On spectral bounds for symmetric Markov chains with coarse Ricci - PowerPoint PPT Presentation

On spectral bounds for symmetric Markov chains with coarse Ricci curvatures Kazuhiro Kuwae (Kumamoto University) Stochastic Analysis and Applications German–Japanese bilateral research project Okayama University 27 September

Aim 1 Under the coarse Ricci curvature lower bound, (1) Upper estimate of (non-linear) spec- tral radius (2) Lower estimate of (non-linear) spec- tral gap (3) Strong L p -Liouville property for P - harmonic maps

Plan of talk 2 (1) Wasserstein distance (Historical Re- mark) (2) Coarse Ricci curvature (3) CAT(0)-space, 2-uniformly convex space (4) Main Theorems

Wasserstein space 3 Def 3.1 (Wasserstein distance) ( E, d ): Polish space, p ∈ [1 , ∞ [. P p ( E ):= { µ ∈ P ( E ) | E d p ( · , ∃ / ∀ ∫ x 0 ) dµ < ∞} , For µ, ν ∈ P p ( E ), 1 /p (∫ ) d p ( x, y ) π ( dxdy ) d W p ( µ, ν ) := inf π ∈ Π( µ,ν ) E × E : p -Wasserstein distance.

Rem 3.1 (1) d W 1 is nothing but the Kantorovich-Rubinstein distance. d W p was (re)discovered by var- ious authors independently: Gini (’14): d W 1 on discrete prob. on R . Kantorovich (’42): d W 1 on prob. on cpt sp Salvemini (’43): For discrete µ, ν ∈ P ( E ), Dall’Aglio (’56): For general µ, ν ∈ P p ( E ), ∫ 1 d W p ( µ, ν ) p = 0 | F − 1 µ ( t ) − F − 1 ν ( t ) | p dt .

Fr´ echet (’57): metric properties of d W p . Kantorovich–Rubinshtein (’58): (∫ ∫ ) d W 1 ( µ, ν ) = sup E fdµ − E fdν f :1-Lip Vasershtein (’69): d W 1 ( µ, ν ) := X ∼ µ,Y ∼ ν E[ d ( X, Y )] inf Dobrushin (’70) named ‘Vasershtein distance’ Mallows (’72): d W 2 in statistical context Tanaka (’73): d W 2 , Boltzmann equation

Bickel–Freedman (’80): d W 2 was named as Mallows metric (2) In English literatures, the German spelling ‘Wasserstein’ 1 is used (attributed to the name ‘Vasershtein’ being of Germanic ori- gin). 1 Vaserstein himself uses the terminology ‘Wasserstein distance’ in http://www.math.psu.edu/vstein/

Coarse Ricci curvature 4 ( E, d ): Polish space, E = B ( E ): Borel field. N 0 := N ∪ { 0 } . X = (Ω , X n , F n , F ∞ , P x ) x ∈ E : conservative Markov chain on ( E, E ). Ω := E N 0 : set of all E -valued sequences ω = { ω ( n ) } n ∈ N 0 . X n ( ω ) := ω ( n ), n ∈ N 0 .

P ( x, dy ) := P x ( X 1 ∈ dy ), x ∈ E : transition kernel of X: P ( x, dy ) satisfies the following: (P1) For each x ∈ E , P ( x, · ) ∈ P ( E ). (P2) For each A ∈ E , P ( · , A ) ∈ E . Further we impose the following: (P3) For each x ∈ E , P ( x, · ) ∈ P 1 ( E ).

We set P x ( A ) := P ( x, A ), A ∈ E and ∫ P f ( x ) := E f ( y ) P x ( dy ) = E x [ f ( X 1 )]. For the given Markov chain X as above and a fixed n ∈ N , a Markov chain X n = (Ω , X n k , F n k , F n ∞ , P n x ) x ∈ E with state space ( E, d ) defined by the transition kernel P n ( x, dy ) is called an n -step Markov chain .

Def 4.1 ( Ollivier (2009)) The coarse Ricci curvature κ ( x, y ) along ( xy ) for x ̸ = y is defined by κ ( x, y ) := 1 − d W 1 ( P x , P y ) ( ≤ 1) d ( x, y ) and κ :=inf { κ ( x, y ) | ( x, y ) ∈ E 2 \ diag } is said to be the lower bound of the coarse Ricci curvature . κ ∈ [ −∞ , 1].

The n -step coarse Ricci curvature κ n ( x, y ) of X along ( xy ) is defined to be d W 1 ( P n x , P n y ) κ n ( x, y ) := 1 − d ( x, y ) and κ n :=inf { κ n ( x, y ) | ( x, y ) ∈ E 2 \ diag } is its lower bound. κ n ( x, y ) is nothing but the coarse Ricci curvature for X n and κ 1 ( x, y ) = κ ( x, y ) for ( x, y ) ∈ E 2 \ diag . Note that κ n ≥ 1 − (1 − κ ) n holds.

