##
**Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians.**
*(English)*
Zbl 1072.35006

Lecture Notes in Mathematics 1862. Berlin: Springer (ISBN 3-540-24200-7/pbk). x, 209 p. (2005).

The aim of this text is to give an account of how the known techniques from partial differential equations and spectral theory can be applied for the analysis of Fokker-Planck operators (which are not self-adjoint and only hypoelliptic) or Witten Laplacians, while completing or referring to existing and sometimes recent results. The main motivation comes from the theory of kinetic equations, statistical physics and differential geometry.

We recall that the Fokker-Planck operator is defined by \(K:= X_0-\Delta_v+{v^2\over 4}-{n\over 2}\) and the Witten Laplacian by \(\Delta^0_{\phi/z,h}: -h^2\Delta+ 1/4|\nabla\phi|^2- h/z\Delta\phi\) \((h> 0)\), where \(\phi(x,v)= v^2/z+ V(x)\) is a classical Hamiltonian on \(\mathbb{R}^{2n}_{x,v}\) and \(X_0:= v\partial_x- (\partial_xV(x))\,\partial_v\) in the corresponding Hamiltonian vector field. The main problem is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. More exactly, one can prove that \((e^{-tP})_{t\geq 0}\) (where \(P= K\) or \(P= \Delta^{(0)}_{\phi/z,h}\)) is a well defined contraction semigroup on \(L^2(\mathbb{R}^{2n},dx\,dv)\) for any \(V\in{\mathcal C}^\infty(\mathbb{R}^n_x)\). The Maxwellian \(M(x,v)= e^{-\phi(x,v)/z}\) if \(e^{-\phi(x,v)/z}\in L^2(\mathbb{R}^{2n})\), \(M(x,v)= 0\) else, is the (unique up to normalization) equilibrium for \(P: PM= 0\). Two questions arise from statistical physics or the theory of kinetic equations:

1. Is there an exponential return to the equilibrium, that there exists a \(\tau> a\) such that \[ \| e^{-tP}u- c_\mu M\|\leq e^{-\tau t}\| u\|,\;u\in L^2(\mathbb{R}^n),\;c_\mu= (u,M/\| M\|)? \] 2. Is it possible to get quantitative estimates of the rate \(\tau\)?

For \(P= \Delta^0_{\phi/z,h}\) which is essentially self-adjoint, it is reduced to the estimate of its first nonzero eigenvalue. For \(K\) the authors strongly use hypoelliptic techniques together with the spectral theory for non-self-adjoint operators. A related and preliminary result in this approach concerns the compactness of the resolvent. Among other things, the authors explore as deeply as possible the validity of the following conjecture: the Fokker-Planck operator has a compact resolvent if and only if the Witten Laplacian has a compact resolvent. Hypoelliptic techniques enter at this level twice:

– In the proof of the equivalence when it is possible.

– In order to get effective criteria for the compactness of the resolvent of \(\Delta^0_{\phi/z}h\).

In this direction the present text provides a review of various techniques due to Hörmander, Kohn, Helffer-Mohamed, Helffer-Nourrigat on the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudodifferential calculus, etc. The hypoelliptic estimates are not only used for the question of the compactness of \((1+K)^{-1}\), but a variant of them permits to give a meaning to the contour integral \[ e^{-tK}= (2\pi i)^{-1} \int_{\partial S_K} e^{-tz}(z- K)^{-1}\,dz,\quad t> 0, \] although one cannot say more on the numerical range of \(K\) than \(\{(u,Ku); u\in D(K)\}\subset \{z\in \mathbb{C}:\text{Re\,}z\geq 0\}\). This last point is crucial in the quantitative analysis of the rate of return to the equilibrium.

The latter part of this text gives an account of the semiclassical analysis of the Witten Laplacian \(\Delta^0_{V/z,h}\). One recalls the relationship with Morse inequalities, after introducing the whole Witten complex and the corresponding Hodge Laplacians \(\Delta^{(p)}_{f,h}\) on all \(p\)-forms. After recalling some basic tools in semiclassical analysis, the authors recall the more accurate results of Helffer-Sjöstrand stating that the \(O(h^{n/2})\) eigenvalues of these Witten Laplacians are actually \(O(e^{-c/h})\). Finally, they discuss and propose some improvements about the accurate asymptotics of those exponentially small eigenvalues given by Bovier-Eckhoff-Gayrard-Klein. This last result will finally be combined with the comparison inequalities of Héron and Nier for the rate of decay for the semigroup associated to the Fokker-Planck operators, which was one of the main motivations of the whole study.

