Schrödinger problem and W 2 -geodesics

Transcription

1 Schrödinger problem and W 2 -geodesics join work wih Nicola Gigli Luca Tamanini Universié Paris Nanerre SISSA En l honneur de P. Caiaux e C. Léonard Toulouse, le 7 juin 2017 L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

2 Firs order differeniaion formula Theorem (Gigli 13) Le (X, d, m) be a RCD(K, ) space; (µ ) a W 2 -geodesic wih µ 0, µ 1 Cm; f W 1,2 (X ). Then f dµ is C 1 ([0, 1]) and d d f dµ = f, φ dµ where (φ ) W 1,2 (X ) is any coninuous choice of locally Lipschiz funcions such ha µ + div( φ µ ) = 0. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

3 Aim Le (X, d, m) be a RCD (K, N) space; (µ ) a W 2 -geodesic wih µ 0, µ 1 Cm; f W 2,2 (X ). Quesion: can we say ha f dµ is C 2 ([0, 1]) and d 2 d 2 f dµ = Hess(f )( φ, φ )dµ where (φ ) W 1,2 (X ) is any coninuous choice of locally Lipschiz funcions such ha µ + div( φ µ ) = 0? L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

4 Aim Le (X, d, m) be a RCD (K, N) space; (µ ) a W 2 -geodesic wih µ 0, µ 1 Cm; f W 2,2 (X ). Quesion: can we say ha f dµ is C 2 ([0, 1]) and d 2 d 2 f dµ = Hess(f )( φ, φ )dµ where (φ ) W 1,2 (X ) is any coninuous choice of locally Lipschiz funcions such ha µ + div( φ µ ) = 0? L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

5 Geodesics in (P 2 (X ), W 2 ) A W 2 -geodesic (µ ) on P 2 (X ) solves for funcions (φ ) such ha µ + div( φ µ ) = 0 φ φ 2 = 0 Problem: no maer how nice µ 0, µ 1 are, in general he φ s are only semiconcave. Quesion: given a geodesic (µ ), can we find curves (µ ε ) which are smooh and produce a second order approximaion of (µ )? L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

6 Geodesics in (P 2 (X ), W 2 ) A W 2 -geodesic (µ ) on P 2 (X ) solves for funcions (φ ) such ha µ + div( φ µ ) = 0 φ φ 2 = 0 Problem: no maer how nice µ 0, µ 1 are, in general he φ s are only semiconcave. Quesion: given a geodesic (µ ), can we find curves (µ ε ) which are smooh and produce a second order approximaion of (µ )? L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

7 Idea Le (µ ε ), (φ ε ) be smooh and such ha µ ε + div( φ ε µ ε ) = 0 φ ε φε 2 = a ε Then for every f smooh we have d f dµ ε = f, φ ε d dµ ε d 2 ( ) d 2 f dµ ε = Hess(f )( φ ε, φ ε ) + f, a ε dµ ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

8 Idea Le (µ ε ), (φ ε ) be smooh and such ha µ ε + div( φ ε µ ε ) = 0 φ ε φε 2 = a ε Then for every f smooh we have d f dµ ε = f, φ ε d dµ ε d 2 ( ) d 2 f dµ ε = Hess(f )( φ ε, φ ε ) + f, a ε dµ ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

9 Rigorous saemen of he problem Given X smooh and µ 0, µ 1 wih bounded densiies and suppors, find (µ ε ) so ha 0h order: (µ ε ) uniformly W 2 -converges o he only W 2 -geodesic (µ ) from µ 0 o µ 1 wih densiies uniformly bounded; 1s order: up o subsequences, φ εn φ in W 1,2 (X ), wih (φ ) a choice of Kanorovich poenials associaed o (µ ); 2nd order: for every f W 2,2 (X ) and δ (0, 1/2) i holds 1 δ The esimaes should depend only on: δ f, a ε ρ ε ddm 0 he L -norms of he densiies of µ 0, µ 1 ; he diameer of heir suppors; he lower bound on he Ricci curvaure of X ; he dimension of X. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

10 Rigorous saemen of he problem Given X smooh and µ 0, µ 1 wih bounded densiies and suppors, find (µ ε ) so ha 0h order: (µ ε ) uniformly W 2 -converges o he only W 2 -geodesic (µ ) from µ 0 o µ 1 wih densiies uniformly bounded; 1s order: up o subsequences, φ εn φ in W 1,2 (X ), wih (φ ) a choice of Kanorovich poenials associaed o (µ ); 2nd order: for every f W 2,2 (X ) and δ (0, 1/2) i holds 1 δ The esimaes should depend only on: δ f, a ε ρ ε ddm 0 he L -norms of he densiies of µ 0, µ 1 ; he diameer of heir suppors; he lower bound on he Ricci curvaure of X ; he dimension of X. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

