Scaling limits for random processes from the point ofview of group cohomology
Makiko Sasada
The University of Tokyo

September 7, 2022, Online
Women in Mathematics
Based on joint work with Kenichi Bannai and Yukio Kametani

1Introduction

2

Hydrodynamic limit from geometric approach and main results

Motivation

Problem : Derive macroscopic dynamics from a microscopic stochastic
process
General story• (St )t≥0 : Microscopic stochastic process• (Stε )t≥0 : Properly scaled stochastic process in space and time withscaling parameter ε > 0• (S̄t )t≥0 = lim (Stε )t≥0 : Macroscopic dynamics
ε↓0

Key ingredient for the convergence
Homogenization (averaging) in space and time : Microscopic state space
has some homogeneity

Motivation
One particle model• Example 1 : Random walk
• (St )t≥0 : Discrete time/Continuous time simple random walk on Zd• Stε := εSε−2 t• (S̄t ) : Brownian motion in Rd with the diﬀusion matrix A = (ajk )dj,k=1 ,ajk = d1 δjk

Motivation
One particle model• Example 1 : Random walk
• (St )t≥0 : Discrete time/Continuous time simple random walk on Zd• Stε := εSε−2 t• (S̄t ) : Brownian motion in Rd with the diﬀusion matrix A = (ajk )dj,k=1 ,ajk = d1 δjk• Example 2 : Diﬀusion process in Rd with periodic coeﬃcient• G (x) = (gjk (x)) : smooth positive deﬁnite, one-periodic ⇔ Riemannian
metric on Td = Rd /Zd , m(x)dx : Riemannian volume measure• (St )t≥0 : Diﬀusion process in Rd with the generator()∂1∂jkL = 12 m(x)(Laplace-Beltrami operator)∂xj m(x)g (x) ∂xkε• St := εSε−2 t• (S̄t ) : Brownian motion in Rd with a constant diﬀusion matrix
A = (ajk )dj,k=1 given by an implicit form and also a variational formulaGeometric interpretation
Example 1 : Random walk
• Microscopic geometric object : (Zd , Ed , p) : weighted graph with the1periodic weight p : Ed → R>0 , p(±ej ) = 2d
• Generator of the “Brownian motion” of the microscopic space :∑Lf (x) = e∈Ex p(e)(f (te) − f (oe)) =1 ∑dj=1 (f (x + ej ) + f (x − ej ) − 2f (x))
2dGeometric interpretation
Example 1 : Random walk
• Microscopic geometric object : (Zd , Ed , p) : weighted graph with the1periodic weight p : Ed → R>0 , p(±ej ) = 2d
• Generator of the “Brownian motion” of the microscopic space :∑Lf (x) = e∈Ex p(e)(f (te) − f (oe)) =1 ∑dj=1 (f (x + ej ) + f (x − ej ) − 2f (x))
2d• Macroscopic geometric object : (Rd , G = (gjk )) : Riemannian
mannifold with the constant metric gjk = dδjk
• Generator of “Brownian motion” of the macroscopic space :1 ∑d∂21Lf = 2d
j=1 ∂u 2 f = 2 ∆G fj

Convergence of a geometric space with Riemannian structure and somehomogeneity!

Motivation : Geometric interpretation
Example 2 : Diﬀusion process in Rd with periodic coeﬃcient• Microscopic geometric object : (Rd , G = (gjk (x))) : Riemannian
mannifold with a periodic metric gjk (x)• Generator of theof the microscopic space :( “Brownian motion”)
1∂∂
jkL = 12 m(x)∂xj m(x)g (x) ∂xk

Motivation : Geometric interpretation
Example 2 : Diﬀusion process in Rd with periodic coeﬃcient• Microscopic geometric object : (Rd , G = (gjk (x))) : Riemannian
mannifold with a periodic metric gjk (x)• Generator of theof the microscopic space :( “Brownian motion”)
1∂∂
jk
L = 12 m(x)∂xj m(x)g (x) ∂xk• Macroscopic geometric object (Rd , Ḡ = (ḡjk )) : Riemannian
mannifold with a constant metric ḡjk
• Generator of “Brownian motion” of the macroscopic space :∑2Lf = 12 dj,k=1 ḡ jk ∂x∂j ∂xk f = 12 ∆Ḡ f
Convergence of a geometric space with Riemannian structure and somehomogeneity!

Period matrix and the macroscopic diﬀusion coeﬃcient
For both examples, G = Zd acts on the microscopic geometric object, and(H 1 (X , R))G ∼= H 1 (G , R) ∼= Zd holds.
Period matrix• In the class of one-forms of microscopic geometric objects, there is a
topological basis dθ1 , . . . dθd ∈ (H 1 (X , R))G .
• Once we introduce a Riemannian structure, which induces an innerproduct ⟨·, ·⟩ in the class of one-forms, there is a harmonic basis
H1 , . . . , Hd ∈ (H 1 (X , R))G so that ⟨dθj , Hk ⟩ = δjk .
• The change-of-basis matrix from dθ1 , . . . dθd to H1 , . . . , Hd is calleda period matrix.
Geometric interpretation of the macroscopic diﬀusion matrix
For these examples and more general random walks on periodic lattices,
the macroscopic diﬀusion matrix A is the inverse of the period matrix. In
other words, the period matrix is the “Riemannian” metric of themacroscopic geometric object.

