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Geometric Kramers- Fokker- Planck operators with boundary conditions Geometric Kramers-Fokker-Planck operators with boundary Francis Nier, conditions IRMAR, Univ. Rennes 1 The Francis Nier, problem IRMAR, Univ. Rennes 1 Main results Applications Elements of proof Microlocal analysis and spectral theory in honor of J. Sj¨ ostrand CIRM sept. 26th 2013

Outline Geometric Kramers- Fokker- Planck operators with boundary conditions Francis Nier, IRMAR, Univ. Presentation of the problem Rennes 1 Main results The problem Applications Main Elements of proofs results Applications Elements of proof

Geometric Kramers- Fokker- Planck operators with boundary conditions Raoul Bott: Morse theory indomitable (IHES 1988) Francis Nier, IRMAR, Univ. Rennes 1 The problem Main results Applications Elements of proof

Geometric Kramers-Fokker-Planck operators Geometric Kramers- Fokker- In the euclidean space, the operator Planck operators P ± = ± p .∂ q − ∂ q V ( q ) .∂ p + − ∆ p + | p | 2 with x = ( q , p ) ∈ Ω × R d , boundary 2 conditions Francis is associated with the Langevin process Nier, IRMAR, Univ. dq = pdt , dp = − ∂ q V ( q ) dt − pdt + dW Rennes 1 The problem Q = Q ⊔ ∂ Q riem. mfld with bdy , X = T ∗ Q , ∂ X = T ∗ ∂ Q Q . Main Metric g = g ij ( q ) dq i dq j , g − 1 = ( g ij ) results Applications − ∆ p + | p | 2 Elements q P ± , Q , g = ±Y E + , ∆ p = g ij ( q ) ∂ p i ∂ p j of proof 2 | p | 2 = g ij ( q ) p i p j q E ( q , p ) = , 2 2 Y E = g ij ( q ) p i ∂ q j − 1 2 ∂ q k g ij ( q ) p i p j ∂ p k = g ij ( q ) p i e j , e j = ∂ q j + Γ ℓ ij p ℓ ∂ p j . acting on C ∞ ( X ; f ) . P ± , Q , g = scalar part of Bismut’s hypoelliptic Laplacian.

Geometric Kramers-Fokker-Planck operators Geometric Kramers- Fokker- In the euclidean space, the operator Planck operators P ± = ± p .∂ q − ∂ q V ( q ) .∂ p + − ∆ p + | p | 2 with x = ( q , p ) ∈ Ω × R d , boundary 2 conditions Francis is associated with the Langevin process Nier, IRMAR, Univ. dq = pdt , dp = − ∂ q V ( q ) dt − pdt + dW Rennes 1 The problem Q = Q ⊔ ∂ Q riem. mfld with bdy , X = T ∗ Q , ∂ X = T ∗ ∂ Q Q . Main Metric g = g ij ( q ) dq i dq j , g − 1 = ( g ij ) results Applications − ∆ p + | p | 2 Elements q P ± , Q , g = ±Y E + , ∆ p = g ij ( q ) ∂ p i ∂ p j of proof 2 | p | 2 = g ij ( q ) p i p j q E ( q , p ) = , 2 2 Y E = g ij ( q ) p i ∂ q j − 1 2 ∂ q k g ij ( q ) p i p j ∂ p k = g ij ( q ) p i e j , e j = ∂ q j + Γ ℓ ij p ℓ ∂ p j . acting on C ∞ ( X ; f ) . P ± , Q , g = scalar part of Bismut’s hypoelliptic Laplacian.