This synthetic text is very shallenging and useful for researches in partial differential equation, probability theory on mathematical physics.

We recall that the Fokker-Planck operator is defined by \(K:= X_0-\Delta_v+{v^2\over 4}-{n\over 2}\) and the Witten Laplacian by \(\Delta^0_{\phi/z,h}: -h^2\Delta+ 1/4|\nabla\phi|^2- h/z\Delta\phi\) \((h> 0)\), where \(\phi(x,v)= v^2/z+ V(x)\) is a classical Hamiltonian on \(\mathbb{R}^{2n}_{x,v}\) and \(X_0:= v\partial_x- (\partial_xV(x))\,\partial_v\) in the corresponding Hamiltonian vector field. The main problem is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. More exactly, one can prove that \((e^{-tP})_{t\geq 0}\) (where \(P= K\) or \(P= \Delta^{(0)}_{\phi/z,h}\)) is a well defined contraction semigroup on \(L^2(\mathbb{R}^{2n},dx\,dv)\) for any \(V\in{\mathcal C}^\infty(\mathbb{R}^n_x)\). The Maxwellian \(M(x,v)= e^{-\phi(x,v)/z}\) if \(e^{-\phi(x,v)/z}\in L^2(\mathbb{R}^{2n})\), \(M(x,v)= 0\) else, is the (unique up to normalization) equilibrium for \(P: PM= 0\). Two questions arise from statistical physics or the theory of kinetic equations:

1. Is there an exponential return to the equilibrium, that there exists a \(\tau> a\) such that \[ \| e^{-tP}u- c_\mu M\|\leq e^{-\tau t}\| u\|,\;u\in L^2(\mathbb{R}^n),\;c_\mu= (u,M/\| M\|)? \] 2. Is it possible to get quantitative estimates of the rate \(\tau\)?

For \(P= \Delta^0_{\phi/z,h}\) which is essentially self-adjoint, it is reduced to the estimate of its first nonzero eigenvalue. For \(K\) the authors strongly use hypoelliptic techniques together with the spectral theory for non-self-adjoint operators. A related and preliminary result in this approach concerns the compactness of the resolvent. Among other things, the authors explore as deeply as possible the validity of the following conjecture: the Fokker-Planck operator has a compact resolvent if and only if the Witten Laplacian has a compact resolvent. Hypoelliptic techniques enter at this level twice:

– In the proof of the equivalence when it is possible.

– In order to get effective criteria for the compactness of the resolvent of \(\Delta^0_{\phi/z}h\).

In this direction the present text provides a review of various techniques due to Hörmander, Kohn, Helffer-Mohamed, Helffer-Nourrigat on the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudodifferential calculus, etc. The hypoelliptic estimates are not only used for the question of the compactness of \((1+K)^{-1}\), but a variant of them permits to give a meaning to the contour integral \[ e^{-tK}= (2\pi i)^{-1} \int_{\partial S_K} e^{-tz}(z- K)^{-1}\,dz,\quad t> 0, \] although one cannot say more on the numerical range of \(K\) than \(\{(u,Ku); u\in D(K)\}\subset \{z\in \mathbb{C}:\text{Re\,}z\geq 0\}\). This last point is crucial in the quantitative analysis of the rate of return to the equilibrium.

The latter part of this text gives an account of the semiclassical analysis of the Witten Laplacian \(\Delta^0_{V/z,h}\). One recalls the relationship with Morse inequalities, after introducing the whole Witten complex and the corresponding Hodge Laplacians \(\Delta^{(p)}_{f,h}\) on all \(p\)-forms. After recalling some basic tools in semiclassical analysis, the authors recall the more accurate results of Helffer-Sjöstrand stating that the \(O(h^{n/2})\) eigenvalues of these Witten Laplacians are actually \(O(e^{-c/h})\). Finally, they discuss and propose some improvements about the accurate asymptotics of those exponentially small eigenvalues given by Bovier-Eckhoff-Gayrard-Klein. This last result will finally be combined with the comparison inequalities of Héron and Nier for the rate of decay for the semigroup associated to the Fokker-Planck operators, which was one of the main motivations of the whole study.

This synthetic text is very shallenging and useful for researches in partial differential equation, probability theory on mathematical physics.

Reviewer: Viorel Iftimie (Bucureşti)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35H10 | Hypoelliptic equations |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

35P15 | Estimates of eigenvalues in context of PDEs |

82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |