degenerate Fokker-Planck equations with linear drift

Transcription

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

2 degenerate Fokker-Planck equations with linear drift evolution of probability density f(x,t), x R n, t > 0: ( ) f t = div D f +Cx f f(x,0) = f 0 (x) (1) 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

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

4 hypocoercive example from plasma physics kinetic Fokker-Planck equation for f(x,v,t), x,v R n : f t + v x f }{{} free transport x V v f }{{} influence of potential V(x) steady state: (x,v) = c e ν σ V(x)... given confinement potential [ v 2 2 +V(x) ] = σ v f }{{} +ν div v (vf) }{{} diffusion, σ>0 friction, ν>0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 3 / 32

5 hypocoercive example from plasma physics kinetic Fokker-Planck equation for f(x,v,t), x,v R n : f t + v x f }{{} free transport x V v f }{{} influence of potential V(x) steady state: (x,v) = c e ν σ V(x)... given confinement potential [ v 2 2 +V(x) ] rewritten (with x, v variables): [ ( ) 0 0 f t = div x,v 0 σi }{{} =:D...diffusion ( x,v f + explicit Green s function for V not quadratic! = σ v f }{{} +ν div v (vf) }{{} diffusion, σ>0 friction, ν>0 ) v x V +νv }{{} drift ] f Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 3 / 32

6 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

7 (hypo)coercivity 1 example 1: standard Fokker-Planck equation on R n : ( ) f t = div f +x f =: Lf... symmetric on H := L 2 (f 1 ) (x) = ce x 2 2, kerl = span( ) 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 L 2 (f 1 ) f { } Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 5 / 32

8 (hypo)coercivity 2 example 2: ( ) f t = div D f +Cx 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

9 (hypo)coercivity 2 example 2: ( ) f t = div D f +Cx f =: Lf (2) with degenerate D is degenerate parabolic; (symmetric part of) L is not coercive. Definition 1 (Villani 2009) Consider L on Hilbert space H with K = kerl; let H K (densely) (e.g. H... weighted L 2, H... weighted H 1 ). L is called hypocoercive on H if λ > 0, c 1: e Lt f H ce λt f H f H typically c > 1 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 6 / 32

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

11 degenerate prototype: degenerate diffusion (1D Fokker-Planck) + rotation [ ( ) ( ) 1 0 x1 ωx f t = div f ωx 1 }{{}}{{} =D =Cx (x) = c e x 2 /2 ω R (unique for ω 0); sharp decay rate = 1 2 (= minrλ C) for fast enough rotation ( ω > 1 2 ) x 2 ] f x 1 equilibration by drift/diffusion Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 8 / 32

12 degenerate prototype: degenerate diffusion (1D Fokker-Planck) + rotation [ ( ) ( ) 1 0 x1 ωx f t = div f ωx 1 }{{}}{{} =D =Cx (x) = c e x 2 /2 ω R (unique for ω 0); sharp decay rate = 1 2 (= minrλ C) for fast enough rotation ( ω > 1 2 ) x 2 ] f 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

13 y degenerate prototype with ω = 1: [ ( 1 0 f t = div 0 0 ) f + ( ) x f x 2 -axis: drift characteristics of ẋ = Cx tangent to level curve of x : ] x = x + y y = x Drift characteristic Level curve of P norm x level curve of distorted vector norm x P x; P = Ref: [Dolbeault-Mouhot-Schmeiser] 2015 ( Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 9 / 32 )

14 coefficients C, D in Fokker-Planck equation ( ) f t = div D f +Cx f =: Lf Condition A: No (nontrivial) subspace of kerd is invariant under C. (equivalent: L is hypoelliptic.) Proposition 1 Let Condition A hold. a) Let f 0 L 1 (R d ) f C (R n R + ). [Hörmander 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

15 coefficients C, D in Fokker-Planck equation ( ) f t = div D f +Cx f =: Lf Condition A: No (nontrivial) subspace of kerd is invariant under C. (equivalent: L is hypoelliptic.) Proposition 1 Let Condition A hold. a) Let f 0 L 1 (R d ) f C (R n R + ). [Hörmander 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. Rλ 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