A simple case Geometric Kramers- Fokker- Take Q = ( −∞ , 0)] with g = ( dq 1 ) 2 . Planck operators p 1 with boundary conditions Francis Nier, IRMAR, Univ. Rennes 1 (0 , 0) q 1 The problem X Main ∂ X results Applications Elements of proof Specular reflection: u (0 , − p 1 ) = u (0 , p 1 ) for p 1 > 0 . It can be written γ odd u = 0 with γ odd u = u (0 , p 1 ) − u (0 , − p 1 ) . 2 Absorption: u (0 , p 1 ) = 0 for p 1 < 0 . It can be written γ odd u = sign( p 1 ) γ ev u with γ ev u = u (0 , p 1 )+ u (0 , − p 1 ) . 2

A simple case Geometric Kramers- Fokker- Take Q = ( −∞ , 0)] with g = ( dq 1 ) 2 . Planck operators p 1 with boundary conditions Francis Nier, IRMAR, Univ. Rennes 1 (0 , 0) q 1 The problem X Main ∂ X results Applications Elements of proof Specular reflection: u (0 , − p 1 ) = u (0 , p 1 ) for p 1 > 0 . It can be written γ odd u = 0 with γ odd u = u (0 , p 1 ) − u (0 , − p 1 ) . 2 Absorption: u (0 , p 1 ) = 0 for p 1 < 0 . It can be written γ odd u = sign( p 1 ) γ ev u with γ ev u = u (0 , p 1 )+ u (0 , − p 1 ) . 2

General BC Metric locally on ∂ Q : ( dq 1 ) 2 ⊕ ⊥ m ( q 1 , q ′ ) . Consider f -valued functions, f Hilbert Geometric Kramers- space. Fokker- � q 1 = 0 � Planck Let j be a unitary involution in f and define along ∂ X = : operators with γ odd = Π odd γ = γ ( q ′ , p 1 , p ′ ) − j γ ( q ′ , − p 1 , p ′ ) boundary , conditions 2 Francis γ ev = Π ev γ = γ ( q ′ , p 1 , p ′ ) + j γ ( q ′ , − p 1 , p ′ ) Nier, . IRMAR, 2 Univ. Rennes 1 � Let the boundary condition on the trace γ u = u ∂ X be � The problem γ odd u = ± sign( p 1 ) A γ ev u , Π ev A = A Π ev . Main results Formal integration by part Applications �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 � ± 1 L 2 ( X , dqdp ; f ) Elements | γ u | ( q ′ , p ) 2 p 1 dq ′ dp Re � u , P ± , Q , g u � = of proof 2 2 ∂ X �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 L 2 ( X , dqdp ; f ) = + Re � γ ev u , A γ ev u � L 2 ( ∂ X , | p 1 | dq ′ dp ; f ) . 2 � �� � Assumptions: A = A ( q , | p | q ) is local in q and | p | q (local elastic collision at the boundary); A ( q , | p | q ) ∈ L ( L 2 ( S ∗ ∂ Q Q , | ω 1 | dq ′ d ω ; f )) with � A ( q , r ) � ≤ C unif. either Re A ( q , r ) ≥ c A > 0 unif. or A ( q , r ) ≡ 0 .

General BC Metric locally on ∂ Q : ( dq 1 ) 2 ⊕ ⊥ m ( q 1 , q ′ ) . Consider f -valued functions, f Hilbert Geometric Kramers- space. Fokker- � q 1 = 0 � Planck Let j be a unitary involution in f and define along ∂ X = : operators with γ odd = Π odd γ = γ ( q ′ , p 1 , p ′ ) − j γ ( q ′ , − p 1 , p ′ ) boundary , conditions 2 Francis γ ev = Π ev γ = γ ( q ′ , p 1 , p ′ ) + j γ ( q ′ , − p 1 , p ′ ) Nier, . IRMAR, 2 Univ. Rennes 1 � Let the boundary condition on the trace γ u = u ∂ X be � The problem γ odd u = ± sign( p 1 ) A γ ev u , Π ev A = A Π ev . Main results Formal integration by part Applications �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 � ± 1 L 2 ( X , dqdp ; f ) Elements | γ u | ( q ′ , p ) 2 p 1 dq ′ dp Re � u , P ± , Q , g u � = of proof 2 2 ∂ X �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 L 2 ( X , dqdp ; f ) = + Re � γ ev u , A γ ev u � L 2 ( ∂ X , | p 1 | dq ′ dp ; f ) . 2 � �� � Assumptions: A = A ( q , | p | q ) is local in q and | p | q (local elastic collision at the boundary); A ( q , | p | q ) ∈ L ( L 2 ( S ∗ ∂ Q Q , | ω 1 | dq ′ d ω ; f )) with � A ( q , r ) � ≤ C unif. either Re A ( q , r ) ≥ c A > 0 unif. or A ( q , r ) ≡ 0 .

