Heat Flows, Geometric and Functional Inequalities

Heat Flows, Geometric and Functional Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France

heat flow and semigroup interpolations Duhamel formula (19th century) pde, probability, dynamics major tool in geometric and functional analysis

last decades developments dynamical proofs of Euclidean and Riemannian geometric, functional and transport inequalities heat kernel bounds, spectral analysis, integral inequalities, measure concentration, statistical mechanics, optimal transport

geometric aspects gradient bounds convexity Ricci curvature lower bounds D. Bakry and M. Émery (1985) hypercontractive diffusions

selected topics logarithmic Sobolev form of the Li-Yau parabolic inequality (Gaussian) isoperimetric-type inequalities transport and Harnack inequalities Brascamp-Lieb inequalities and noise stability joint with D. Bakry Analysis and geometry of Markov diffusion operators D. Bakry, I. Gentil, M. L. (2014)

selected topics logarithmic Sobolev form of the Li-Yau parabolic inequality (Gaussian) isoperimetric-type inequalities transport and Harnack inequalities Brascamp-Lieb inequalities and noise stability to start with: a basic model example the logarithmic Sobolev inequality

classical logarithmic Sobolev inequality L. Gross (1975) γ standard Gaussian (probability) measure on R n dγ(x) = e x 2 /2 dx (2π) n/2 f > 0 smooth, R f dγ = 1 n f log f dγ 1 Rn f 2 dγ R n 2 f

classical logarithmic Sobolev inequality L. Gross (1975) γ standard Gaussian (probability) measure on R n dγ(x) = e x 2 /2 dx (2π) n/2 f > 0 smooth, R f dγ = 1 n (relative) entropy f log f dγ 1 Rn f 2 dγ R n 2 f

classical logarithmic Sobolev inequality f log f dγ 1 Rn f 2 dγ R n 2 f f f R 2 f 2 log f 2 dγ 2 f 2 dγ n R n Wiener analysis, hypercontractivity, convergence to equilibrium of kinetic equations and Markov chains, measure concentration, information theory, optimal transport (C. Villani) Perelman s Ricci flow

heat flow interpolation D. Bakry, M. Émery (1985) simplest proof? (P t ) t 0 heat semigroup on R n P t f (x) = f (y) e x y 2 /4t dy R n (4πt) n/2 = R n f ( x + 2t y ) dγ(y) u(x, t) = P t f (x), t 0, x R n solves the heat equation t u = u, u(x, 0) = f (x)

heat kernel inequality f > 0, t > 0, at any point: P t (f log f ) P t f log P t f P t f (x) = t = 1 2 R n f ( x + 2t y ) dγ(y) (x = 0) : P t γ P t (f log f ) P t f log P t f = R n f log f dγ ( R n f dγ = 1 )

heat flow interpolation P t (f log f ) P t f log P t f = t 0 d ds P ( s Pt s f log P t s f ) ds heat equation t P t = P t chain rule (both in time and space) d ds P ( s Pt s f log P t s f ) ( Pt s f 2 ) = P s P t s f

P t (f log f ) P t f log P t f = t 0 ( Pt s f 2 ) P s ds P t s f gradient commutation P u f = P u ( f ) (curvature) P u f 2 [P u ( f ) ] 2 Pu ( f 2 f ) P u f u = t s P t (f log f ) P t f log P t f t 0 ( ( f 2 ( f 2 ) P s P t s ))ds = t P t f f

classical logarithmic Sobolev inequality L. Gross (1975) γ standard Gaussian (probability) measure on R n dγ(x) = e x 2 /2 dx (2π) n/2 f > 0 smooth, R f dγ = 1 n f log f dγ 1 Rn f 2 dγ R n 2 f

P t (f log f ) P t f log P t f = heat flow interpolation t φ(s) = P s ( Pt s f 2 P t s f 0 ) φ(s) ds gradient commutation P u f = P u ( f ) ( f 2 ) φ increasing: s t, φ(s) φ(t) = P t f differentiate φ, φ 0?

differentiate φ, φ 0? ( Pt s f 2 ) φ(s) = P s P t s f = P s ( Pt s f log P t s f 2) φ (s) = 2 P s ( Pt s f Γ 2 (log P t s f ) )