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

17 normalization of Fokker-Planck equations f t = div ( D f +Cx f ) with (x) = c K e x K 1 x 2 transformations: 1 y := K 1 x g t = div y ( D y g + Cy g ) with g (x) = ce y 2 2, D = C S Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 12 / 32

18 normalization of Fokker-Planck equations f t = div ( D f +Cx f ) with (x) = c K e x K 1 x 2 transformations: 1 y := K 1 x g t = div y ( D y g + Cy g ) with g (x) = ce y 2 2, D = C S 2 rotation of y Ď = 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

19 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(x,0) = f 0 (x); f 0 L 1 +(R n ), f 0 dx = 1 f(x,t) 0 R n Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 13 / 32

20 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(x,0) = f 0 (x); f 0 L 1 +(R n ), f 0 dx = 1 f(x,t) 0 R n (x) = e A(x)... (unique) normalized steady state ( Lf = div D f )... symmetric in L 2 (R n,f 1 ) 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

21 admissible relative entropies (for entropy method) for probability densities f 1,2 : e ψ (f 1 f 2 ) := ( f1 ψ R n ) f 2 dx 0 f 2 ψ : R + 0 R entropy generators ψ 0, ψ(1) = 0, ψ > 0, (ψ ) ψ ψ IV... relative entropy Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 14 / 32

22 admissible relative entropies (for entropy method) for probability densities f 1,2 : e ψ (f 1 f 2 ) := ( f1 ψ R n ) f 2 dx 0 f 2 ψ : R + 0 R entropy generators ψ 0, ψ(1) = 0, ψ > 0, (ψ ) ψ ψ IV... relative entropy examples: 1) ψ 1 (σ) = σlnσ σ +1 2) ψ p (σ) = σ p 1 p(σ 1), 1 < p 2 ψ(σ) ϕ...quadr. e ψ (f 1 f 2 ) = 0 f 1 = f 2 σ 1 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 14 / 32

23 entropy dissipation Lemma 1 ( ) Let f(t) solve Fokker-Planck equation f t = div D [ f +f A(x)] d dt e ψ(f(t) ) = ψ R n ( f(t) ) f(t) D f(t) dx =: I ψ (f(t) ) 0... (negative) Fisher information Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 15 / 32

24 Step 1: exponent. decay of entropy dissipation for D const D = const. in x, (x) = e A(x) Theorem 3 Let I ψ (f 0 ) <. Let D,A satisfy a Bakry - Emery condition 2 A(x) x 2 λ 1 D 1 x R n (4) >0 I ψ (f(t) ) e 2λ 1t I ψ (f 0 ), t 0 A...uniformly convex if D = I Ref s: [Bakry-Emery] 1984/85; [Arnold-Markowich-Toscani-Unterreiter] Comm. PDE 2001 robust w.r.t. many nonlinear perturbations Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 16 / 32

25 Step 2: exponential decay of relative entropy for D const Theorem 4 Let D, A satisfy BEC 2 A(x) x 2 λ 1 D 1 e ψ (f(t) ) e 2λ 1t e ψ (f 0 ), t 0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 17 / 32

26 Step 2: exponential decay of relative entropy for D const Theorem 4 Let D, A satisfy BEC 2 A(x) x 2 λ 1 D 1 e ψ (f(t) ) e 2λ 1t e ψ (f 0 ), t 0 Proof: from proof of Theorem 3 : d dt I(t) 2λ 1 I(t) }{{} = e (t) t... dt Since I(t), e(t) t 0: d dt e(t) 2λ 1e(t) (5) (+ density argument) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 17 / 32

27 problem of entropy decay (cp. standard entropy method) decay of quadratic entropy e 2 (t) = f(t) 2 L 2 (f 1 ) : standard Fokker-Planck equation: non-degenerate e(t) is convex; entropy dissip. e (t) < 0 f ; e µe possible (with µ > 0) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 18 / 32

