Generalized Gamma z calculus

A drift-diffusion process with a non-degenerate diffusion coefficient matrix possesses good properties: convergence to equilibrium, entropy dissipation rate, etc. The degenerate drift-diffusion possesses a non-positive definite diffusion coefficient matrix, which makes it difficult to govern the convergence property and entropy dissipation rate by drift-diffusion coefficients on its own because of lacking control for the system. In general, the degenerate drift-diffusion is intrinsically equipped with a sub-Riemannian structure defined by the diffusion coefficients. We propose a new methodology to systematically study the general drift-diffusion process through sub-Riemannian geometry and Wasserstein geometry. We generalize the Bakry-Emery calculus and Gamma z calculus to define a new notion of sub-Riemannian Ricci curvature tensor. With the new Ricci curvature tensor, we are able to establish generalized curvature dimension bounds on sub-Riemannian manifolds which goes beyond step two condition. As an application, for the first time, we establish analytical bounds for logarithmic Sobolev inequalities for the weighted measure on the displacement group and Engel group. Our result also provides an entropy dissipation rate for Langevin dynamics with gradient drift and variable temperature matrix. The talk is based on joint works with Qi Feng.

Transcript

1. Sub-Riemannian Ricci curvature via generalized Gamma z calculus and functional inequalities Qi Feng Joint work with Wuchen Li April 24th, 2020 Wuchen Li This is based on a joint work with Qi Feng
2. Outline of the talk. SDEs and Bakry-´ Emery calculus Generalized sub-Riemannian Ricci curvature tensor Functional inequalities.
3. Toy model I. Consider the 3-dimensional Brownian motion Xt which satisfies dXt = 0 @ 1 0 0 0 1 0 0 0 1 1 A 0 @ dB1 t dB2 t dB3 t 1 A , Xx t = x, then we know Xt = (B1 t , B2 t , B3 t ).
4. Toy model I. Consider the 3-dimensional Brownian motion Xt which satisfies dXt = 0 @ 1 0 0 0 1 0 0 0 1 1 A 0 @ dB1 t dB2 t dB3 t 1 A , Xx t = x, then we know Xt = (B1 t , B2 t , B3 t ). In particular, I E(f (Xt )) = Z f (y)pt (x, dy) =: Ptf (x);
5. Toy model I. Consider the 3-dimensional Brownian motion Xt which satisfies dXt = 0 @ 1 0 0 0 1 0 0 0 1 1 A 0 @ dB1 t dB2 t dB3 t 1 A , Xx t = x, then we know Xt = (B1 t , B2 t , B3 t ). In particular, I E(f (Xt )) = Z f (y)pt (x, dy) =: Ptf (x); I Pt = e 1 2 tL, L = = @2 @x2 1 + @2 @x2 2 + @2 @x2 3 ;
6. Toy model I. Consider the 3-dimensional Brownian motion Xt which satisfies dXt = 0 @ 1 0 0 0 1 0 0 0 1 1 A 0 @ dB1 t dB2 t dB3 t 1 A , Xx t = x, then we know Xt = (B1 t , B2 t , B3 t ). In particular, I E(f (Xt )) = Z f (y)pt (x, dy) =: Ptf (x); I Pt = e 1 2 tL, L = = @2 @x2 1 + @2 @x2 2 + @2 @x2 3 ; I pt (x, dy) = 1 (2⇡t)n/2 e |x y|2/2tdy;
7. Toy model I. Consider the 3-dimensional Brownian motion Xt which satisfies dXt = 0 @ 1 0 0 0 1 0 0 0 1 1 A 0 @ dB1 t dB2 t dB3 t 1 A , Xx t = x, then we know Xt = (B1 t , B2 t , B3 t ). In particular, I E(f (Xt )) = Z f (y)pt (x, dy) =: Ptf (x); I Pt = e 1 2 tL, L = = @2 @x2 1 + @2 @x2 2 + @2 @x2 3 ; I pt (x, dy) = 1 (2⇡t)n/2 e |x y|2/2tdy; I @ tu = 1 2 u, with u(t, x) = Ptf (x) and u(0, x) = f (x).