Recent works on coarse Ricci curvature: Lin-Yau (2010): locally finite graphs 1 − 1 − 1 ( ) κ ( x, y ) ≥ − 2 d x d y Lin-Lu-Yau (2011): New def for κ ( x, y ). Jost-Liu (2011): locally finite graphs 1 − 1 − 1 ( ) κ ( x, y ) ≥ − 2 d x d y + Bauer-Jost-Liu (2011): graphs with loops 1 1 n ≤ λ 1 ≤ · · · ≤ λ N − 1 ≤ 1+(1 − κ n ) 1 − (1 − κ n ) n

Kitabeppu (2011): Lower estimate for κ ( x, y ) under CD( K, N ) Veysseire (2012): m -sym Markov process 1 1 − d W 1 ( P t ( x, · ) , P t ( y, · ) ( ) κ ( x, y ):=lim ≥ κ ∈ R t d ( x, y ) t → 0 ⇒ d W 1 ( P t ( x, · ) , P t ( y, · )) ≤ e − κt d ( x, y ) , E ( f ) m ( E ) < ∞ , κ ≤ if κ > 0 . ∥ f − 〈 m, f 〉∥ 2 2

Ex 4.1 (Sym. simple random walk on Z n ) E := Z n , d Z n ( x, y ) := ∑ n i =1 | x i − y i | : x, y ∈ Z n : i =1 | x i − y i | 2 ) 1 (∑ n 2 : x, y ∈ Z n d R n ( x, y ) := X: symmetric simple random walk on Z n . P ( x, dy ) := 1 ∑ δ z ( dy ) . 2 n | x − z | =1 ,z ∈ Z n ⇒ κ ( x, y ) = 0 w.r.t. either of d Z n or d R n . =

Ex 4.2 (RW on locally finite graph) Jost-Liu (2011): G = ( V, E ): a locally finite graph d x : degree at vertex x ∈ V def x ∼ y ⇐ ⇒ xy ∈ E 1 ∑ P ( x, dz ) := x ∼ y δ y ( dz ) d x 1 − 1 − 1 ( ) κ ( x, y ) ≥ − 2 d x d y + Equality holds if G = ( V, E ) is a tree.

Ex 4.3 (RW on Riemannian mfd) E = M : C ∞ compl. N -dim Riem mfd. ε > 0. m = vol: volume measure. X : ε -step Random walk on E defined by 1 P x ( dy ) = m ( B ε ( x ))1 B ε ( x ) ( y ) m ( dy ) . ⇒ κ ( x, y ) = ε 2 Ric( v,v ) Ollivier(09) 2( N +2) + O ( ε 3 + ε 2 d ( x, y )) = for v ∈ U x M and y ∈ exp x tv with t = d ( x, y ) small enough.

Ex 4.4 (Circle graph) G = ( V, E ): a circle graph of size N ; V := { x i } N i =1 : vertices, E := { x i x i +1 } N i =1 ( x N + i = x i ( i ∈ N )): edges, d x ( G ) = 2 for x ∈ V : degree at x ∈ V , P x i ( dy ) := 1 2 δ x i − 1 ( dy ) + 1 2 δ x i +1 ( dy ). κ ( x, y ) = 0 for ( x, y ) ∈ V × V \ diag , κ n ( x, y ) ≥ 0 for ( x, y ) ∈ V × V \ diag ,

X (hence X n ) is m -symmetric w.r.t. ∑ N 1 m ( dy ) := i =1 δ x i ( dy ). N We take N = 5. 3-step Markov chain X 3 is associated with G 3 := ( V 3 , E 3 ) defined by V 3 := V and E 3 := { x i x j | 1 ≤ i, j ≤ 5 with i ̸ = j } . The transition kernel P 3 x ( dy ) is given by x i = 1 8 δ x i − 2 + 3 8 δ x i − 1 + 3 8 δ x i +1 + 1 P 3 8 δ x i +2 .

d x ( G 3 ) = 4. The 3-step coarse Ricci curvature κ 3 ( x, y ) for xy ∈ E 3 can be estimated by use of Bauer-Jost-Liu (2011). κ 3 ( x i , x i +1 ) = 3 5 8 ≤ κ 3 ( x i , x i +2 ) ≤ 7 8 , 8 . 3 Therefore, κ 3 ( x, y ) ≥ 8 for all ( x, y ) ∈ V × V \ diag .

