Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Tunnel effect for semiclassical random walk F . Hérau (joint work with J.-F. Bony and L. Michel) Laboratoire Jean Leray, Université de Nantes Microlocal Analysis and Spectral Theory Conference in honor of J. Sjöstrand CIRM, September 27, 2013
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Plan Introduction 1 Supersymmetry and Witten Laplacian 2 Supersymmetry for random walks 3 Final remarks 4
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Introduction 1 Supersymmetry and Witten Laplacian 2 Supersymmetry for random walks 3 Final remarks 4
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Semiclassical random walk Let φ ∈ C ∞ ( R d ) be a real function such that d µ h = e − φ ( x ) / h dx is a probability measure. We are interested in the random-walk operator defined on the space C 0 of continuous function going to 0 at infinity by � 1 T h f ( x ) = f ( y ) d µ h ( y ) , µ h ( B h ( x )) B h ( x ) where B h ( x ) = B ( x , h ) . By duality, this defines an operator T ⋆ h on the set M b of bounded Borel measures ∀ f ∈ C 0 , ∀ ν ∈ M b , T ⋆ h ( ν )( f ) = ν ( T h f )
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Invariant measure Observe that if d ν has a density with respect to Lebesgue measure d ν = ρ ( x ) dx ,then �� � 1 T ⋆ e − φ ( y ) / h dy h ( d ν ) = µ h ( B h ( x )) ρ ( x ) dx | x − y | < h As a consequence, the measure d ν h , ∞ = µ h ( B h ( x )) e − φ ( x ) / h dx := M h ( x ) dx Z h where Z h is chosen so that d ν h , ∞ is a probability on R d satisfies T ⋆ h ( d ν h , ∞ ) = d ν h , ∞ . We say that d ν h , ∞ is an invariant measure for T h and M h is sometimes called the Maxwellian.
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Convergence to equilibrium Question For d ν ∈ M b , what is the behavior of ( T ⋆ h ) n ( d ν ) when n → ∞ ? Under suitable assumptions on φ we can easily prove the following : Theorem For any probability measure d ν , we have n → + ∞ ( T ⋆ h ) n ( d ν ) = d ν h , ∞ lim We are willing to compute the speed of convergence in the above limit. The answer is closely related to the spectral theory of T ⋆ h , at least when we restrict to a stable Hilbertian subspace of T ⋆ h in M b .
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Reduction and Some elementary properties For the coming analysis, we restrict to the following Hilbertian subspace of measures (with density) H h = L 2 ( R d , d ν h , ∞ ) ֒ → M b : f − → fd ν h , ∞ We denote again by T ∗ h this restriction. We have the following elementary properties : Proposition The following hold true : T ∗ h is bounded and self-adjoint on H h 1 is an eigenvalue of T ⋆ h (Markov property)
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Assumptions on φ We make the following assumptions on φ : there exists c , R > 0 and some constants C α > 0, α ∈ N d such that : ∀ α ∈ N d \ { 0 } , ∀ x ∈ R d | ∂ α x φ ( x ) | ≤ C α and ∀| x | ≥ R , |∇ φ ( x ) | ≥ c and φ ( x ) ≥ c | x | . φ is a Morse function (i.e. φ the critical points of φ are non-degenerate). We denote by U ( k ) the set of critical points,of φ of index k , n k = ♯ U ( k ) , U ( 0 ) = { m k , k = 1 . . . n 0 } and for convenience U ( 1 ) = { s j , j = 1 . . . n 1 + 1 } with s 1 = ∞ . We suppose that the values φ ( s j ) − φ ( m k ) , s j ∈ U ( 1 ) , m k ∈ U ( 0 ) are distincts. (recall that the index of a critical point c is the number of negative eigenvalues of Hess ( φ )( c ) ).
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Description of small eigenvalues Theorem [Bony-Hérau-Michel] Suppose that the previous assumptions are fullfilled. Then There exists κ 0 > 0 such that : - σ ess ( T ⋆ h ) ∩ [ 1 − κ 0 , 1 ] = ∅ - σ ( T ⋆ h ) ∩ [ − 1 , − 1 + κ 0 ] = ∅ There exists ε > 0 such that there are exactly n 0 eigenvalues of T ⋆ h in the interval [ 1 − ε h , 1 ] . One of them is 1 and the other enjoy the following asymptotic h θ k , 0 λ ⋆ 2 ( d + 2 ) e − S k / h ( 1 + O ( h )) k , h = 1 − where the coefficient θ k , S k are defined later.