Γ 2 Bakry-Émery operator Γ 2 (h) = 1 2 ( h 2) h ( h) = Hess(h) 2 + Ric g ( h, h) on R n or (M, g) n-dimensional Riemannian manifold Bochner s formula (M, g) non-negative Ricci curvature Γ 2 (h) 0, φ 0 P t (f log f ) P t f log P t f = t 0 ( f 2 ) φ(s) ds t P t (P t ) t 0 heat semigroup on (M, g) f

heat flow interpolation under Γ 2 0 heat kernel inequalities, functional inequalities (Sobolev, logarithmic Sobolev, Poincaré, Nash...) under geometric (Ricci) curvature conditions more generally under Ric g K, K R equivalent gradient bounds (commutation) P t f e Kt P t ( f )

reverse logarithmic Sobolev inequality heat kernel inequality ( f 2 ) P t (f log f ) P t f log P t f t P t f reverse form t P tf 2 P t f P t (f log f ) P t f log P t f gradient bounds (both equivalent to Γ 2 0 as t 0)

P t (f log f ) P t f log P t f = dimensional improvement t 0 φ(s) ds φ (s) = 2 P s ( Pt s f Γ 2 (log P t s f ) ) actually under Ric g 0 Γ 2 (h) = Hess(h) 2 + Ric g ( h, h) Hess(h) 2 1 n ( h)2 φ (s) 2 n P ( s Pt s f [ log P t s f ] 2)

dimensional logarithmic Sobolev inequality P t (f log f ) P t f log P t f t P t f + n ( 2 P tf log 1 2t n (log(1 + x) x or n ) ) P t (f log f ) P t f reverse form P t (f log f ) P t f log P t f t P t f n ( 2 P tf log 1 + 2t ) n (log P tf )

reverse form P t (f log f ) P t f log P t f t P t f n ( 2 P tf log 1 + 2t ) n (log P tf ) in particular 1 + 2t n (log P tf ) > 0 f : M R, f > 0, Ric g 0, t > 0 P t f 2 (P t f ) 2 P tf P t f n 2t Li-Yau (1986) parabolic inequality maximum principle D. Bakry, M. L. (2006)

Li-Yau (1986) parabolic inequality P t f 2 (P t f ) 2 P tf P t f (Ric g 0) n 2t ( t + s P t f (x) P t+s f (y) t Harnack inequalities ) n/2e d(x,y) 2 /4s d(x, y) Riemannian distance heat kernel bounds

weighted Riemannian manifold (M, g) weighted Riemannian manifold dµ = e V dx, V smooth potential on M L = V, (P t ) t 0 semigroup non-negative curvature-dimension CD(0, N) Γ 2 = Ric g + Hess(V ) 1 N (Lh)2 curvature-dimension CD(K, N), K R, N 1 Γ 2 (h) K h 2 + 1 N (Lh)2

Markov Triple Markov Triple (E, µ, Γ) semigroup (P t ) t 0, generator L, invariant measure µ carré du champ operator Γ(f, g) = 1 L(f g) f Lg g Lf, 2 Γ (f, g) A A Bakry-Émery Γ 2 operator Γ 2 (f, g) = 1 LΓ(f, g) Γ(f, Lg) Γ(g, Lf ) 2 curvature-dimension CD(K, N) Γ 2 (h, h) K Γ(h, h) + 1 N (Lh)2, h A

logarithmic Sobolev form of the Li-Yau parabolic inequality (Gaussian) isoperimetric-type inequalities transport and Harnack inequalities Brascamp-Lieb inequalities and noise stability

Gaussian isoperimetric inequality isoperimetric inequality for the standard Gaussian measure dγ(x) = e x 2 /2 dx (2π) n/2 on R n C. Borell (1975), V. Sudakov, B. Tsirelson (1974) half-spaces extremal sets for the isoperimetric problem H = { x R n ; x, u a }, u = 1, a R if A R n, γ(a) = γ(h) then γ + (A) γ + (H) γ + (A) = lim inf ε 0 1 [ γ(aε ) γ(a) ] ε