28 problem of entropy decay (cp. standard entropy method) decay of quadratic entropy e 2 (t) = f(t) 2 L 2 (f 1 ) : standard Fokker-Planck equation: non-degenerate e(t) is convex; entropy dissip. e (t) < 0 f ; e µe possible (with µ > 0) degenerate prototype ex.: e(t) is not convex; e (t) = 0 for some f ; e µe wrong (in general) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 18 / 32

29 entropy decay in inhomogeneous Boltzmann equation simulation of 1+2 D Boltzmann equation: wavy entropy decay H g (t): relative entropy w.r.t. the global Maxwellian Ref: [Filbet-Mouhot-Pareschi] 2006 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 19 / 32

30 new entropy method for degen. FP: f t = div(d f +Cx f) e (t) = 0 for some f entropy dissipation: ( ) d f dt e ψ = ψ f D R n }{{} f dx =: I ψ (f) 0 is useless as Lyapunov functional. 0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 20 / 32

31 new entropy method for degen. FP: f t = div(d f +Cx f) e (t) = 0 for some f entropy dissipation: ( ) d f dt e ψ = ψ f D R n }{{} f dx =: I ψ (f) 0 is useless as Lyapunov functional. define modified entropy dissipation as auxiliary functional: ( ) f S ψ (f) := f P }{{} f dx 0 R n ψ d goal: estimate between S(f(t)), dts(f(t)) for good choice of P > 0. Then: P c P D S ψ (f) c P I ψ (f) ց 0 0 >0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 20 / 32

32 modified entropy dissipation S ψ (f) : choice of P Lemma 2 Let Q be positively stable, i.e. µ := min{rλ Q } > 0. 1 If all λ min Q {λ σ(q) Rλ = µ} are non-defective (i.e. geometric = algebraic multiplicity) P R n n, P > 0 : PQ+Q P 2µP. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 32

33 modified entropy dissipation S ψ (f) : choice of P Lemma 2 Let Q be positively stable, i.e. µ := min{rλ Q } > 0. 1 If all λ min Q {λ σ(q) Rλ = µ} are non-defective (i.e. geometric = algebraic multiplicity) P R n n, P > 0 : PQ+Q P 2µP. 2 If (at least) one λ min Q is defective ε > 0 P = P(ε) > 0 : PQ+Q P 2(µ ε)p. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 32

34 modified entropy dissipation S ψ (f) : choice of P Lemma 2 Let Q be positively stable, i.e. µ := min{rλ Q } > 0. 1 If all λ min Q {λ σ(q) Rλ = µ} are non-defective (i.e. geometric = algebraic multiplicity) P R n n, P > 0 : PQ+Q P 2µP. 2 If (at least) one λ min Q is defective ε > 0 P = P(ε) > 0 : PQ+Q P 2(µ ε)p. Proof: P can be constructed explicitly; e.g. for Q non-defective / diagonalizable: n P := z j z j ; z j... eigenvectors of Q j=1 P not unique; but the decay rates µ (or µ ε) are independent of P. application with Q := C. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 32

35 Step 1: exponential decay of auxiliary functional S ψ (f) S ψ (f) := R n ψ ( f ) f P f dx 0 Proposition 2 µ := min{rλ C }. Let f 0 satisfy: ( ) ψ f0 f 0 2 dx < ( weighted H 1 seminorm) 1 If all λ min C are non-defective S(f(t)) e 2µt S(f 0 ), t 0; Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 22 / 32

36 Step 1: exponential decay of auxiliary functional S ψ (f) S ψ (f) := R n ψ ( f ) f P f dx 0 Proposition 2 µ := min{rλ C }. Let f 0 satisfy: ( ) ψ f0 f 0 2 dx < ( weighted H 1 seminorm) 1 If all λ min C are non-defective S(f(t)) e 2µt S(f 0 ), t 0; 2 If one λ min C is defective S(f(t),ε) e 2(µ ε)t S(f 0,ε), t 0. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 22 / 32

