^\tau-c, \tau\in(0,1],c\in\R \}$.

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