8. Bakry-´ Emery calculus on toy model I. We introduce the “carr´ e du champ”operator for operator L: (f , g) := 1 2 (L(fg) fLg gLf ) (1.1)
9. Bakry-´ Emery calculus on toy model I. We introduce the “carr´ e du champ”operator for operator L: (f , g) := 1 2 (L(fg) fLg gLf ) (1.1) Take L = , we have (f , g) = 3 X i=1 @f @xi @g @xi = hrf , rgiR3 .
10. Bakry-´ Emery calculus on toy model I. We introduce the “carr´ e du champ”operator for operator L: (f , g) := 1 2 (L(fg) fLg gLf ) (1.1) Take L = , we have (f , g) = 3 X i=1 @f @xi @g @xi = hrf , rgiR3 . The iterative Gamma is the second order di↵erential operator: 2 (f , g) := 1 2 (L (f , g) (f , Lg) (g, Lf )) (1.2)
11. Bakry-´ Emery calculus on toy model I. Take L = , we have 2 (f , f ) = 1 2 ( (f , f ) (f , f ) (f , f )) = 1 2 3 X i=1 @ ii 3 X j=1 |@ j f |2 3 X i=1 @ i f ⇣ @ i ⇣ 3 X j=1 @ jj f ⌘⌘ = 3 X i,j=1 |@ ij f |2 = kHessf k2 HS 1 3 ( f )2
12. Bakry-´ Emery calculus on toy model I. Take L = , we have 2 (f , f ) = 1 2 ( (f , f ) (f , f ) (f , f )) = 1 2 3 X i=1 @ ii 3 X j=1 |@ j f |2 3 X i=1 @ i f ⇣ @ i ⇣ 3 X j=1 @ jj f ⌘⌘ = 3 X i,j=1 |@ ij f |2 = kHessf k2 HS 1 3 ( f )2 2 (f , f ) = kHessf k2 HS 1 3 ( f )2 (1.3)
13. Logarithmic Sobolev inequality: I L. Gross’s LSI For any smooth positive function f : Rn ! R such that R Rn fd = 1, Z Rn f log fd  1 2 Z Rn (f ) f d , where d = e |x|2/2/(2⇡)n/2. [Amer. J. Math. 97, 1061-1083 (1975)]
14. Logarithmic Sobolev inequality: II Consider the entropy (on f ) along the semigroup Pt , we have, Pt (f log f ) Ptf log Ptf = Z t 0 d ds Ps (Pt sf log Pt sf )ds
15. Logarithmic Sobolev inequality: II Consider the entropy (on f ) along the semigroup Pt , we have, Pt (f log f ) Ptf log Ptf = Z t 0 d ds Ps (Pt sf log Pt sf )ds d ds Ps (Pt sf log Pt sf ) = 1 2Ps ( (Pt sf log Pt sf ) Pt sf log Pt sf Pt sf ) = 1 2Ps ⇣|rPt sf |2 Pt sf ⌘ =: 1 2 (s) = 1 2Ps (Pt sf 1 (log Pt sf , log Pt sf ))
16. Logarithmic Sobolev inequality: II Consider the entropy (on f ) along the semigroup Pt , we have, Pt (f log f ) Ptf log Ptf = Z t 0 d ds Ps (Pt sf log Pt sf )ds d ds Ps (Pt sf log Pt sf ) = 1 2Ps ( (Pt sf log Pt sf ) Pt sf log Pt sf Pt sf ) = 1 2Ps ⇣|rPt sf |2 Pt sf ⌘ =: 1 2 (s) = 1 2Ps (Pt sf 1 (log Pt sf , log Pt sf )) 0 (s)=Ps (Pt s f 2(log Pt s f )) Ps (Pt s f |Hess(log Pt s f )|2) 0. Pt (f log f ) Ptf log Ptf = 1 2 Z t 0 (s)ds  1 2tPt ( (f ) f ).
17. Bakry-´ Emery calculus and curvature dimension inequality. Bochner’s formula For a a more general Laplacian operator (e.g. on a Riemannian manifold) 2 (f , f ) = kHessf k2 H.S. + Ric(rf , rf ).