Gaussian isoperimetric profile I(v) = inf { γ + (A) ; γ(a) = v }, A R n, v (0, 1) half-spaces: one-dimensional I = ϕ Φ 1 Φ(s) = s e x2 /2 dx, ϕ = Φ 2π isoperimetric inequality A R n, I ( γ(a) ) γ + (A)

heat flow and isoperimetric-type inequalities Γ 2 0 (CD(0, ) non-negative curvature) M (weighted) Riemannian manifold f : M [0, 1] smooth I (P t f ) P t ( I 2 (f ) + 2t f 2 ) (stronger than logarithmic Sobolev inequality, εf, ε 0) D. Bakry, M. L. (1996)

heat flow and isoperimetric-type inequalities f : M [0, 1] smooth I (P t f ) P t ( I 2 (f ) + 2t f 2 ) M = R n t = 1 2 : P t γ Bobkov s inequality ( ) I f dγ R n R n I 2 (f ) + f 2 dγ f = 1 A (I(0) = I(1) = 0) I ( γ(a) ) γ + (A)

(spherical) isoperimetric-type inequalities similar under CD(K, ) on (E, µ, Γ) Markov triple comparison to the Gaussian isoperimetric profile open problem: heat flow proof of isoperimetry on the sphere on S n R n+1 spherical caps are the isoperimetric extremizers Lévy-Gromov (1980) theorem (M, g), µ normalized volume element, CD(n 1, n) isoperimetric profile I µ I S n

reverse isoperimetric-type inequalities f : M [0, 1] smooth, Γ 2 0 [ I (Pt f ) ] 2 [ Pt ( I (f ) )] 2 2t Pt f 2 Φ 1 P t f 1 2t - Lipschitz Φ 1 P t f (x) Φ 1 P t f (y) + d(x, y) 2t sharp gradient bounds, Harnack-type inequalities

new isoperimetric Harnack inequality Γ 2 0 (CD(0, ) non-negative curvature) P t (1 A )(x) P t (1 Ad(x,y) )(y) d(x, y) distance from x to y A ε = { z M ; d(z, A) < ε } similar under CD(K, ), K R on (E, µ, Γ) Markov triple D. Bakry, I. Gentil, M. L. (2013)

Wang s Harnack inequality (1997) Γ 2 0 (CD(0, ) non-negative curvature) ( Pt f ) 2 (x) Pt ( f 2 ) (y) e d(x,y)2 /2t (infinite dimensional version of the Li-Yau Harnack inequality) variant: log-harnack inequality P t (log f )(x) log P t f (y) + d(x, y)2 4t links to optimal transport inequalities

HWI inequality F. Otto, C. Villani (2000) M (weighted) Riemannian manifold CD(0, ) dν = f dµ, f > 0, M f dµ = 1 ( f 2 ) 1/2 f log f dµ W 2 (ν, µ) dµ f M M Kantorovich-Wasserstein distance W 2 (ν, µ) 2 = inf d(x, y) 2 dπ(x, y) π µ ν M M

Kantorovich-Rubinstein duality ( ) 1 2 W 2(ν, µ) 2 = sup Q 1 ϕ dµ ϕ dν ϕ M M Hofp-Lax infimum convolution semigroup Q s f (x) = inf [f (y) + y M ] d(x, y)2, x M, s > 0 2s log-harnack inequality P t (log f ) Q 2t (log P t f ) P t f log P t f dµ 1 4t W 2 2 (ν, µ), dν = f dµ M S. Bobkov, I. Gentil, M. L. (2001)

commutation property (P t ) t 0 heat semigroup (Q s ) s 0 Hopf-Lax semigroup P t (Q s f ) Q s (P t f ) heat flow contraction in Wasserstein distance W 2 (µ t, ν t ) W 2 (µ 0, ν 0 ) dµ t = P t f dµ dν t = P t g dµ M. K. von Renesse, K. T. Sturm (2005), F. Otto, M. Westdickenberg (2006), K. Kuwada (2010)

logarithmic Sobolev form of the Li-Yau parabolic inequality (Gaussian) isoperimetric-type inequalities transport and Harnack inequalities Brascamp-Lieb inequalities and noise stability

γ standard Gaussian (probability) measure on R n T ρ f (x) = dγ(x) = e x 2 /2 dx (2π) n/2 R n f ( ρ x + 1 ρ 2 y ) dγ(y), ρ (0, 1) T ρ f = P t f, ρ = e t (P t ) t 0 Ornstein-Uhlenbeck semigroup generator L = x invariant measure γ