Motivation
Our goal : Generalize these ideas to the case for a microscopic systemwith many particles!
Interacting particle systems• Example 3 : Exclusion processes• (ηt )t≥0 : Continuous time Markov process on {0, 1}Z∑• Lf (η) = x,y ∈Zd rx,y (η)(f (η x,y ) − f (η)) where η x,y is obtained fromby exchanging ηx and ηy∑d
• π : {0, 1}Z → M(Rd ) : ⟨π(η), f ⟩ := x∈Zd ηx f (x)∑d• π ε : {0, 1}Z → M(Rd ) : ⟨π ε (η), f ⟩ := εd x∈Zd ηx f (εx)• Stϵ := π ε (ηε−2 t ).
• (S̄t ) : Deterministic dynamic given by S̄t = ρ(t, u)du where ρ(t, u) is
the solution of the diﬀusion equationd∂t ρ =d∑∂uj (Djk (ρ)∂uk ρ).

j,k=1

Can we construct a good microscopic geometric object andunderstand Djk (ρ) as a period matrix? ⇒ Yes!

Remarks• The convergence of Markov processes ≒ The convergence of thegenerator + tightness + the existence and uniqueness of the processwith the generators• The scaling limit of random walks on general periodic lattices (crystallattices) are not trivial as the case for Zd and the discrete harmonic
analysis plays a role to describe the macroscopic diﬀusion matrix.
• The scaling limit like Example 2 is called the homogenization
problem. There have been many studies on this topic.
• The scaling limit like Example 3 is called the hydrodynamic limit.
There have been many studies on this topic too, but there was not auniversal framework to unify diﬀerent models.
• We introduced a universal framework for the microscopic geometricobject.
• The role of the group action was not understood well in the theory of
the hydrodynamic limit. (Even not for the one-particle case.)• By introducing a general framework and its geometric interpretation,
we also obtain new hydrodynamic limits for speciﬁc models.

1Introduction

2

Hydrodynamic limit from geometric approach and main results

Typical example : Exclusion process on Zd• {0, 1}Z : State space = Conﬁguration spaced• η = (ηx ) ∈ {0, 1}Z ,dηx : number of particle at x ∈ Zd

• Exclusion process : Continuous time Markov process {η(t)}t≥0 withthe generator L∑
Lf (η) =rx,y (η)ηx (1 − ηy ){f (η x,y ) − f (η)}
x,y ∈Zd• Jump rate : rx,y : {0, 1}Z → R≥0 : “frequency of jump from x to y ”dTypical example : Exclusion process on Zd
We always assume :• Translation invariant : rx,y (η) = r0,y −x (τ−x η)• Locality of interaction : rx,y are local functions∑• Finite range : ∃R > 0 s.t. rx,y ≡ 0 if ∥x − y ∥ := di=1 |xi − yi | > R• Non degenerate ⇒ the density of particles ρ characterizes theinvariant measures {νρ }Typical example : Exclusion process on Zd
We always assume :• Translation invariant : rx,y (η) = r0,y −x (τ−x η)• Locality of interaction : rx,y are local functions∑• Finite range : ∃R > 0 s.t. rx,y ≡ 0 if ∥x − y ∥ := di=1 |xi − yi | > R• Non degenerate ⇒ the density of particles ρ characterizes theinvariant measures {νρ }Reversible or Mean-zero case : Expected HDL equation∂t ρ = ∇ · D(ρ)∇ρ =d∑()∂ui Dij (ρ)∂uj ρi,j=1

Rigorous results :• Symmetric (not necessarily nearest neighbor) simple :∑rx,y (η) = ry ,x (η) = cx,y , Dij (ρ) = Dij = x∈Zd c0,x xi xj• Reversible and nearest neighbor (Funaki-Uchiyama-Yau (productmeasure), Varadhan-Yau (non-product measure with mixingcondition)) : D(ρ) is given by a variational formulaOther typical microscopic models
Generalized exclusion process : state space {0, 1, 2, . . . , κ}ZdMulti-color (species) exclusion process : state space {0, 1, 2, . . . , κ}ZdOpen problems of hydrodynamic limits (before our work)
Speciﬁc modelsd• Multi-species exclusion process {0, 1, 2, . . . , κ}Z• Energy exchange model RZ+ : Mesoscopic model obtained from somedeterministic model
dGeneral extensions• Finite range interaction (not nearest neighbor) models on Zd , where
the underlying graph is (Zd , EdR := {(x, y ) : |x − y | ≤ R})• Models on crystal lattices, such as hexagonal lattice, diamondlattice...
• Stationary measures which are not product (except for the exclusionprocess)Main result 1 : Framework of microscopic models
Microscopic models are deﬁned bygeometric data and stochastic data
• Geometric (spatial/topological) data : the triple (S, ϕ, X )• Local state space (Set S) (ex. {0, 1}, {0, 1, 2}, N, R, R+ )• Local interaction (Map ϕ : S × S → S × S) (ex. ϕ(s1 , s2 ) = (s2 , s1 ))• Underlying spatial space (Graph X = (X , E )) (ex. (Zd , Ed ), (Zd , EdR ),triangular lattice, diamond lattice)• Stochastic (spatial/metric) data
• Speed of local interaction r : Φ → R>0 (ex. rx,y (η)))• Equilibrium measures : µ (ex. Bernoulli product measures νρ )Symmetry data also plays an essential role
• Symmetry data :G• Symmetry of the underlying space space (Group G acting on X ) (ex.
G∼
= Zd )