18. Bakry-´ Emery calculus and curvature dimension inequality. Bochner’s formula For a a more general Laplacian operator (e.g. on a Riemannian manifold) 2 (f , f ) = kHessf k2 H.S. + Ric(rf , rf ). Curvature dimension inequality (elliptic) [D. Bakry, M. Emery: 12, 177-206, LNM 1123 (1985)] 2 (f )  (f ) + 1 n (Lf )2, CD(, n); 2 (f )  (f ), CD(, 1).
19. Bakry-´ Emery calculus and curvature dimension inequality. Bochner’s formula For a a more general Laplacian operator (e.g. on a Riemannian manifold) 2 (f , f ) = kHessf k2 H.S. + Ric(rf , rf ). Curvature dimension inequality (elliptic) [D. Bakry, M. Emery: 12, 177-206, LNM 1123 (1985)] 2 (f )  (f ) + 1 n (Lf )2, CD(, n); 2 (f )  (f ), CD(, 1). I Poincare, Hypercontractivity, Transport cost, etc.. I K.T. Sturm(06), J. Lott-C. Villani(09).
20. “Analysis and Geometry of Markov Di↵usion Operators” —Bakry-Gentil-Ledoux
21. How about degenerate operators?
22. Brownian motion on Heisenberg group We look at the following Stratonovich SDE: dWt = a(Wt ) ✓ dB1 t dB2 t ◆ , a = 0 @ 1 0 0 1 y/2 x/2 1 A
23. Brownian motion on Heisenberg group We look at the following Stratonovich SDE: dWt = a(Wt ) ✓ dB1 t dB2 t ◆ , a = 0 @ 1 0 0 1 y/2 x/2 1 A which is the same as dWt = X dB1 t + Y dB2 t with X = @ @x 1 2 y @ @z , Y = @ @y + 1 2 x @ @z , Z = @ @z
24. Brownian motion on Heisenberg group We look at the following Stratonovich SDE: dWt = a(Wt ) ✓ dB1 t dB2 t ◆ , a = 0 @ 1 0 0 1 y/2 x/2 1 A which is the same as dWt = X dB1 t + Y dB2 t with X = @ @x 1 2 y @ @z , Y = @ @y + 1 2 x @ @z , Z = @ @z I Look at the video of horizontal Brownian motion; I {X, Y , Z} forms an orthonormal basis for the Lie algebra of Heisenberg group; I [X, Y ] = XY YX = Z; (bracket generating, H¨ ormander) I L = X2 + Y 2, horizontal Laplacian operator.
25. Gamma calculus on Heisenberg group L = X2 + Y 2, r Hf = XfX + YfY = aaT rf = 2 X i=1 ai fai 1 (f , f ) = XfXf + YfYf = (r Hf , r Hf )
26. Gamma calculus on Heisenberg group L = X2 + Y 2, r Hf = XfX + YfY = aaT rf = 2 X i=1 ai fai 1 (f , f ) = XfXf + YfYf = (r Hf , r Hf ) 2 (f , f ) = kHessHf k2 H.S. + 1 2 (Zf )2 2(Xf )(YZf ) + 2(Yf )(XZf )
27. Gamma calculus on Heisenberg group L = X2 + Y 2, r Hf = XfX + YfY = aaT rf = 2 X i=1 ai fai 1 (f , f ) = XfXf + YfYf = (r Hf , r Hf ) 2 (f , f ) = kHessHf k2 H.S. + 1 2 (Zf )2 2(Xf )(YZf ) + 2(Yf )(XZf ) The curvature dimension inequality does not work!!!
28. Generalized Curvature dimension inequality F. Baudoin and N. Garofalo (09) [Journal of the EMS, Vol. 19, Issue 1, 2017] 2 (f , f ) + ⌫ z 2 (f , f ) 1 d (Lf )2 + (⇢ 1  ⌫ ) 1 (f , f ) + ⇢ 2 z 1 (f , f ). with z 2 (f , f ) := 1 2 (L z 1 (f , g) z 1 (f , Lg) z 1 (g, Lf )). This is called CD(⇢ 1 , ⇢ 2 , , d) condition.