R n heat flow and noise stability E. Mossel, J. Neeman (2012) ρ (0, 1), J : R 2 R smooth, f, g : R n R R n J ( f (x), g ( ρ x + 1 ρ 2 y )) dγ(x)dγ(y) ( ) J f dγ, R n g dγ R n if and only if ( ) 11 J ρ 12 J 0 ρ 12 J 22 J (J ρ-concave)

(P t ) t 0 Ornstein-Uhlenbeck semigroup ψ(t) = R n R n J ( P t f (x), P t g ( ρx + 1 ρ 2 y )) dγ(x)dγ(y), t 0 ψ (t) = R n R n [ ( 11 J) P t f 2 + ( 22 J) P t g 2 2 ρ 12 J P t f P t g ] dγ dγ J ρ-concave = ψ (t) 0 ψ(0) ψ( )

first example of J- function J(u, v) = Jρ H (u, v) = u α v β, u, v > 0, α, β [0, 1] ρ-concave if ρ 2 αβ (α 1)(β 1) f α (x)g β( ρ x + 1 ρ 2 y ) dγ(x)dγ(y) R n R n ( ) α ( ) β f dγ g dγ R n R n duality T ρ g q g p ρ 2 q 1 p 1 (1 < p < q < ) Nelson s hypercontractivity (1966-73)

multidimensional extensions f k : R p R, k = 1,..., m A k p n matrix, A k t A k = Id R p Γ kl = A l t A k (p p matrix), k, l = 1,..., m smooth J : R m R J(f 1 A 1,..., f m A m ) dγ J R n ( f 1 A 1 dγ,..., R n f m A m dγ R n ) m if and only if kl J Γ kl v k, v l 0 for all v k R p k,l=1

geometric Brascamp-Lieb inequalities K. Ball (1989) decomposition of the identity m c k A k A k = Id R n, c k 0, A k unit vectors in R n k=1 f k : R R +, k = 1,..., m m R n k=1 J(u 1,..., u m ) = u c 1 1 ucm m f c ( k k Ak, x ) dx m ( f k dx R k=1 ) ck

multilinear inequalities Brascamp-Lieb inequalities H. Brascamp, E. Lieb (1976) maximize m R n k=1 f c ( k k Ak, x ) dx maximizers: Gaussian kernels heat flow methodology E. Carlen, E. Lieb, M. Loss (2004) J. Bennett, A. Carbery, M. Christ, T. Tao (2008)

second example of J-function E. Mossel, J. Neeman (2012) J(u, v) = J B ρ (u, v) ( Jρ B (u, v) = P X 1 Φ 1 (u), ρ X 1 + ) 1 ρ 2 Y 1 Φ 1 (v), u, v [0, 1] X 1, Y 1 N (0, 1) independent, distribution Φ Φ(s) = J B ρ s e x2 /2 ρ-concave dx 2π

Jρ B R n R n ( f (x), g ( ρ x + 1 ρ 2 y )) dγ(x)dγ(y) ( ) Jρ B f dγ, R n g dγ R n f = 1 A g = 1 B, A, B R n P ( X A, ρ X + 1 ρ 2 Y B ) P ( X H, ρ X + 1 ρ 2 Y K ) X, Y independent with distribution γ in R n H, K parallel half-spaces γ(h) = γ(a), γ(k) = γ(b) C. Borell (1985) Gaussian rearrangements

Borell s noise stability theorem noise stability of A R n S ρ (A) = P ( X A, ρ X + 1 ρ 2 Y A ) S ρ (A) S ρ (H), γ(a) = γ(h) half-spaces are the most noise stable Max-Cut and Majority is Stablest E. Mossel, R. O Donnell, K. Oleszkiewicz (2010)

from noise stability to isoperimetry S ρ (A) = P ( X A, ρ X + 1 ρ 2 Y A ) Borell s theorem S ρ (A) S ρ (H) (γ(a) = γ(h) half-space) heat flow description of the perimeter γ + (A) lim sup ρ 1 π log 1 ρ [ γ(a) Sρ (A) ] equality when A = H half-space Gaussian isoperimetry γ + (A) γ + (H)

extensions log-concave mesures dµ = e V dx, Hess(V ) K > 0 (E, µ, Γ) Markov Triple, CD(K, ), K > 0 deficit (Gaussian noise stability and isoperimetry) R. Eldan (2013) (stochastic calculus) developments in Boolean analysis discrete cube and structures (Majority is Stablest) A. De, E. Mossel, J. Neeman (2013)

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