11 Inerpolaion beween probabiliy densiies via he hea flow: he Schrödinger sysem Le X be smooh and ρ 0, ρ 1 bounded probabiliy densiies wih bounded suppor. Find funcions f, g on X such ha { ρ0 = f P 1 (g) ρ 1 = P 1 (f ) g The enropic inerpolaion beween ρ 0 and ρ 1 is hen defined by ρ := P (f )P 1 (g), [0, 1]. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

12 Inerpolaion beween probabiliy densiies via he hea flow: he Schrödinger sysem Le X be smooh and ρ 0, ρ 1 bounded probabiliy densiies wih bounded suppor. Find funcions f, g on X such ha { ρ0 = f P 1 (g) ρ 1 = P 1 (f ) g The enropic inerpolaion beween ρ 0 and ρ 1 is hen defined by ρ := P (f )P 1 (g), [0, 1]. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

13 Inerpolaion beween probabiliy densiies via he hea flow: slowing down Le X be smooh and ρ 0, ρ 1 bounded probabiliy densiies wih bounded suppor. Find funcions f ε, g ε on X such ha { ρ0 = f ε P ε (g ε ) ρ 1 = P ε (f ε ) g ε The enropic inerpolaion beween ρ 0 and ρ 1 is hen defined by ρ ε := P ε (f ε )P ε(1 ) (g ε ), [0, 1]. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

14 Inerpolaion beween probabiliy densiies via he hea flow: slowing down Le X be smooh and ρ 0, ρ 1 bounded probabiliy densiies wih bounded suppor. Find funcions f ε, g ε on X such ha { ρ0 = f ε P ε (g ε ) ρ 1 = P ε (f ε ) g ε The enropic inerpolaion beween ρ 0 and ρ 1 is hen defined by ρ ε := P ε (f ε )P ε(1 ) (g ε ), [0, 1]. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

15 How o find he funcions f ε, g ε : he Schrödinger problem Le R ε be he measure on X 2 given by dr ε (x, y) := dp ε(δ x ) (y)d(m m)(x, y) dm Then (f ε, g ε ) is a soluion o he Schrödinger sysem if and only if where f ε g ε (x, y) := f ε (x)g ε (y). f ε g ε R ε Adm(ρ 0 m, ρ 1 m) L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

16 How o find he funcions f ε, g ε : he Schrödinger problem Le π ε be he unique minimum of Is Euler equaion is inf H(π R ε) π Adm(ρ 0 m,ρ 1 m) ( ) dπ ε log dσ = 0 dr ε for every σ such ha π# 1 σ = π2 # = 0. This forces ( ) dπ ε log = a ε b ε dr ε for some a ε, b ε, where a ε b ε (x, y) := a ε (x) + b ε (y). Thus for f ε := exp(a ε ), g ε := exp(b ε ) we have π ε = f ε g ε R ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

17 How o find he funcions f ε, g ε : he Schrödinger problem Le π ε be he unique minimum of Is Euler equaion is inf H(π R ε) π Adm(ρ 0 m,ρ 1 m) ( ) dπ ε log dσ = 0 dr ε for every σ such ha π# 1 σ = π2 # = 0. This forces ( ) dπ ε log = a ε b ε dr ε for some a ε, b ε, where a ε b ε (x, y) := a ε (x) + b ε (y). Thus for f ε := exp(a ε ), g ε := exp(b ε ) we have π ε = f ε g ε R ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

18 How o find he funcions f ε, g ε : he Schrödinger problem Le π ε be he unique minimum of Is Euler equaion is inf H(π R ε) π Adm(ρ 0 m,ρ 1 m) ( ) dπ ε log dσ = 0 dr ε for every σ such ha π# 1 σ = π2 # = 0. This forces ( ) dπ ε log = a ε b ε dr ε for some a ε, b ε, where a ε b ε (x, y) := a ε (x) + b ε (y). Thus for f ε := exp(a ε ), g ε := exp(b ε ) we have π ε = f ε g ε R ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

19 The dual problem Wih some manipulaions one can show ha he dual problem of is { sup ϕ,ψ C(X ) inf εh(π R ε) π Adm(ρ 0 m,ρ 1 m) ϕρ 0 dm + ( ψρ 1 dm ε log e ϕ ψ ε dr ε )} Moreover, if π ε is a minimizer and ϕ ε, ψ ε maximizers (Schrödinger poenials), we have π ε = e ϕε ψ ε ε R ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

