## Semiclassical analysis for the Kramers-Fokker-Planck equation.(English)Zbl 1083.35149

The authors study accurate semiclassical estimates of the resolvent of a class of pseudodifferential operators that include the Kramers-Fokker-Planck operator $P=v\cdot h\partial_x-V'(x)\cdot h\partial_v+{\gamma\over{2}}(-(h\partial_v)^2+v^2-hn),$ in $${\mathbb R}^{2n},$$ where $$V$$ is a smooth potential, $$x,v\in{\mathbb R}^n$$ and $$h>0$$ is essentially the temperature (so that in this case the semiclassical limit is given by a low-temperature limit). The class they consider is made of pseudodifferential operators that are neither elliptic nor self-adjoint, but that satisfy certain subelliptic conditions: if $$p=p_1+ip_2$$ is the symbol of $$p^w(x,D)$$ (semiclassical Weyl-Hörmander quantization), it is assumed that $$p_1\geq 0$$ and that:
$$\bullet$$ $$p\in S(\lambda^2,g_0),$$ $$\partial p\in S(\lambda,g_0),$$ $$\partial^2p_1\in S(1,g_0)$$ and $$\partial H_{p_2}p_1\in S(\lambda,g_0)$$, where $$1\leq\lambda(x,\xi)=\lambda\in C^\infty$$ is a $$g_0$$-admissible weight such that $$\lambda\in S(\lambda,g_0)$$ and $$\partial\lambda\in S(1,g_0)$$, where $$g_0=| dx| ^2+\lambda^{-2}| d\xi| ^2$$ is an admissible Hörmander’s metric on $${\mathbb R}^{2n}$$, and where $$H_{p_2}$$ is the Hamilton vector-field associated with $$p_2$$;
$$\bullet$$ $$p$$ has finitely many critical points $$C=\{\rho_1,\ldots,\rho_N\}$$ (where, for simplicity only, $$p(\rho_j)=0$$) such that, with $$c_0>0$$ sufficiently small, in a fixed ball $$B$$ containing $$C$$ one has $p_1+c_0H_{p_2}^2p_1\approx\text{ dist}_C^2;$
$$\bullet$$ outside $$B$$ one has $p_1+c_0H_{p_2}^2p_1\approx\lambda^2.$ Under these assumptions, the authors obtain precise resolvent estimates inside the pseudo-spectrum and, when $$p^w+Ch^2$$ is $$m$$-accretive, a precise description of the spectrum in a ball centered at the origin and radius $$Ch.$$ They are then able to apply the resolvent estimates and the description of the eigenspaces to the large time behavior of the semigroup $$e^{-tp^w/h}$$, $$t\geq 0,$$ associated with the Cauchy problem $(h\partial_t+p^w(x,D))u=0,\,\,\,\, u\bigl| _{t=0}=u_0.$ Near the critical points of $$p$$, the main technical tool is the use of the FBI-transform and of suitable weighted $$L^2$$-spaces of holomorphic functions; whereas away from the critical points, it is the use of the semiclassical Weyl-Hörmander calculus.

### MSC:

 35S10 Initial value problems for PDEs with pseudodifferential operators 35P20 Asymptotic distributions of eigenvalues in context of PDEs 47D06 One-parameter semigroups and linear evolution equations 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 47A10 Spectrum, resolvent 47G30 Pseudodifferential operators
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