29. Generalized Curvature dimension inequality F. Baudoin and N. Garofalo (09) [Journal of the EMS, Vol. 19, Issue 1, 2017] 2 (f , f ) + ⌫ z 2 (f , f ) 1 d (Lf )2 + (⇢ 1  ⌫ ) 1 (f , f ) + ⇢ 2 z 1 (f , f ). with z 2 (f , f ) := 1 2 (L z 1 (f , g) z 1 (f , Lg) z 1 (g, Lf )). This is called CD(⇢ 1 , ⇢ 2 , , d) condition. Come back to Heisenberg group: z 2 (f , f ) = (XZf )2 + (YZf )2 The Heisenberg group satisfies Generalized CD(0, 1 2 , 1, 2).
30. Generalized Curvature dimension inequality to LSI Recall in the Riemannian case: (s) = Ps (Pt sf 1 (Pt sf , Pt sf )) 0(s) = Ps (Pt sf 2 (Pt sf , Pt sf )).
31. Generalized Curvature dimension inequality to LSI Recall in the Riemannian case: (s) = Ps (Pt sf 1 (Pt sf , Pt sf )) 0(s) = Ps (Pt sf 2 (Pt sf , Pt sf )). In the sub-Riemannian setting: a (s) = Pt sf 1 (Pt sf , Pt sf ) z (s) = Pt sf z 1 (Pt sf , Pt sf ). We have (denote g(s) = Pt sf ) (@ s + L)( a + z ) =g 2 (g, g) + g Z 2 (g, g) + 1 (g, z 1 (g, g)) z 1 (g, 1 (g, g))
32. Generalized Curvature dimension inequality to LSI Recall in the Riemannian case: (s) = Ps (Pt sf 1 (Pt sf , Pt sf )) 0(s) = Ps (Pt sf 2 (Pt sf , Pt sf )). In the sub-Riemannian setting: a (s) = Pt sf 1 (Pt sf , Pt sf ) z (s) = Pt sf z 1 (Pt sf , Pt sf ). We have (denote g(s) = Pt sf ) (@ s + L)( a + z ) =g 2 (g, g) + g Z 2 (g, g) + 1 (g, z 1 (g, g)) z 1 (g, 1 (g, g)) Require the assumption: 1 (f , z 1 (f , f )) = z 1 (f , 1 (f , f ))
33. Totally geodesic foliations. In general, totally geodesic foliation satisfies the generalized CD inequality. I Hopf fibration U(1) ! S2n+1 ! CPn. I Riemannian submersion, ⇡ : Mn+m ! Bn. I K-contact manifold, foliated by Reeb vector field. I Sasakian manifold I Generalized Hopf fibration. I ...
34. Totally geodesic foliations. In general, totally geodesic foliation satisfies the generalized CD inequality. I Hopf fibration U(1) ! S2n+1 ! CPn. I Riemannian submersion, ⇡ : Mn+m ! Bn. I K-contact manifold, foliated by Reeb vector field. I Sasakian manifold I Generalized Hopf fibration. I ... 1 (f , z 1 (f , f )) = z 1 (f , 1 (f , f )) Applies to step two condition and totally geodesic foliation.
35. How about 1 (f , z 1 (f , f )) 6= z 1 (f , 1 (f , f ))?
36. Displacement group For the vector fields: X = @ @✓ , Y = e ✓ @ @x + @ @y , R = @ @y , The horizontal Brownian motion is: dWt = X dB1 t + Y dB2 t with [X, Y ] = Y + R, [X, R] = 0, [Y , R] = 0. Satisfying the step two condition.
37. Displacement group For the vector fields: X = @ @✓ , Y = e ✓ @ @x + @ @y , R = @ @y , The horizontal Brownian motion is: dWt = X dB1 t + Y dB2 t with [X, Y ] = Y + R, [X, R] = 0, [Y , R] = 0. Satisfying the step two condition. However, 1 (f , z 1 (f , f )) z 1 (f , 1 (f , f )) 6= 0.