20 The dual problem Wih some manipulaions one can show ha he dual problem of is { sup ϕ,ψ C(X ) inf εh(π R ε) π Adm(ρ 0 m,ρ 1 m) ϕρ 0 dm + ( ψρ 1 dm ε log e ϕ ψ ε dr ε )} Moreover, if π ε is a minimizer and ϕ ε, ψ ε maximizers (Schrödinger poenials), we have π ε = e ϕε ψ ε ε R ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

21 Link wih opimal ranspor Theorem (Mikami-Thieullen 04 and Léonard 12) As ε 0 he curves ρ ε converge o he (unique) W 2 -geodesic beween ρ 0 m and ρ 1 m. Moreover εh(π ε 1 R ε ) inf d 2 (x, y)dπ. π Adm(ρ 0 m,ρ 1 m) 2 The precise saemen involves: absrac spaces; Γ-convergence; a large deviaion assumpion on he hea kernel. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

22 Link wih opimal ranspor Theorem (Mikami-Thieullen 04 and Léonard 12) As ε 0 he curves ρ ε converge o he (unique) W 2 -geodesic beween ρ 0 m and ρ 1 m. Moreover εh(π ε 1 R ε ) inf d 2 (x, y)dπ. π Adm(ρ 0 m,ρ 1 m) 2 The precise saemen involves: absrac spaces; Γ-convergence; a large deviaion assumpion on he hea kernel. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

23 Building he approximaion ρ ε := f ε g ε ϕ ε := ε log f ε f ε f ε := P ε/2 f ε g ε := P ε(1 )/2 g ε = ε 2 f ε ϕ ε = 1 2 ϕε 2 + ε 2 ϕε g ε = ε 2 g ε ψ ε := ε log g ε ψ ε = 1 2 ψε 2 + ε 2 ψε ε log ρ ε = ϕ ε + ψ ε ϑ ε := 1 2 (ψε ϕ ε ) ρ ε + div( ϑ ε ρ ε ) = 0 ϑ ε ϑε 2 = 1 8 ε2 (2 log ρ ε + log ρ ε 2 ) }{{} =:a ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

24 Building he approximaion ρ ε := f ε g ε ϕ ε := ε log f ε f ε f ε := P ε/2 f ε g ε := P ε(1 )/2 g ε = ε 2 f ε ϕ ε = 1 2 ϕε 2 + ε 2 ϕε g ε = ε 2 g ε ψ ε := ε log g ε ψ ε = 1 2 ψε 2 + ε 2 ψε ε log ρ ε = ϕ ε + ψ ε ϑ ε := 1 2 (ψε ϕ ε ) ρ ε + div( ϑ ε ρ ε ) = 0 ϑ ε ϑε 2 = 1 8 ε2 (2 log ρ ε + log ρ ε 2 ) }{{} =:a ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

25 Building he approximaion ρ ε := f ε g ε ϕ ε := ε log f ε f ε f ε := P ε/2 f ε g ε := P ε(1 )/2 g ε = ε 2 f ε ϕ ε = 1 2 ϕε 2 + ε 2 ϕε g ε = ε 2 g ε ψ ε := ε log g ε ψ ε = 1 2 ψε 2 + ε 2 ψε ε log ρ ε = ϕ ε + ψ ε ϑ ε := 1 2 (ψε ϕ ε ) ρ ε + div( ϑ ε ρ ε ) = 0 ϑ ε ϑε 2 = 1 8 ε2 (2 log ρ ε + log ρ ε 2 ) }{{} =:a ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

26 Building he approximaion ρ ε := f ε g ε ϕ ε := ε log f ε f ε f ε := P ε/2 f ε g ε := P ε(1 )/2 g ε = ε 2 f ε ϕ ε = 1 2 ϕε 2 + ε 2 ϕε g ε = ε 2 g ε ψ ε := ε log g ε ψ ε = 1 2 ψε 2 + ε 2 ψε ε log ρ ε = ϕ ε + ψ ε ϑ ε := 1 2 (ψε ϕ ε ) ρ ε + div( ϑ ε ρ ε ) = 0 ϑ ε ϑε 2 = 1 8 ε2 (2 log ρ ε + log ρ ε 2 ) }{{} =:a ε L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