CAT(0)-space, 2-unif. convex sp 5 Def 5.1 (CAT(0)-space) ( Y, d Y ): CAT(0)-space ⇐ ⇒ For ∀ z, x, y ∈ Y , ∃ γ : [0 , 1] → Y with γ 0 = x , γ 1 = y s.t. for t ∈ [0 , 1] d 2 Y ( z, γ t ) ≤ (1 − t ) d 2 Y ( z, x ) + td 2 Y ( z, y ) − t (1 − t ) d 2 Y ( x, y ) . Cartan-Alexandrov-Toponogov

Ex 5.1 (Examples of CAT(0)-spaces) • Hadamard manifold; simply connected smooth compl Riem mfd with NPC. • products of CAT(0)-sp • Hilbert space • convex subset of CAT(0)-space • Tree • Euclidean Buildings • CAT(0)-space valued L 2 -maps �

Def 5.2 (2-Uniformly Convex Space) ( Y, d ): 2 -uniformly convex with k > 0 def ⇐ ⇒ ( Y, d ): geodesic space & ∀ x, y, z ∈ Y , ∀ γ := ( γ t ) t ∈ [0 , 1] : min. geo. in Y from x to y & ∀ t ∈ [0 , 1], d 2 ( z, γ t ) ≤ (1 − t ) d 2 ( z, x ) + td 2 ( z, y ) − k 2 t (1 − t ) d 2 ( x, y ) . z = γ t = ⇒ k ∈ ]0 , 2]. �

Ex 5.2 (Examples of 2-Unif. Conv. Spaces) • Convex subset of a 2-uniformly convex space. • CAT(0)-space • CAT(1)-space with diam < π 2 is 2-uniformly convex (see Ohta (2007)) • L 2 -maps into a CAT(1)-sp. with diam < π 2 �

( Y, d Y ): complete 2-unif, convex space γ , η ( ⊂ Y ): minimal geodesic segments def γ ⊥ p η ⇐ ⇒ p ∈ γ ∩ η , d Y ( x, p ) ≤ d Y ( x, y ) ∀ x ∈ γ, y ∈ η . (B): γ ⊥ p η ↔ η ⊥ p γ . Ex 5.3 (Examples satisfying (B)) • complete CAT(0)-space. • complete CAT(1)-sp with diam < π/ 2.

Def 5.3 (Barycenter) ( Y, d Y ): complete sep. 2-unif. convex space µ ∈ P 1 ( Y ) ⇒ b ( µ ): ∃ 1 unique minimizer (independent of w ∈ Y ) of ∫ ( d 2 Y ( z, y ) − d 2 z �→ Y ( w, y )) µ ( dy ) . Y We call b ( µ ) the barycenter of µ .

Lem 5.1 (Jensen’s inequality, K. (2010)) ( Y, d Y ): complete sep. 2-unif. convex space. µ ∈ P 1 ( Y ). (B): γ ⊥ p η ↔ η ⊥ p γ . Then for any l.s.c. convex func ϕ on Y ∫ ϕ ( b ( µ )) ≤ ϕ ( x ) µ ( dx ) . Y

Ass 5.1 m ∈ P 1 ( E ), supp[ m ] = E , p ≥ 1, X : m -sym Markov chain on E with (P3), ( Y, d Y ): compl sep. 2-unif. convex space, (B): γ ⊥ p η ↔ η ⊥ p γ , (CG): Convex Geometry: ∃ Φ : Y 2 → R convex s.t. C − 1 d Y ≤ Φ ≤ Cd Y on Y × Y for C > 0.

L p ( E, Y, m ): space of L p -maps, L p ( E, Y ; m ) := { u : E → Y m’ble map | ∫ m d p Y ( u ( x ) , o ) m ( dx ) < ∞∃ / ∀ o ∈ Y } / ∼ , E ∫ d L p ( u, v ) p := d p Y ( u ( x ) , v ( x )) m ( dx ) , E ∫ ∫ C p d p ( x, y ) m ( dx ) m ( dy ) ≤ ∞ p := E E ( C p < ∞ ⇔ m ∈ P p ( E )) . �

def Def 5.4 u ∈ S ( E, Y ) ⇔ ♯ (Im( u )) < ∞ . def d Y ( u ( x ) ,u ( y )) u ∈ Lip( E, Y ) ⇔ Lip( u ):=sup < ∞ . d ( x,y ) x ̸ = y m ∈ P p ( E ) ⇒ Lip( E, Y ) ⊂ L p ( E, Y ; m ) u ∈ S ( E, Y ) ∪ Lip( E, Y ) ⇒ u ∗ P x ∈ P 1 ( Y ) ⇒ P u ( x ) := b ( u ∗ P x ). dense → L p ( E, Y ; m ) and Thm 5.1 S ( E, Y ) ֒ dense → L p ( E, Y ; m ) if m ∈ P p ( E ) . Lip( E, Y ) ֒ Lem 5.2 κ n ∈ R , u ∈ Lip( E, Y ) ⇒ Lip( P n u ) ≤ C 2 (1 − κ n )Lip( u ) .