38. Displacement group For the vector fields: X = @ @✓ , Y = e ✓ @ @x + @ @y , R = @ @y , The horizontal Brownian motion is: dWt = X dB1 t + Y dB2 t with [X, Y ] = Y + R, [X, R] = 0, [Y , R] = 0. Satisfying the step two condition. However, 1 (f , z 1 (f , f )) z 1 (f , 1 (f , f )) 6= 0. One step three example: [X, Y ] = W , [X, W ] = Z, [X, Z] = 0
39. Generalized Gamma z calculus Theorem ( F. and Li (19) ) For a general hypoelliptic operator L, (satisfies some assumptions), we have 2 (f , f )+ z,⇢⇤ 2 (f , f ) = |HessG a,z f |2 +“R(rf , rf )”+R⇢⇤ (rf , rf ) | {z } new tensor .
40. Generalized Gamma z calculus Theorem ( F. and Li (19) ) For a general hypoelliptic operator L, (satisfies some assumptions), we have 2 (f , f )+ z,⇢⇤ 2 (f , f ) = |HessG a,z f |2 +“R(rf , rf )”+R⇢⇤ (rf , rf ) | {z } new tensor . with new Gamma z 2 defined as: z,⇢⇤ 2 (f , f ) = z 2 (f , f ) + div ⇢⇤ z ⇣ 1,r(aaT ) (f , g) ⌘ div ⇢⇤ a ⇣ 1,r(zzT ) (f , g) ⌘ , div ⇢⇤ a (F) = 1 ⇢⇤ r · (⇢⇤aaT F), 1,r(aaT) (f , g) = hrf , r(aaT )rgi = (hrf , @ @xˆ k (aaT )rgi)n+m ˆ k=1 ,
41. Generalized Gamma z calculus Lemma (IBP for transition kernel) For the transition kernel ⇢(t, x, y) for the semigroup Pt, we have E[g 1(log g, z 1 (log g,log g)) g z 1 (log g, 1(log g,log g))] =R r·(⇢(s,x,y)zzT r(aaT) (log g(s,y),log g(s,y))) ⇢(s,x,y) g(s,y)⇢(s,x,y)dy R r·(⇢(s,x,y)aaT r(zzT) (log g(s,y),log g(s,y))) ⇢(s,x,y) g(s,y)⇢(s,x,y)dy.
42. Generalized Gamma z calculus Given the z-Bochner’s formula, we get the generalized Curvature dimension inequality Generalized CD inequality For any f 2 C1(Rn+m), if we have 2 (f , f ) + z,⇢⇤ 2 (f , f ) ⌫ ( 1 (f , f ) + z 1 (f , f )), Then the operator L satisfies generalized CD(, 1).
43. Generalized Gamma z calculus Given the z-Bochner’s formula, we get the generalized Curvature dimension inequality Generalized CD inequality For any f 2 C1(Rn+m), if we have 2 (f , f ) + z,⇢⇤ 2 (f , f ) ⌫ ( 1 (f , f ) + z 1 (f , f )), Then the operator L satisfies generalized CD(, 1). I Our generalized CD applies to non-totally geodesic foliation, and goes beyond step 3 condition. (SE(2), Engel group, etc..) I The new CD(, n) can be computed for L with drift, and general weighted volume. I Implies Poincar´ e, and LSI inequalities, and more other functional inequalities.
44. Wasserstein Geometry and Generalized Gamma z calculus Denote ⇢⇤ as the density function for the invariant measure dµ. We introduce the Kullback–Leibler divergence by DKL (⇢k⇢⇤) = Z Rn+m ⇢ log ⇢ ⇢⇤ dx, and the a, z–Fisher information functional I a,z (⇢k⇢⇤) = Z Rn+m ⇣ r log ⇢ ⇢⇤ , (aaT + zzT )r log ⇢ ⇢⇤ ⌘ ⇢dx. Proposition For any smooth density ⇢ and  > 0, d dt I a,z (⇢ t ) = 2 Z ⇣ 2 ( D, D) + z,⇢⇤ 2 ( D, D)) ⌘ ⇢ tdx, and CD(, 1) with  > 0 implies DKL (⇢k⇢⇤)  1 2 I a,z (⇢k⇢⇤). (zLSI)
45. LSI on Heisenberg group and Displacement group Consider a more general drift-di↵usion proess, dXt = a(Xt )a(Xt )T rV (Xt )dt + p 2a(Xt ) dBt , (3.4) The ⇢⇤ has the form (Vol is the canonical volum) ⇢⇤ = e V Vol R Rn+m e V Voldx .