27 Saemen of he convergence resuls Given X smooh and µ 0, µ 1 wih bounded densiies and suppors, wih he noaions previously inroduced we have ha: 0h order: (ρ ε m) uniformly W 2 -converges o he only W 2 -geodesic (µ ) from µ 0 o µ 1 wih densiies uniformly bounded; 1s order: up o subsequences, ϑ εn ϑ in W 1,2 loc (X ), wih (ϑ ) a choice of Kanorovich poenials associaed o (µ ); 2nd order: for every f H 2,2 (X ) and δ (0, 1/2) i holds 1 δ δ f, a ε ρ ε ddm 0 The esimaes only depend on ρ 0 L, ρ 1 L, K, N and on he diameer of he suppors of ρ 0, ρ 1. Acually, he saemen holds on RCD (K, N) spaces. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

28 Saemen of he convergence resuls Given X smooh and µ 0, µ 1 wih bounded densiies and suppors, wih he noaions previously inroduced we have ha: 0h order: (ρ ε m) uniformly W 2 -converges o he only W 2 -geodesic (µ ) from µ 0 o µ 1 wih densiies uniformly bounded; 1s order: up o subsequences, ϑ εn ϑ in W 1,2 loc (X ), wih (ϑ ) a choice of Kanorovich poenials associaed o (µ ); 2nd order: for every f H 2,2 (X ) and δ (0, 1/2) i holds 1 δ δ f, a ε ρ ε ddm 0 The esimaes only depend on ρ 0 L, ρ 1 L, K, N and on he diameer of he suppors of ρ 0, ρ 1. Acually, he saemen holds on RCD (K, N) spaces. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

29 A couple of esimaes Le (u ) be a soluion of he hea equaion wih posiive compacly suppored L 1 iniial daum. Then: Hamilon s gradien esimae Li-Yau Laplacian esimae log u C ( ) d(, x), (0, 1]; log u C 2, (0, 1]. The consans C 1, C 2 are posiive and only depend on K, N and on he diameer of supp(u 0 ) (on x oo for C 1 ). his leads o 0h and 1s order approximaion L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

30 A couple of esimaes Le (u ) be a soluion of he hea equaion wih posiive compacly suppored L 1 iniial daum. Then: Hamilon s gradien esimae Li-Yau Laplacian esimae log u C ( ) d(, x), (0, 1]; log u C 2, (0, 1]. The consans C 1, C 2 are posiive and only depend on K, N and on he diameer of supp(u 0 ) (on x oo for C 1 ). his leads o 0h and 1s order approximaion L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

31 Second order approximaion We sar from Theorem (Léonard 13) d 2 d 2 H(ρε m m) = 1 ( ) Γ 2 (ϕ ε 2 ) + Γ 2 (ψ ε ) ρ ε dm ( = Γ 2 (ϑ ε ) + ε ) 2 Γ 2(log ρ ε ) ρ ε dm where Γ 2 (h) := h 2 2 h, h L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

32 Second order approximaion Then from Léonard s formula we deduce ha sup 1 δ ε (0,1) δ 1 δ sup ε (0,1) δ for every δ (0, 1/2). Indeed, in he case K = 0, ( Hess(ϑ ε ) 2 + ε 2 Hess(log ρ ε ) 2) ρ ε ddm < + ( ϑ ε 2 + ε 2 log ρ ε 2) ρ ε ddm < + Γ 2 (h) Hess(h) 2 Γ 2 (h) ( h)2 N L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

33 Second order approximaion Then from Léonard s formula we deduce ha sup 1 δ ε (0,1) δ 1 δ sup ε (0,1) δ for every δ (0, 1/2). Indeed, in he case K = 0, ( Hess(ϑ ε ) 2 + ε 2 Hess(log ρ ε ) 2) ρ ε ddm < + ( ϑ ε 2 + ε 2 log ρ ε 2) ρ ε ddm < + Γ 2 (h) Hess(h) 2 Γ 2 (h) ( h)2 N L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

34 Second order differeniaion formula Theorem (Gigli-T. 17) Le (X, d, m) be a RCD (K, N) space; (µ ) a W 2 -geodesic wih µ 0, µ 1 Cm and bounded suppors; f H 2,2 (X ). Then f dµ is C 2 ([0, 1]) and d 2 d 2 f dµ = Hess(f )( φ, φ )dµ where (φ ) W 1,2 (X ) is any coninuous choice of locally Lipschiz funcions such ha µ + div( φ µ ) = 0. In paricular, he choice of evolved Kanorovich poenial does he job. L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19

35 Merci de vore aenion! L. Tamanini (Paris Nanerre & SISSA) Schrödinger pb. and W 2 -geodesics Toulouse, / 19