TECHNISCHE UNIVERSIT AT WIEN Entropy method for hypocoercive - PowerPoint PPT Presentation

TECHNISCHE UNIVERSIT¨ AT WIEN Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with linear drift Anton ARNOLD with Jan Erb; Franz Achleitner Trieste, May 2017 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 1 / 32

degenerate Fokker-Planck equations with linear drift evolution of probability density f ( x , t ) , x ∈ R n , t > 0: � � f t = div D ∇ f + C x f (1) f ( x , 0) = f 0 ( x ) D ∈ R n × n ... symmetric, const in x , degenerate C ∈ R n × n ... const in x Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 2 / 32

degenerate Fokker-Planck equations with linear drift evolution of probability density f ( x , t ) , x ∈ R n , t > 0: � � f t = div D ∇ f + C x f (1) f ( x , 0) = f 0 ( x ) D ∈ R n × n ... symmetric, const in x , degenerate C ∈ R n × n ... const in x goals: existence & uniqueness of steady state f ∞ ( x ); convergence f ( t ) t →∞ − → f ∞ with sharp rates; complete theory for the equation class (1) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 2 / 32

hypocoercive example – from plasma physics kinetic Fokker-Planck equation for f ( x , v , t ) , x , v ∈ R n : f t + v · ∇ x f − ∇ x V · ∇ v f = σ ∆ v f + ν div v ( vf ) � �� � � �� � � �� � � �� � free transport influence of potential V ( x ) diffusion, σ> 0 friction, ν> 0 � | v | 2 � − ν 2 + V ( x ) steady state : f ∞ ( x , v ) = c e σ V ( x )... given confinement potential Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 3 / 32

hypocoercive example – from plasma physics kinetic Fokker-Planck equation for f ( x , v , t ) , x , v ∈ R n : f t + v · ∇ x f − ∇ x V · ∇ v f = σ ∆ v f + ν div v ( vf ) � �� � � �� � � �� � � �� � free transport influence of potential V ( x ) diffusion, σ> 0 friction, ν> 0 � | v | 2 � − ν 2 + V ( x ) steady state : f ∞ ( x , v ) = c e σ V ( x )... given confinement potential rewritten (with x , v variables): � � 0 � � � � 0 − v f t = div x , v ∇ x , v f + f 0 σ I ∇ x V + ν v � �� � � �� � =: D ... diffusion drift • � ∃ explicit Green’s function for V not quadratic! Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 3 / 32

Outline: 1 hypocoercivity, prototypic examples 2 review of standard entropy method for non-degenerate Fokker-Planck equations 3 decay of modified “entropy dissipation” functional 4 mechanism of new method Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 4 / 32

(hypo)coercivity 1 example 1: standard Fokker-Planck equation on R n : � � =: Lf . . . symmetric on H := L 2 ( f − 1 f t = div ∇ f + x f ∞ ) c e − | x | 2 2 , f ∞ ( x ) = ker L = span( f ∞ ) • L is dissipative, i.e. � Lf , f � H ≤ 0 ∀ f ∈ D ( L ) • − L is coercive (has a spectral gap), in the sense: �− Lf , f � H ≥ � f � 2 ∀ f ∈ { f ∞ } ⊥ L 2 ( f − 1 ∞ ) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 5 / 32

(hypo)coercivity 2 example 2: � � f t = div D ∇ f + C x f =: Lf (2) with degenerate D is degenerate parabolic; (symmetric part of) − L is not coercive. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 6 / 32

(hypo)coercivity 2 example 2: � � f t = div D ∇ f + C x f =: Lf (2) with degenerate D is degenerate parabolic; (symmetric part of) − L is not coercive. Definition 1 (Villani 2009) → K ⊥ (densely) Consider L on Hilbert space H with K = ker L ; let ˜ H ֒ (e.g. H ... weighted L 2 , ˜ H ... weighted H 1 ). − L is called hypocoercive on ˜ H if ∃ λ > 0 , c ≥ 1: ∀ f ∈ ˜ � e Lt f � ˜ H ≤ c e − λ t � f � ˜ H H • typically c > 1 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 6 / 32

Steady state of (non)degenerate FP equations: standard Fokker-Planck equation f t = div( ∇ f + x f ) : unique steady state f ∞ ( x ) = c e −| x | 2 / 2 as a balance of drift & diffusion; sharp decay rate = 1 n = 2: x 2 drift x 1 diffusion Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 7 / 32

degenerate prototype: degenerate diffusion (1D Fokker-Planck) + rotation � � 1 � x 1 − ω x 2 � � 0 � f t = div ∇ f + f 0 0 ω x 1 � �� � � �� � = D = C x f ∞ ( x ) = c e −| x | 2 / 2 ∀ ω ∈ R (unique for ω � = 0); sharp decay rate = 1 2 (= min ℜ λ C ) for fast enough rotation ( | ω | > 1 2 ) x 2 x 1 equilibration by drift/diffusion Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 8 / 32