Topological structure constructed by geometric data
Suppose the triple (S, ϕ, X ) is given.
• The data (S, ϕ) deﬁnes the space of conserved quantities Consvϕ (S),which is a subspace of function {f : S → R}
• The data (S, ϕ, X ) deﬁnes a graph structure (S X , Φ), which we call a
conﬁguration space with transition structure :Φ = {(η, η e ) : η ∈ S X , e ∈ E }.
• We introduce a uniform cohomology on the graph (S X , Φ)0
• Cunif(S X ) : set of uniform functions
1• Cunif (S X ) : set of uniform one forms
01• ∂ : Cunif(S X ) → Cunif(S X ) : diﬀerential (usual graph diﬀerential)
1X• Zunif (S ) : set of uniform closed forms0• ∂Cunif(S X ) : set of uniform exact forms0• Hunif (S X ) := ker ∂110• Hunif(S X ) := Zunif(S X )/∂Cunif(S X )Main result 2 : Characterization of “smooth” cohomology• Assumption 1• (S, ϕ) is irreducibly quantiﬁed (∼ the dynamics is non-degenerate)• X is transferable ((Zd , EdR ), d ≥ 2 satisﬁes the condition)• Assumption 2• (S, ϕ) is simple (Consvϕ (S) is the one-dimensional space, and somemore)• X is weakly transferable ((Zd , EdR ), d ≥ 1 satisﬁes the condition)

Theorem (Bannai-Kametani-S)
Under the assumptions 1 or 20Hunif(S X ) ∼= Consvϕ (S),1Hunif(S X ) ∼= {0}.

• X must be an inﬁnite graph under the assumption.

Main result 3 : De Rham cohomology for S X /G
Assume that a group G acts freely on the locale X .
• Action of G on functions and forms are naturally induced.
G0• E := ∂(Cunif(S X ) ) : set of G -invariant uniform exact formsG1• C := Zunif(S X ) : set of G -invariant uniform closed forms• H 1 (G , Consvϕ (S)) : the ﬁrst group cohomology of G with coeﬃcientsin Consvϕ (S)Theorem (Bannai-Kametani-S)
Under the assumptions 1 or 2C/E ∼= H 1 (G , Consvϕ (S)).
In particular, if G ∼= Zd , thenC∼=E⊕d⊕
k=1Consvϕ (S).

Main result 4 : A version of Hodge-Kodaira theorem• Using the stochastic data, an inner product is deﬁned on
1 (S X ))G . (analogy to Riemannian metric)(Cunif0 (S X )G ) : completion of set of G -invariant uniform exact• EL2 := ∂(Cunifforms• CL2 := ZL12 (S X )G : set of G -invariant L2 closed forms• Assume : S is a ﬁnite set and G ∼= Zd , and the induced measure on
X
S is product.

Theorem (Bannai-S)
Under the assumptions 1 or 2, and several essential assumptions including
above,d⊕∼C L 2 = E L2 ⊕Consvϕ (S).
k=1

New interpretation of the macroscopic diﬀusion matrix

There are two natural decomposition of closed forms
C L2 ∼= E L2 ⊕d⊕

Consvϕ (S) :topological (Varadhan’s) decomposition

k=1C L2 ∼= E L2 ⊕d⊕

Consvϕ (S) :orthogonal decomposition

k=1Diﬀusion matrix
Macroscopic diﬀusion matrix D(ρ) = Transition matrix of two diﬀerent
decomposition under the measure νρ = The inverse of the period matrix

Summary• The decomposition of the space of (H 1 )G (X ) or (H 1 )G (S X ) is thekey to understand the macroscopic diﬀusion matrix• The group cohomology of G play an essential role as (H 1 )G (S X ) can
be computed easily once we prove H 1 (S X ) = {0}• Uniform functions and uniform forms are “smooth functions” on
S X /G• Hodge-Kodaira theorem is generalized to conﬁguration spaces• The macroscopic diﬀusion matrix is the inverse of the period matrix
universally.