46. LSI on Heisenberg group and Displacement group Consider a more general drift-di↵usion proess, dXt = a(Xt )a(Xt )T rV (Xt )dt + p 2a(Xt ) dBt , (3.4) The ⇢⇤ has the form (Vol is the canonical volum) ⇢⇤ = e V Vol R Rn+m e V Voldx . Applying our generalized Gamma z calculus and generalized CD condition, we find the analytical condition for matrix A  = min (A) > 0 ) zLSI. Where U=((aTr)1f ,(aTr)2f ,(zTr)1f )3⇥1 , and “R(rf ,rf )”=UT·A·U,
47. LSI on Heisenberg group Corollary (F. and Li, 20) The matrix A associated with Heisenberg group has the following form A11 = h @2V @x@x + y2 4 @2V @z@z y @2V @x@z i + ⇣ @2V @x@z y 2 @2V @z@z ⌘ 1; A22 = h @2V @y@y + x2 4 @2V @z@z +x @2V @y@z i 1; A33= 1 2 ; A12 = h @2V @x@y + x 2 @2V @x@z y 2 @2V @y@z xy 4 @2V @z@z i + 1 2 ⇣ @2V @y@z + x 2 @2V @z@z ⌘ ; A13 = A31= 1 2 (aTr)2V ; A23=A32= 1 2 (aTr)1V . See video.
48. LSI on Displacement group Corollary (F. and Li, 20) The matrix A associated with SE(2) has the following representation A11 = @2V @✓@✓ g @2V @✓@y 2(1+ 1 g2 ); A22 = h e2 ✓ @2V @x@x +2e ✓ @2V @x@y + @2V @y@y i 2(1+ 1 g2 ) (aTr)1V ; A33 = 2 2g2 1(log g,log g) 2 1(log ⇢⇤,log g) 1 g h @2g @✓@✓ +e2 ✓ @2g @x@x + @2g @y@y +2e ✓ @2g @x@y i ; A12 = A21= 1 2 ⇣ e ✓ @V @x +2(e ✓ @2V @✓@x + @2V @✓@y ) g(e ✓ @2V @x@y + @2V @y@y )+ (aTr)2V ⌘ ; A13 = A31= 1 2 ⇣ g (aTr)2V 1(log g,V ) ⌘ ; A23 = A32= 1 2 ⇣ g (aTr)1V + 2 g e ✓ ⌘ . Recall that: X= @ @✓ , Y =e ✓ @ @x + @ @y , R= @ @y . ( ) g(✓, x, y))
49. Application and future works The (overdamped, underdamped ) Langevin dynamic is covered by ( dxt = vtdt, dvt = (xt )vtdt rU(xt )dt + p 2 u(xt )dBt . (3.5) With matrices b = ✓ v (x)v rU(x) ◆ , a = ✓ 0 p 2 u(x) ◆
50. Application and future works The (overdamped, underdamped ) Langevin dynamic is covered by ( dxt = vtdt, dvt = (xt )vtdt rU(xt )dt + p 2 u(xt )dBt . (3.5) With matrices b = ✓ v (x)v rU(x) ◆ , a = ✓ 0 p 2 u(x) ◆ I Asymptotic analysis I Non-asymptotic analysis I Algorithm design (MCMC, Gibbs measure, etc.) I Gradient drift to non-gradient drift.
51. Summary I Generalized Gamma z calculus; (F. and Li, 19) I Generalized CD(, n); I z-Logrithmic-Sobolev-inequality; I We also derive matrix A for Martinet flat sub-Riemannian structure;(F. and Li, 20) I The results goes beyond step 2 condition, e.g. Engel group, etc. I Wasserstein geometry and entropy. I More is coming...
52. Reference Q. Feng and W. Li. Generalized gamma z calculus via sub-Riemannian density manifold. arXiv preprint arXiv:1910.07480, 2019. Q. Feng and W. Li. Sub-Riemannian Ricci curvature via Generalized gamma z calculus. arXiv preprint arXiv:2004.01863, 2020