 02332 J.P. Eckmann and M. Hairer
 Spectral Properties of Hypoelliptic Operators
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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 FokkerPlanck 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 $\{x+iy~~ x\ge y^\tauc, \tau\in(0,1],c\in\R \}$.
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