Heat Flow on Non-Riemannian Spaces Karl-Theodor Sturm Universit¨ at Bonn
L 2 -Wasserstein Space Let ( M , d ) complete separable metric space, define � � � d 2 ( x , x 0 ) µ ( dx ) < ∞ P 2 ( M ) = prob. meas. µ on M with M and �� � 1 / 2 d 2 ( x , y ) d q ( x , y ) W 2 ( µ 0 , µ 1 ) = inf . q M × M Then ( P 2 ( M ) , W 2 ) is a complete separable metric space. ( P 2 ( M ) , W 2 ) is a compact space or a length space or an Alexandrov space 1) with curvature ≥ 0 if and only if ( M , d ) is so. 1) Pythagorean inequality a 2 + b 2 ≥ c 2
L 2 -Wasserstein Space for Riemannian M Given compl. Riem. manifold M and µ 0 , µ 1 ∈ P 2 ( M ) with d µ 0 ≪ d vol . There exists a unique geodesic ( µ t ) 0 ≤ t ≤ 1 connecting µ 0 , µ 1 , given as µ t := ( F t ) ∗ µ 0 , where F t ( x ) = exp x ( t ∇ ϕ ( x )) with suitable d 2 / 2-convex ϕ : M → R . In the case M = R n this states that there exists a F t ( x ) ��� ��� F 1 ( x ) convex function ϕ 1 such that ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� x ��� ��� ��� ��� F t ( x ) = (1 − t ) x + t ∇ ϕ 1 ( x ) . In particular, F 1 ( x ) = ∇ ϕ 1 ( x ) . µ 0 µ t µ 1 The ϕ from above is ϕ ( x ) = ϕ 1 ( x ) − | x | 2 / 2.
Riemannian Structure of P 2 ( M ) Tangent space: � M |∇ ϕ | 2 d µ 0 < ∞} T µ 0 P 2 = closure of { Φ = ∇ ϕ : M → TM , Riemannian tensor: � �∇ ϕ, ∇ ψ � T µ 0 P 2 = M �∇ ϕ, ∇ ψ � T x d µ 0 ( x ) Exponential map: exp µ 0 ( t ∇ ϕ ) = [exp( t ∇ ϕ )] ∗ µ 0 F t ( x ) F 1 ( x ) ��� ��� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� x ��� ��� ��� ��� µ 0 µ t µ 1
Gradient Flows on P 2 ( M ) The gradient ∇ S ( ν ) ∈ T ν P 2 ( M ) of the relative entropy � � ρ log ρ dm , if d ν = ρ dm S ( ν ) = + ∞ , if d ν �≪ dm as a function on P 2 ( M ) is given by ∇ S ( ν ) = ∇ log ρ. The gradient flow ∂ν ∂ t = −∇ S ( ν ) on P 2 ( M ) for the relative entropy S is given by ν t ( dx ) = ρ t ( x ) m ( dx ) where ρ solves the heat equation ∂ ∂ t ρ = △ ρ on M . R n : Otto ’01, Riemann ( M , g ) : Ohta ’09, Savare ’09, Villani ’09, Erbar ’09, Alexandrov space: Gigli/Kuwada/Ohta ’10, Finsler ( M , F , m ) : Ohta/Sturm ’09, Wiener space: Fang/Shao/Sturm ’09 Heisenberg group: Juillet ’09,
Gradient Flows on P 2 ( M ) M = C ( R + , R d ), m = Wiener measure, d = Cameron-Martin distance �� ∞ � 1 / 2 y ( t ) | 2 dt d ( x , y ) = | ˙ x ( t ) − ˙ 0 Transport cost / concentration inequalities Talagrand, Ledoux, Wang, Fang, Shao, . . . (1996, . . . ) Existence & uniqueness of optimal transport map between m and ρ m Feyel/Ustunel (2004) Gradient flow for the relative entropy Ent( . | m ) on P 2 ( M , d ) = Ornstein-Uhlenbeck semigroup on M . Fang/Shao/St.: PTRF (2009)
Gradient Flows on P 2 ( M ) Consider � � � � 1 ρ s dx + S ( ν ) = Vd ν + Wd ν d ν s − 1 for d ν = ρ dx + d ν sing . V : R n → R some external potential and Here s > 0 real, W : R n × R n → R some interaction potential. Theorem. (Jordan/Kinderlehrer/Otto ’98, Otto ’01, Villani ’03, Ambrosio/Gigli/Savare ’05, . . . ) The gradient flow ∂ν ∂ t = −∇ S ( ν ) on P 2 ( R n ) is given by ν t ( dx ) = ρ t ( x ) dx where ρ solves the nonlinear PDE � ∂ ∂ t ρ = △ ( ρ s ) + ∇ ( ρ · ∇ V ) − ∇ ( ρ · ( ∇ W ρ )) This includes porous medium equation, fast diffusion, Fokker-Planck, McKean-Vlasov. Other examples: quantum-drift diffusion (Fisher information), Ginzburg-Landau dynamics (squared H − 1 -norm), p -Laplacian.