37 Proof of Proposition 2 modified entropy method d dt S(f(t)) = 2 ψ ( f ) [ ] u PC+C P }{{} 2µP... replaces BEC u dx Tr(XY) dx 2µS(f(t)); u = f }{{} 0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 32

38 Proof of Proposition 2 modified entropy method d dt S(f(t)) = 2 ψ ( f ) [ ] u PC+C P }{{} 2µP... replaces BEC u dx Tr(XY) dx 2µS(f(t)); u = f }{{} 0 use X := ( ψ ψ ψ 1 2 ψiv ) ( f ) 0, since det X = 1 2 ψ ψ IV (ψ ) 2 0 (for admissible rel. entropies) ; Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 32

39 Proof of Proposition 2 modified entropy method d dt S(f(t)) = 2 ψ ( f ) [ ] u PC+C P }{{} 2µP... replaces BEC u dx Tr(XY) dx 2µS(f(t)); u = f }{{} 0 use X := ( ψ ψ ψ 1 2 ψiv ) ( f ) 0, since det X = 1 2 ψ ψ IV (ψ ) 2 0 (for admissible rel. entropies) ; ( Tr(D u x P u x ) u D u x Pu ) Y := with Cauchy-Schwarz u D u x Pu (u Pu)(u Du) 0, Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 32

40 Step 2: exponential decay of relative entropy Theorem 5 Let f 0 satisfy: ( ) ψ f0 u 0 2 dx <. e(f(t) ) c S(f(t)) ce 2µt S(f 0 ), t 0 ( reduced rate for a defective λ min C : 2(µ ε) ) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 24 / 32

41 Step 2: exponential decay of relative entropy Theorem 5 Let f 0 satisfy: ( ) ψ f0 u 0 2 dx <. e(f(t) ) c S(f(t)) ce 2µt S(f 0 ), t 0 ( reduced rate for a defective λ min C : 2(µ ε) ) Proof: Consider non-degenerate (auxiliary) symmetric FP equation: ( g t = div }{{} P ( g ) ) ; g = = c e x 2 /2 = c e A(x) (6) >0 It satisfies the Bakry-Emery condition 2 A x 2 = I λ P P 1. convex Sobolev inequality: e ψ (g ) 1 2λ P S ψ (g) g Remark: S ψ (g) is the true entropy dissipation for (6)! Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 24 / 32

42 (parabolic) regularization of semigroup e Lt Proposition 3 The Hörmander order m [1,n k] (k =rank D) is the minimum such that m C j D(C ) j κi for some κ > 0. j=0 (Existence of m is equivalent to Condition A, i.e. hypoellipticity of L.) S ψ (f(t)) c t (2m+1) e ψ (f 0 ), 0 < t 1. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 25 / 32

43 (parabolic) regularization of semigroup e Lt Proposition 3 The Hörmander order m [1,n k] (k =rank D) is the minimum such that m C j D(C ) j κi for some κ > 0. j=0 (Existence of m is equivalent to Condition A, i.e. hypoellipticity of L.) S ψ (f(t)) c t (2m+1) e ψ (f 0 ), 0 < t 1. Ref s: Prop. 3 is generalization to all admissible relative entropies of: [Hérau] JFA 2007; [Villani] book 2009 (only for quadratic & logarithmic entropies) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 25 / 32

44 exp. decay of rel. entropy for f t = div(d f +Cx f) =: Lf combination of regularization for initial time with Th.5 (entropy decay) Theorem 6 (Arnold-Erb 2014) Let L satisfy Condition B; µ := min{rλ C }. c > 0: e ψ (f(t) ) ce 2µt e ψ (f 0 ), t 0 ( reduced rate for a defective λ min C : 2(µ ε) ) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 32

45 exp. decay of rel. entropy for f t = div(d f +Cx f) =: Lf combination of regularization for initial time with Th.5 (entropy decay) Theorem 6 (Arnold-Erb 2014) Let L satisfy Condition B; µ := min{rλ C }. c > 0: e ψ (f(t) ) ce 2µt e ψ (f 0 ), t 0 ( reduced rate for a defective λ min C : 2(µ ε) ) Proof: e(t) CSI 1 S(f(t)) decay 2λ P 1 e 2µ(t δ) S(f(δ)) regularization c(δ)e 2µt e(0) 2λ P Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 32

