Brownian motion, evolving geometries and entropy formulas Talk 4 - PowerPoint PPT Presentation

Brownian motion, evolving geometries and entropy formulas Talk 4 Anton Thalmaier Universit´ e du Luxembourg School on Algebraic, Geometric and Probabilistic Aspects of Dynamical Systems and Control Theory ICTP, Trieste July 4 to 15, 2016 Anton Thalmaier Brownian motion, evolving geometries and entropy

Outline 1 Stochastic Calculus on manifolds (stochastic flows) 2 Analysis of evolving manifolds 3 Heat equations under Ricci flow and functional inequalities 4 Geometric flows and entropy formulas Anton Thalmaier Brownian motion, evolving geometries and entropy

I. Entropy under Ricci flow Consider positive solutions to ∂  ∂ t u − ∆ g ( t ) u = 0   ∂  ∂ t g t = − 2 Ric g ( t )  It is convenient to let time run backwards in both equation. Then: Backward heat equation under backward Ricci flow Thus ∂  ∂ t u + ∆ u = 0   ∂  ∂ t g = 2 Ric  Let ( X t ( x ) , t ) the space-time Brownian motion starting at ( x , 0). Then X t ( x ) is a g ( t )-Brownian motion on M . For simplicity always start at time s = 0. Anton Thalmaier Brownian motion, evolving geometries and entropy

Let X t ( x ) be a g ( t )-Brownian motion on M . Consider the heat kernel measure m t ( dy ) := P { X t ( x ) ∈ dy } . We are interested in the entropy of µ t := u ( · , t ) dm t ≡ u ( X t ( x ) , t ) d P The quantity � u ( y , t ) m t ( dy ) = E [ u ( X t ( x ) , t )] M stays constant along the flow, since u ( X t ( x ) , t ) is a martingale. Anton Thalmaier Brownian motion, evolving geometries and entropy

Theorem Denote by E ( t ) = E [( u log u )( X t ( x ) , t )] � = ( u log u )( y , t ) m t ( dy ) M the entropy of µ t = u ( · , t ) dm t ≡ u ( X t ( x ) , t ) d P . The first two derivatives of E ( t ) are given by � |∇ u | 2 � E ′ ( t ) = E ( X t ( x ) , t ) u �� u | Hess log u | 2 � � E ′′ ( t ) = 2 E ( X t ( x ) , t ) . Anton Thalmaier Brownian motion, evolving geometries and entropy

Applications to the classification of ancient solutions to the heat equation. (Hongxin Guo, Robert Philipowski, A.Th. 2015) With the substitution τ := − t , solutions to the backward equation above defined for all t ≥ 0 correspond to ancient solutions of the (forward) heat equation, τ ≤ 0, under forward Ricci flow. Let t →∞ E ′ ( t ) ∈ [0 , + ∞ ] . θ := lim Anton Thalmaier Brownian motion, evolving geometries and entropy

Example Consider u ( t , y ) = e y − t on R with the standard metric. Then E ( t ) = t and θ = 1 . Proposition Assume that ∂ g ∂ t = 2 Ric (or ∂ g ∂ t ≤ 2 Ric ) and let u be a positive solution of the backward heat equation. Then u is constant if and only if θ = 0. If the entropy E ( t ) grows sublinearly, i.e. t →∞ E ( t ) / t = 0 , lim then θ = 0 and u is constant. Anton Thalmaier Brownian motion, evolving geometries and entropy

II. Ricci flow under conjugate backward heat equation Consider ∂  ∂ t g = − 2 Ric   ∂  ∂ t u + ∆ u = Ru .  Now � t � � � � exp − R ( X s ( x ) , s ) ds u ( X t ( x ) , t ) = u ( x , 0) indep. of t . E 0 Take � t � � P x , t := exp − R ( X s ( x ) , s ) ds d P 0 as reference measure. Anton Thalmaier Brownian motion, evolving geometries and entropy

Consider the entropy of the measure µ x , t := u ( X t ( x ) , t ) d P x , t defined as � � E ( t ) = E x , t ( u log u )( X t ( x ) , t ) where E x , t denotes expectation w/r to P x , t . The derivative of E ( t ) is given by �� � 2 ) u � � E ′ ( t ) = E x , t � � ( R + � ∇ log u ( X t ( x ) , t ) . Anton Thalmaier Brownian motion, evolving geometries and entropy

Theorem Consider the following entropy functional � � Ent ( g , u , t ) := E x , t ( u log u )( X t ( x ) , t ) � t � � − 2 E x , s ∆ u ( X s ( x ) , s ) ds . 0 Then � 2 ��� � � � � ∇ u d dt Ent ( g , u , t ) = E x , t − 2∆ u + Ru ( X t ( x ) , t ) , u d 2 � 2 u ��� � � � dt 2 Ent ( g , u , t ) = 2 E x , t � Ric − Hess log u ( X t ( x ) , t ) . Anton Thalmaier Brownian motion, evolving geometries and entropy

We observe that F ( g , u , t ) := d dt Ent ( g , u , t ) is non-decreasing in time and monotonicity is strict unless Ric + Hess f = 0 (steady Ricci soliton) where f = log u . Anton Thalmaier Brownian motion, evolving geometries and entropy

