Spectral Properties of Hypoelliptic Operators
J.-P. Eckmann, Martin Hairer · 2003
We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = ∑m i=1 XT i Xi + X0 + f, where the Xj denote first order differential operators, f is a function with at most polynomial growth, and X T i denotes the formal adjoint of Xi in L 2. For any ε> 0 we show that an inequality of the form ‖u‖δ,δ ≤ C(‖u‖0,ε + ‖(K + iy)u‖0,0) holds for suitable δ and C which are independent of y ∈ 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 Hérau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy | x ≥ |y | τ −c, τ ∈ (0,1], c ∈ R}. 1