46 exp. decay of rel. entropy for f t = div(d f +Cx f) =: Lf combination of regularization for initial time with Th.5 (entropy decay) Theorem 6 (Arnold-Erb 2014) Let L satisfy Condition B; µ := min{rλ C }. c > 0: e ψ (f(t) ) ce 2µt e ψ (f 0 ), t 0 ( reduced rate for a defective λ min C : 2(µ ε) ) Proof: e(t) CSI 1 S(f(t)) decay 2λ P 1 e 2µ(t δ) S(f(δ)) regularization c(δ)e 2µt e(0) 2λ P Remark: Rate µ is sharp, but constant c is not. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 32

47 kinetic Fokker-Planck eq. with non-quadratic potential f t +v x f x V(x) v f = σ v f +ν div v (vf); x,v R n steady state factors in x, v: (x,v) = c e ν σ [ v 2 2 +V(x) ] Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 32

48 kinetic Fokker-Planck eq. with non-quadratic potential f t +v x f x V(x) v f = σ v f +ν div v (vf); x,v R n steady state factors in x, v: (x,v) = c e ν σ f t = div x,v [ ( 0 σ ν I σ ν I σi [ v 2 2 +V(x) ] ) x,v ( f ) f ] Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 32

49 kinetic Fokker-Planck eq. with non-quadratic potential f t +v x f x V(x) v f = σ v f +ν div v (vf); x,v R n steady state factors in x, v: (x,v) = c e ν σ f t = div x,v [ ( 0 σ ν I σ ν I σi [ v 2 2 +V(x) ] ) x,v ( f ) f ] Theorem 7 (Arnold-Erb, Achleitner-Arnold 2015) Let n = 1; V(x) = ω2 0 2 x 2 +Ṽ(x)... given confinement potential with maxv (x) minv (x) ν e ψ (f(t) ) c S ψ (f 0 )e 2κt, t 0 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 32

50 kinetic Fokker-Planck eq. with non-quadratic potential f t +v x f x V(x) v f = σ v f +ν div v (vf); x,v R n steady state factors in x, v: (x,v) = c e ν σ f t = div x,v [ ( 0 σ ν I σ ν I σi [ v 2 2 +V(x) ] ) x,v ( f ) f ] Theorem 7 (Arnold-Erb, Achleitner-Arnold 2015) Let n = 1; V(x) = ω2 0 2 x 2 +Ṽ(x)... given confinement potential with maxv (x) minv (x) ν e ψ (f(t) ) c S ψ (f 0 )e 2κt, t 0 Ṽ... O(1) perturbation Greens function not explicit Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 32

51 local vs. global decay rate in non-symmetric FP equations decay of logarithmic entropy for the non-symmetric FP equation [ ( ) ( ) 1/4 0 1/4x1 4x ] f t = div f + 2 f 0 1 4x 1 +x 2 with (x) = ce x 2 /2 = e A(x) : find e(f(t) ) c e(0)e λt Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 28 / 32

52 local vs. global decay rate in non-symmetric FP equations decay of logarithmic entropy for the non-symmetric FP equation [ ( ) ( ) 1/4 0 1/4x1 4x ] f t = div f + 2 f 0 1 4x 1 +x 2 with (x) = ce x 2 /2 = e A(x) : find e(f(t) ) c e(0)e λt standard entropy method: BEC 2 A = I λd 1 yields sharp x 2 local decay rate: λ l = 1 4 ; c = 1 Hypocoercive method yields sharp global decay rate: λ g = 5 8 ; c > 1 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 28 / 32

