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

Outline of the talk. SDEs and Bakry-´ Emery calculus Generalized sub-Riemannian Ricci curvature tensor Functional inequalities.

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 ).

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);

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 ;

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;

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).

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)

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 .

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)

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

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)

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)]

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

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 ))

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 ).

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 ).

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).

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).

“Analysis and Geometry of Markov Di↵usion Operators” —Bakry-Gentil-Ledoux

How about degenerate operators?

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

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

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.

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 )

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 )

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!!!

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.

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).

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 )).

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))

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 ))

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 ...

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.

How about 1 (f , z 1 (f , f )) 6= z 1 (f , 1 (f , f ))?

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.

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.

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

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 .

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 ,

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.

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).

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.

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)

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 .

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,

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.

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))

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) ◆

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.

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...

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