Outline Introduction Proof of the main results Main references Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator Wei-Xi Li wei-xi.li@whu.edu.cn Wuhan University January 9, 2010 Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Introduction Proof of the main results Main references Outline Introduction 1 Problems considered Previous results and the approaches involved Our main result Proof of the main results 2 the approach Regularity in velocity variable Regularity in spacial variable Main references 3 Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Outline Introduction 1 Problems considered Previous results and the approaches involved Our main result Proof of the main results 2 the approach Regularity in velocity variable Regularity in spacial variable Main references 3 Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references In this talk we consider the Fokker-Planck operator P = y · ∂ x − ∂ x V ( x ) · ∂ y − △ y + | y | 2 − n ( x , y ) ∈ R 2 n , (1) 2 , 4 where x denotes the space variable and y denotes the velocity vari- able, and V ( x ) is a real-valued potential defined in the whole space R n x . Elementary Properties of Fokker-Planck Operator 1) Observe P is a non-selfadjoint operator: √ P = Re P + − 1 Im P , where Re P = −△ y + | y | 2 4 − n 2 and Im P = y · D x − ∂ x V ( x ) · D y are operators with real symbols. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references 2) If we rewrite P = √− 1( y · D x − ∂ x V ( x ) · D y ) + � n j =1 L ∗ j L j with L j = ∂ y j + y j 2 , then ∀ u ∈ C ∞ 0 , Re � Pu , u � L 2 ≥ 0 . Hence P is accretive in C ∞ 0 . And the spectral σ ( P ) of P is contained in { z ∈ C ; Re z ≥ 0 } . Moreover the closure ¯ P of P is maximally accretive. Hence − ¯ P is a � e − t ¯ P � generator of a contractive semi-group t ≥ 0 . Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references One question arising from statistical physics or the theory of kinetic equations is the exponential return to equilibrium. For a essential self-adjoint operator, it reduces to the estimate of its first nonzero eigenvalue. Unfortunately Fokker-Planck operate is only maximally accretive but non-selfadjoint. However the hypoellipticity could give us a hand. This ideal has been used by H´ erau-Nier. It concerns the compactness of resolvent. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references The link between Fokker-Planck operator and Witten Laplace oper- ator The Witten Laplace operator △ (0) V / 2 defined by V / 2 = −△ x + 1 4 | ∂ x V ( x ) | 2 − 1 △ (0) 2 △ x V ( x ) . Helffer-Nier’s Conjecture The Fokker-Planck operator has a compact resolvent if and only if the Witten Laplacian has a compact resolvent. The ”Only if” part can be deduced directly, while the ”if” part remains substantially open. Some positive answers have been given for some kind of potentials. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references In this talk we are mainly concerned with the conditions imposed on the potential V ( x ) , so that the Fokker-Planck operator P admits a global hypoelliptic estimate and has a compact resolvent, and recall the works of Helffer,H´ erau and Nier on this problem. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Outline Introduction 1 Problems considered Previous results and the approaches involved Our main result Proof of the main results 2 the approach Regularity in velocity variable Regularity in spacial variable Main references 3 Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references The approach used by Helffer-Nier and H´ erau-Nier: Hypoelliptic techniques To analyze the compactness of resolvent of the operator P , the hypoellipticity techniques play an efficient role, one of which is re- ferred to Kohn’s method due to H¨ ormander, Kohn, · · · , and another is based on nilpotent Lie group technique developed by Rothschild- Stein. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references erau-Nier 1 to study such a po- Kohn’s method had been used by H´ tential V ( x ) that behaves at infinity as a high-degree homogeneous function: ∂ x V ( x ) ≈ � x � 2 M − 1 and ∀ | γ | ≥ 1 , | ∂ γ x V ( x ) | ≤ C γ � x � 2 M −| γ | erau-Nier, Helffer-Nier 2 considered By developing the approach of H´ more general potential V ( x ) . 1 F. H´ erau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential , Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. 2 B. Helffer and F. Nier, Hypoelliptic estimates and spectral theory for Fokker- Planck operators and Witten Laplacians ,Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = 1 k ≤ � ∂ x V ( x ) � ≤ C � x � k ; 1 C � x � Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = 1 k ≤ � ∂ x V ( x ) � ≤ C � x � k ; 1 C � x � ∀ | γ | ≥ 2 , | ∂ γ x V ( x ) | ≤ C γ � ∂ x V ( x ) � . Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = 1 k ≤ � ∂ x V ( x ) � ≤ C � x � k ; 1 C � x � ∀ | γ | ≥ 2 , | ∂ γ x V ( x ) | ≤ C γ � ∂ x V ( x ) � . Assumption II Suppose V is real-valued smooth function satisfying that for some C , k ≥ 1 , Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = 1 k ≤ � ∂ x V ( x ) � ≤ C � x � k ; 1 C � x � ∀ | γ | ≥ 2 , | ∂ γ x V ( x ) | ≤ C γ � ∂ x V ( x ) � . Assumption II Suppose V is real-valued smooth function satisfying that for some C , k ≥ 1 , � ∂ x V ( x ) � ≤ C � x � k ; Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
Outline Problems considered Introduction Previous results and the approaches involved Proof of the main results Our main result Main references Assumption I Suppose V is real-valued smooth function satisfying 1 + | ∂ x V ( x ) | 2 � 1 2 , � that with C , k some constants and � ∂ x V ( x ) � = k ≤ � ∂ x V ( x ) � ≤ C � x � k ; 1 1 C � x � ∀ | γ | ≥ 2 , | ∂ γ x V ( x ) | ≤ C γ � ∂ x V ( x ) � . Assumption II Suppose V is real-valued smooth function satisfying that for some C , k ≥ 1 , � ∂ x V ( x ) � ≤ C � x � k ; x V ( x ) | ≤ C α � ∂ x V ( x ) � � x � − κ . | ∂ α ∃ κ > 0 , ∀ | α | = 2 , Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent
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