53 local vs. global decay rate in non-symmetric FP equations decay of logarithmic entropy for the non-symmetric FP equation [ ( ) ( ) 1/4 0 1/4x1 4x ] f t = div f + 2 f 0 1 4x 1 +x 2 with (x) = ce x 2 /2 = e A(x) : find e(f(t) ) c e(0)e λt standard entropy method: BEC 2 A = I λd 1 yields sharp x 2 local decay rate: λ l = 1 4 ; c = 1 Hypocoercive method yields sharp global decay rate: λ g = 5 8 ; c > 1 Lemma: 2D, admissible P: multiplicative constant S(0) 2λ P is sharp. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 28 / 32

54 Why does the hypocoercive method work? algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Q s := 1 2 (Q+QT ). motivation: entropy method operates mostly with quadratic functionals (e.g. e 2 (f ); I ψ = ψ ( f ) T f D f dx ) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 32

55 Why does the hypocoercive method work? algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Q s := 1 2 (Q+QT ). motivation: entropy method operates mostly with quadratic functionals (e.g. e 2 (f ); I ψ = ψ ( f ) T f D f dx ) Lemma 3 Let Q be positively stable, i.e. µ := min{rλ Q } > 0. λ min (Q s ) µ ( typically strict inequality) Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 32

56 Why does the hypocoercive method work? algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Q s := 1 2 (Q+QT ). motivation: entropy method operates mostly with quadratic functionals (e.g. e 2 (f ); I ψ = ψ ( f ) T f D f dx ) Lemma 3 Let Q be positively stable, i.e. µ := min{rλ Q } > 0. λ min (Q s ) µ ( typically strict inequality) Lemma 4 ( = Lemma 2 for choice of P) P > 0 such that the similar matrix Q := PQ P 1 satisfies: λ min ( Q s ) = µ ( λ min ( Q s ) = µ ε in the defective case ) So: a similarity transformation such that Q, Q, Q s have the same spectral gap. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 32

57 Application to f t = div(d f +Cxf) = Lf Decay estimate of the drift characteristics ẋ = Cx: Let x 2 P := x,px. d dt x 2 P = 2x T PCx = ( Px) T ( PC P 1 + P 1 C T ) P ( Px) }{{} :=2 C s 2µI 2µ x 2 P carries over to all invariant eigenspaces of L Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 30 / 32

58 Conclusion new entropy method for degenerate Fokker-Planck eq. (+ linear drift) key tool: modified entropy dissipation functional diff. inequal. Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 31 / 32

59 Conclusion new entropy method for degenerate Fokker-Planck eq. (+ linear drift) key tool: modified entropy dissipation functional diff. inequal. extension to kinetic Fokker-Planck eq. (with non-quadratic potential) extension to non-degenerate, non-symmetric Fokker-Planck equations sharp envelope for global decay 2017: (1+t 2 )e 2κt decay for defective case [AA-Einav-Wöhrer] Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 31 / 32

60 Conclusion new entropy method for degenerate Fokker-Planck eq. (+ linear drift) key tool: modified entropy dissipation functional diff. inequal. extension to kinetic Fokker-Planck eq. (with non-quadratic potential) extension to non-degenerate, non-symmetric Fokker-Planck equations sharp envelope for global decay 2017: (1+t 2 )e 2κt decay for defective case [AA-Einav-Wöhrer] References A. Arnold, J. Erb: Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, arxiv 2014 F. Achleitner, A. Arnold, D. Stürzer: Large-time behavior in non-symmetric Fokker-Planck equations, Rivista di Matematica della Univ. di Parma, 2015 F. Achleitner, A. Arnold, E. Carlen: On linear hypocoercive BGK models, Springer Proc. in Math. & Stat., 2016 Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 31 / 32

61 References: [Villani] 2009: exponential decay in weighted H 1, but no sharp rates [Dolbeault-Mouhot-Schmeiser] Trans. AMS 2015: kinetic models, exponential decay in modified L 2 norm; m = 1 [Gadat-Miclo] KRM 2013: sharp rates for 2 Fokker-Planck toy models [Baudoin] 2014: Γ 2 formalism, includes auxiliary gradient functional Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 32 / 32