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Reformulation of the problem Since we prefer to work in the standard L 2 ( dx ) space, we pose for the following u = M 1 / 2 def = U − 1 U : L 2 ( d ν h , ∞ ) → L 2 ( dx ) unitary f h f where h and T h = U ∗ h T ⋆ h U which expression is � 1 T h f ( x ) = a h ( x ) a h ( y ) f ( y ) dy α d h d | x − y | < h where � 1 a h ( x ) − 2 = e ( φ ( x ) − φ ( y )) / h dy . α d h d | x − y | < h
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks We now have to study the spectral properties of the selfadjoint operator T h on L 2 ( dx ) � 1 T h u ( x ) = a h ( x ) a h ( y ) u ( y ) dy α d h d | x − y | < h � 1 Observe that the operator u �→ | x − y | < h u ( y ) dy is a fourier α d h d multiplier G ( hD x ) with � G ( ξ ) = 1 e ix · ξ dx α d | x | < 1 We can then notice that a − 2 = e φ/ h G ( hD x )( e − φ/ h ) T h = a h G ( hD x ) a h and h In order to study the spectrum of T h near 1, we can study the spectrum near 0 of def P h = 1 − T h = a h ( V h ( x ) − G ( hD x )) a h where V h ( x ) = a − 2 h ( x ) = e φ/ h G ( hD x )( e − φ/ h ) .
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Short heuristics Let u ∈ C ∞ 0 ( R d ) be fixed, using the change of variable y = x + hz and Taylor expansion for G in the expression of P h , we show easily that P h u ( x ) = a h ( V h ( x ) − G ( hD x )) a h u ( x ) � �� � 2 ( d + 2 ) P W 1 h + O ( h 3 ) where h = − h 2 ∆ + |∇ φ | 2 − h ∆ φ P W is the semiclassical Witten Lapacian. Here the term O ( h 3 ) is not an error term from a spectral point of view. Anyway questions P W h widely studied : can we benefit from this knowledge to compute the ev’s of P h ? Is there a supersymmetric structure for P h as for P W h (recall P h ( a − 1 h e − φ/ h ) = 0) ?
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Some biblio and known results The spectrum of semiclassical Witten laplacian has been analyzed by many authors : Witten 85, Helffer-Sjöstrand 85, Cycon-Froese-Kirch-Simon 87, Bovier-Gayrard-Klein 04, Helffer-Klein-Nier 04. In the last article, a complete asymptotic of exponentially small ones is given (under the above assumptions) The spectrum Metropolis operator has also been recently studied (using the connections with Witten). In bounded domains with Neumann conditions, Diaconis-Lebeau-Michel 12, and various geometries, Christianson-Guillarmou-Michel 13, Lebeau-Michel 10 (with an other scalling). No study of exponentially close to 1 spectrum for Metropolis (and "tunneling effect") so far...
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Introduction 1 Supersymmetry and Witten Laplacian 2 Supersymmetry for random walks 3 Final remarks 4
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Description of small eigenvalues h = − h 2 ∆ + |∇ φ | 2 − h ∆ φ . We recall some facts about P W h has n 0 := ♯ U ( 0 ) eigenvalues It is rather easy to show that P W 0 = λ 1 ≤ . . . ≤ λ n 0 , in the interval [ 0 , h 3 / 2 ] . The most accurate result in [HKN04] gives an approximation of these eigenvalues (for k ≥ 2) : � λ k = h θ k ( h ) e − S k / h h l θ k , l , θ k ( h ) = with l ≥ 0 The quantities, S k , θ k , 0 can be computed : there exists a labelling of U ( 0 ) and an application j : { 1 , . . . , n 0 } → { 1 , . . . , n 1 + 1 } such that (for k ≥ 2) : � θ k , 0 = | ˆ λ 1 ( s j ( k ) ) | det ( Hess φ ( m k )) S k = 2 ( φ ( s j ( k ) ) − φ ( m k )) and π det ( Hess φ ( s j ( k ) )) where ˆ λ 1 ( s j ( k ) ) is the negative eigenvalue of Hess φ ( s j ( k ) ) .
Introduction Supersymmetry and Witten Laplacian Supersymmetry for random walks Final remarks Interaction matrix The strategy of Helffer-Klein-Nier (see also Helffer-Sjostrand 84 and Hérau-Hitrik-Sjostrand 11 for Kramers-Fokker-Planck) is the following : Introduce F ( 0 ) = eigenspace associated to the n 0 low lying eigenvalues on 0-forms Π ( 0 ) = projector on F ( 0 ) . M = restriction of ∆ φ, h to F ( 0 ) . We have to compute the eigenvalues of M . We compute suitable quasimodes f ( 0 ) , show that k e ( 0 ) = Π ( 0 ) f ( 0 ) = f ( 0 ) + error k k k and compute the matrix of M in the base e ( 0 ) k . Doing that leads to error terms which are too big. In order to do that, use the supersymmetric structure.
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