degenerate prototype: degenerate diffusion (1D Fokker-Planck) + rotation � � 1 � x 1 − ω x 2 � � 0 � f t = div ∇ f + f 0 0 ω x 1 � �� � � �� � = D = C x f ∞ ( x ) = c e −| x | 2 / 2 ∀ ω ∈ R (unique for ω � = 0); sharp decay rate = 1 2 (= min ℜ λ C ) for fast enough rotation ( | ω | > 1 2 ) x 2 x 1 equilibration by drift/diffusion | ω | = 1 2 : C has a Jordan block ⇒ (sharp) decay rate = 1 2 − ε Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 8 / 32

degenerate prototype with ω = 1: � � 1 � 1 � � � 0 − 1 f t = div ∇ f + x f 0 0 1 0 x 2 -axis: drift characteristics of ˙ x = − C x tangent to level curve of | x | : x ’ = − x + y Drift characteristic Level curve of P−norm y ’ = − x 3 2 1 0 y −1 −2 −3 −4 −3 −2 −1 0 1 2 3 4 x � � √ 2 − 1 level curve of “distorted” vector norm x · P · x ; P = − 1 2 Ref: [Dolbeault-Mouhot-Schmeiser] 2015 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 9 / 32

coefficients C , D in Fokker-Planck equation � � f t = div D ∇ f + C x f =: Lf Condition A: No (nontrivial) subspace of ker D is invariant under C ⊤ . (equivalent: L is hypoelliptic.) Proposition 1 Let Condition A hold. f ∈ C ∞ ( R n × R + ) . a) Let f 0 ∈ L 1 ( R d ) ⇒ [H¨ ormander 1969] b) Let f 0 ∈ L 1 + ( R d ) ⇒ f ( x , t ) > 0 , ∀ t > 0 . (Green’s fct > 0 ) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 10 / 32

coefficients C , D in Fokker-Planck equation � � f t = div D ∇ f + C x f =: Lf Condition A: No (nontrivial) subspace of ker D is invariant under C ⊤ . (equivalent: L is hypoelliptic.) Proposition 1 Let Condition A hold. f ∈ C ∞ ( R n × R + ) . a) Let f 0 ∈ L 1 ( R d ) ⇒ [H¨ ormander 1969] b) Let f 0 ∈ L 1 + ( R d ) ⇒ f ( x , t ) > 0 , ∀ t > 0 . (Green’s fct > 0 ) Condition B: Condition A + let C be positively stable (i.e. ℜ λ C > 0) → ∃ confinement potential; drift towards x = 0. • hypoelliptic + confinement = hypocoercive (for FP eq.) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 10 / 32

steady state � � f t = div D ∇ f + C x f (3) Theorem 2 (3) has a unique (normalized) steady state f ∞ ∈ L 1 ( R n ) iff Condition B holds. f ∞ ( x ) = c K e − x ⊤ K − 1 x Then: . . . non-isotropic Gaussian 2 0 < K ∈ R n × n 2 D = CK + KC ⊤ . . . unique solution of (continuous Lyapunov equation) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 11 / 32

normalization of Fokker-Planck equations with f ∞ ( x ) = c K e − x ⊤ K − 1 x � � f t = div D ∇ f + C x f 2 transformations: √ − 1 x ⇒ 1 y := K with g ∞ ( x ) = c e − | y | 2 � ˜ � D ∇ y g + ˜ 2 , g t = div y C y g ˜ ˜ D = C S Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 12 / 32

normalization of Fokker-Planck equations with f ∞ ( x ) = c K e − x ⊤ K − 1 x � � f t = div D ∇ f + C x f 2 transformations: √ − 1 x ⇒ 1 y := K with g ∞ ( x ) = c e − | y | 2 � ˜ � D ∇ y g + ˜ 2 , g t = div y C y g ˜ ˜ D = C S ˇ 2 rotation of y ⇒ D = diag( d 1 , ..., d k , 0 , ..., 0 ) � �� � n − k [normalization from now on assumed] Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 12 / 32

review of entropy method: linear symmetric Fokker-Planck equations evolution of probability density f ( x , t ) , x ∈ R n , t > 0: � � f t = div D · [ ∇ f + f ∇ A ( x )] =: Lf � f 0 ∈ L 1 + ( R n ) , f ( x , 0) = f 0 ( x ); R n f 0 dx = 1 ⇒ f ( x , t ) ≥ 0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 13 / 32

review of entropy method: linear symmetric Fokker-Planck equations evolution of probability density f ( x , t ) , x ∈ R n , t > 0: � � f t = div D · [ ∇ f + f ∇ A ( x )] =: Lf � f 0 ∈ L 1 + ( R n ) , f ( x , 0) = f 0 ( x ); R n f 0 dx = 1 ⇒ f ( x , t ) ≥ 0 e − A ( x ) . . . (unique) normalized steady state f ∞ ( x ) = � � f ∞ D ∇ f ... symmetric in L 2 ( R n , f − 1 Lf = div ∞ ) f ∞ D > 0 ... positive definite matrix A ( x ) ... scalar confinement potential, i.e. A ( x ) → ∞ as | x | → ∞ ; idea : A ( x ) � c | x | 2 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 13 / 32