General BC Metric locally on ∂ Q : ( dq 1 ) 2 ⊕ ⊥ m ( q 1 , q ′ ) . Consider f -valued functions, f Hilbert Geometric Kramers- space. Fokker- � q 1 = 0 � Planck Let j be a unitary involution in f and define along ∂ X = : operators with γ odd = Π odd γ = γ ( q ′ , p 1 , p ′ ) − j γ ( q ′ , − p 1 , p ′ ) boundary , conditions 2 Francis γ ev = Π ev γ = γ ( q ′ , p 1 , p ′ ) + j γ ( q ′ , − p 1 , p ′ ) Nier, . IRMAR, 2 Univ. Rennes 1 � Let the boundary condition on the trace γ u = u ∂ X be � The problem γ odd u = ± sign( p 1 ) A γ ev u , Π ev A = A Π ev . Main results Formal integration by part Applications �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 � ± 1 L 2 ( X , dqdp ; f ) Elements | γ u | ( q ′ , p ) 2 p 1 dq ′ dp Re � u , P ± , Q , g u � = of proof 2 2 ∂ X �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 L 2 ( X , dqdp ; f ) = + Re � γ ev u , A γ ev u � L 2 ( ∂ X , | p 1 | dq ′ dp ; f ) . 2 � �� � Assumptions: A = A ( q , | p | q ) is local in q and | p | q (local elastic collision at the boundary); A ( q , | p | q ) ∈ L ( L 2 ( S ∗ ∂ Q Q , | ω 1 | dq ′ d ω ; f )) with � A ( q , r ) � ≤ C unif. either Re A ( q , r ) ≥ c A > 0 unif. or A ( q , r ) ≡ 0 .

General BC Metric locally on ∂ Q : ( dq 1 ) 2 ⊕ ⊥ m ( q 1 , q ′ ) . Consider f -valued functions, f Hilbert Geometric Kramers- space. Fokker- � q 1 = 0 � Planck Let j be a unitary involution in f and define along ∂ X = : operators with γ odd = Π odd γ = γ ( q ′ , p 1 , p ′ ) − j γ ( q ′ , − p 1 , p ′ ) boundary , conditions 2 Francis γ ev = Π ev γ = γ ( q ′ , p 1 , p ′ ) + j γ ( q ′ , − p 1 , p ′ ) Nier, . IRMAR, 2 Univ. Rennes 1 � Let the boundary condition on the trace γ u = u ∂ X be � The problem γ odd u = ± sign( p 1 ) A γ ev u , Π ev A = A Π ev . Main results Formal integration by part Applications �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 � ± 1 L 2 ( X , dqdp ; f ) Elements | γ u | ( q ′ , p ) 2 p 1 dq ′ dp Re � u , P ± , Q , g u � = of proof 2 2 ∂ X �∇ p u � 2 L 2 ( X , dqdp ; f ) + �| p | q u � 2 L 2 ( X , dqdp ; f ) = + Re � γ ev u , A γ ev u � L 2 ( ∂ X , | p 1 | dq ′ dp ; f ) . 2 � �� � Assumptions: A = A ( q , | p | q ) is local in q and | p | q (local elastic collision at the boundary); A ( q , | p | q ) ∈ L ( L 2 ( S ∗ ∂ Q Q , | ω 1 | dq ′ d ω ; f )) with � A ( q , r ) � ≤ C unif. either Re A ( q , r ) ≥ c A > 0 unif. or A ( q , r ) ≡ 0 .