III. Ricci solitons A complete Riemannian manifold ( M , g ) is said to be a gradient Ricci soliton if there exists f ∈ C ∞ ( M ; R ) such that Ric + Hess ( f ) = ρ g for some ρ ∈ R . The function f is called a potential function of the Ricci soliton. ρ = 0: steady soliton; ρ > 0: shrinking soliton; ρ < 0: expanding soliton. Note that if f = const, then ( M , g ) is Einstein. Anton Thalmaier Brownian motion, evolving geometries and entropy

Ricci solitons are special solutions to the Ricci flow If ( M , g ) is Einstein with Ric = ρ g , then g ( t ) := (1 − 2 ρ t ) g solves the Ricci flow equation. Likewise, if ( M , g , f ) is a gradient Ricci soliton with Ric + Hess ( f ) = ρ g , then g ( t ) := (1 − 2 ρ t ) ϕ ∗ t g solves the Ricci flow equation. Here ϕ t is the 1-parameter family of diffeomorphisms generated by ∇ f / (1 − 2 ρ t ). Anton Thalmaier Brownian motion, evolving geometries and entropy

IV. Perelman’s W -entropy Let M again be a compact manifold. To study shrinking solitons, Perelman introduced the so-called W -functional. Instead of the F -functional one considers W : M × C ∞ ( M ) × R ∗ + → R , e − f � � � τ ( R + |∇ f | 2 ) + f − n W ( g , f , τ ) : = (4 πτ ) n / 2 d vol g M One studies the gradient flow of W ( g , f , τ ). This leads to evolutions g ( t ), f ( t ) and τ ( t ) where τ is then a strictly positive smooth function τ ( t ). Anton Thalmaier Brownian motion, evolving geometries and entropy

Theorem (Perelman 2002) Let g ( t ) , f ( t ) and τ ( t ) develop according to ∂  ∂ t g = − 2 Ric      ∂ t f = − ∆ f − R + |∇ f | 2 + n ∂  2 τ   ∂   ∂ t τ = − 1 .   Then e − f d � � Ric + Hess f − g 2 � � dt W ( g , f , τ ) = 2 τ (4 πτ ) n / 2 d vol g . � � 2 τ � M In particular, W ( g , f , τ ) is non-decreasing in time and monotonicity is strict unless ( M , g ) satisfies Ric + Hess f = g (shrinking Ricci soliton). 2 τ Anton Thalmaier Brownian motion, evolving geometries and entropy

Let e − f log u + n � � u := or f = − 2 log(4 πτ ) . (4 πτ ) n / 2 Then g ( t ), u ( t ) and τ ( t ) evolve according to ∂  ∂ t g = − 2 Ric ,      ∂  ∂ t u + ∆ u = Ru ,   ∂    ∂ t τ = − 1 .  Let � τ ( R + |∇ log u | 2 ) − log u − n � � W ( g , u , τ ) = 2 log(4 πτ ) − n u d vol g . M Anton Thalmaier Brownian motion, evolving geometries and entropy

Then d � � Ric − Hess log u − g 2 � � dt W ( g , u , τ ) = 2 τ u d vol g . � � 2 τ � M In particular, W ( g , u , τ ) is non-decreasing in time and monotonicity is strict unless Ric − Hess log u = g 2 τ . Anton Thalmaier Brownian motion, evolving geometries and entropy

Entropy of the Gaussian measure on R n Let d µ t ( y ) = (4 π t ) − n / 2 e −| y | 2 / 4 t dy =: γ t ( y ) dy be the standard Gaussian measure on R n . The Boltzmann-Shannon entropy of µ τ is given as � R n ( γ τ log γ τ )( y ) dy = − n � � E 0 ( t ) := 1 + log(4 πτ ) . 2 Anton Thalmaier Brownian motion, evolving geometries and entropy

Relative entropy Let g ( t ), u ( t ) and τ ( t ) evolve according to ∂  ∂ t g = − 2 Ric ,      ∂  ∂ t u + ∆ u = Ru ,   ∂   ∂ t τ = − 1 .   We normalize u such that � u ( t ) d vol g ( t ) ≡ 1 . M Anton Thalmaier Brownian motion, evolving geometries and entropy

Theorem (Relative entropy) Let H ( g , u , t ) : = E ( t ) − E 0 ( t ) � − n � �� � ≡ u log u d vol g − 1 + log(4 πτ ) . 2 M Then d � R + |∇ log u | 2 − n � � dt H ( g , u , t ) = u d vol g 2 τ M d and dt τ H ( g , u , t ) = W ( g , u , τ ) . Anton Thalmaier Brownian motion, evolving geometries and entropy

Excursion Lei Ni’s entropy formula for positive solutions of the heat equation on a static Riemannian manifold. Lei Ni (2004) Let u > 0 be a positive solution of the heat equation � ∂ � ∂ t − ∆ u = 0 on a compact static Riemannian manifold ( M , g ). Let � − n � �� � H ( u , t ) := u log u d vol − 1 + log(4 π t ) 2 M be the difference between the Boltzmann entropy of the measure u ( x ) vol ( dx ) on M (normalized to be a probability measure) and the Boltzmann entropy of the standard Gaussian measure µ ( dy ) on R n . Anton Thalmaier Brownian motion, evolving geometries and entropy

Then � d ∆ log u + n � � dt H ( u , t ) = u d vol . 2 t M Observation Suppose that Ric ≥ 0 . Then, by the differential Harnack inequality, |∇ log u | 2 − ∆ u ≤ n 2 t , u equivalently ∆ log u + n 2 t ≥ 0 . In this case H ( u , t ) non-decreasing as function of t . Anton Thalmaier Brownian motion, evolving geometries and entropy