Optimal Transport, Heat Flow and Ricci Curvature M complete Riemannian manifold, m Riemannian volume measure, � S ( ρ dm ) = ρ log ρ dm . ∂ Recall that the gradient flow of S satisfies ∂ t ρ = △ ρ . Theorem. (Otto ’01, Otto/Villani ’00, Cordero/McCann/Schmuckenschl¨ ager ’01, vRenesse/Sturm ’05) Ric M ≥ K ⇔ Hess S ≥ K W 2 ( p t µ, p t ν ) ≤ e − Kt W 2 ( µ, ν ) ⇔ � Ricci bounds for Markov chains (Ollivier ’08) �� �� �� �� � � � � � � � � � � � � � � � � � � � � �� �� � � � � �� �� � � � � � � � � � � � � � � � � �� �� �� �� � � � � � � �� �� � � �� �� � � � � � � � � � � � � �� �� � � �� �� � � �� �� � � � � � � �� �� �� �� � � �� �� �� �� � � � � � � � � � � � � � � � � � � � � �� �� � � � � �� �� � � � � �� �� � � � � �� �� � � � � �� �� � � � � � � � � � � � � � � �� �� �� �� � � �� �� � � � � � � � � �� �� �� �� � � � � � � �� �� � � �� �� � � � � � � � � � � � � �� �� � � �� �� �� �� � � � � �� �� � � � � � � �� �� � � �� �� � � � � �� �� � � �� �� � � � � � � � � �� �� �� �� � � ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ����� ����� ����� ����� ����� ����� ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ����� ����� ����� ����� ����� ����� ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ Theorem. (Bakry/´ Emery ’84, Kendall, Cranston, Wang ’97) ∇| p t u | 2 ( x ) ≤ e − Kt · p t � |∇ u | 2 � Ric M ≥ K ⇔ ( x ) d ( X t , Y t ) ≤ e − Kt d ( x , y ) ⇔ ∀ x , y : ∃ BMs X t , Y t s.t.
Optimal Transport, Heat Flow and Ricci Curvature Let ( M , g ( t )) evolve under backward Ricci flow ∂ ∂ t g ( t ) = 2 Ric g ( t ) . Theorem (McCann/Topping ’08). W ( t ) 2 ( p t 0 , t µ, p t 0 , t ν ) ≤ W ( t 0 ) ( µ, ν ) 2 with W ( t ) = Wasserstein distance for d g ( t ) and p t 0 , t µ = solution to forward 2 ∂ heat flow ∂ t η = △ g ( t ) η with η ( t 0 ) = µ . Extension to L -distance: Monotonicity formula for Perelman’s L -functional (Topping ’09, Lott ’09) Probabilistic/robust def. of Ricci flow
Optimal Transport, Heat Flow and Ricci Curvature ∂ Let ( M , g ( t )) evolve under backward Ricci flow ∂ t g ( t ) = 2 Ric g ( t ) . Detailed, pathwise version: Theorem (Arnaudon/Coulibaly/Thalmaier ’09). For each pair x , y ∈ M : ∃ coupling of Brownian motions ( X t ) , ( Y t ) with X t 0 = x , Y t 0 = y s.t. d ( t ) ( X t , Y t ) ≤ d ( t 0 ) ( x , y ) P -a.s. for all t ≥ t 0 . Extension: Coupling of BMs s.t. L -distance becomes supermartingale (Kuwada/Philipowski ’10)
Ricci Bounds for Metric Measure Spaces M complete Riemannian manifold, m Riem. volume measure, dim M = n � Let S ( ρ dm ) = ρ log ρ dm . Then Hess S ≥ K ⇔ Ric M ≥ K � Ricci bound for metric measure spaces � logarithmic Sobolev inequality, concentration of measures
Ricci Bounds for Metric Measure Spaces M complete Riemannian manifold, m Riem. volume measure, dim M = n � M ρ s dm . Then 1 Let S ( ρ dm ) = s − 1 � 1 − 1 ≥ s Hess S ≥ 0 ⇔ n Ric M ≥ 0 � Curvature-Dimension condition CD(K,N) for mms � Sobolev inequality, Bishop-Gromov volume growth estimate ⇒ vol 1 / n concave ric ≥ 0 ⇐ sec ≥ 0 ⇒ dist concave ⇐
Ricci Bounds for Metric Measure Spaces ( M , d ) complete separable metric space, m locally finite measure on M Definition. Ric ( M , d , m ) ≥ K or ” CD ( K , ∞ )” ⇐ ⇒ ∀ µ 0 , µ 1 ∈ P 2 ( M ) : ∃ geodesic µ t s.t. ∀ t ∈ [0 , 1]: Ent ( µ t | m ) ≤ (1 − t ) Ent ( µ 0 | m ) + t Ent ( µ 1 | m ) − K 2 t (1 − t ) W 2 2 ( µ 0 , µ 1 ) Recall relative entropy � � M ρ log ρ dm , if ν = ρ · m Ent ( ν | m ) = + ∞ , if ν �≪ m
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