Nonlinear Fokker-Planck equations on graphs and their asymptotic properties

0.26 MB·

by  Shui-Nee Chow

NONLINEAR FOKKER-PLANCK EQUATIONS ON FINITE GRAPHS
AND THEIR ASYMPTOTIC PROPERTIES
7
1
0 SHUI-NEECHOW,WUCHEN LI, ANDHAOMINZHOU
2

r
a
M
Abstract. The notion of Fokker-Planck equations on finite graphs developed in [10]
3
2 is extended to account for the present of nonlinear interactive potentials in the free
energy setting. We show that the solution of the Fokker-Planck equation converges

] to one of the Gibbs measures exponentially fast. And we give a new formula for the
S
asymptotic dissipation rate. The continuous analog of this asymptotic rate is provided,
D
which connects with Yano’s formula [28, 29]. Several numerical examples are provided
. to illustrate theconvergence behavior.
h
t
a
m
[ 1. Introduction

2
Optimaltransporttheoryhasbecomeoneofthemostactivebranchesinmathematicsin
v
1 the past few decades, with many remarkable results revealing the deep connections among
4 the theory of partial differential equations (PDEs), geometry and stochastic processes.
8
For example, in the seminal work [15], Jordan, Kinderlehrer and Otto showed that the
4
0 Fokker-Planck equation,aclassicalPDEdescribingtheevolutionoftheprobabilitydensity
. function for stochastic diffusion processes, is actually the gradient flow of a free energy
1
0 in the probability space equipped with 2-Wasserstein metric. The results have profound
7 impact and sparkled tremendous developments in the field [6, 7, 19, 23]. Much more can
1
be found in the books by Villani [26, 27].
:
v
i Motivated by applications in various fields such as game theory, RNA folding and nu-
X
merical schemes for PDEs, similar matters have been studied in discrete settings, such as
r
a finite graphs, in recent years. It turns out that the optimal transport theory in discrete
space is more subtle than a merely direct discretization of its counterpart in continuous
space, due to reasons that might not be obvious at the first sight. On one hand, a graph
with more than one nodeis not a length space [5], a crucial concept relied on by many im-
portant tools used in the optimal transporttheory in continuous setting. Simple geodesics
in the 2-Wasserstein space often donot exist if theunderlyingmetric space is discrete [17].
On the other hand, graphs may have more complicated neighborhood structures that can-
not be embedded into a continuous space, such as Rd. That makes the discretization
impossible.
Key words and phrases. Fokker-Planckequation;Optimaltransport;Gradientflow; Graph;Dynamical
system; Entropydissipation; Yano’s formula.
This work is partially supported by NSF Awards DMS–1419027, DMS-1620345, and ONR Award
N000141310408.
1
2 Chow,Li,Zhou
Despite of the challenges, metrics that mimic the 2-Wasserstein distance in continuous
space have been proposed for the probability spaces on discrete sets by several groups
independently. In [17], Maas constructed a discrete Wasserstein metric, based on the
properties of an irreducible and reversible Markov kernel on a finite set and Benamou-
Brenier formula [3], and showed that the heat flow is gradient flow of the entropy with
respect to the metric. In a study of the gradient flow of free energy on graphs, authors
of [10] introduced a class of metrics on probability spaces on finite graphs that can be
consideredasananalogueof2-Wassersteinmetric,andthenconstructedtheFokker-Planck
equations on graphs using the metrics. In the context of reaction-diffusion equations, a
similar metric structure is also discovered in [20]. More recently, many studies related to
the optimal transport theory in discrete settings and applications have been carried out.
Readers are referred to [12, 13, 14, 18, 21, 22, 25] for more information.
Based on results developed in [10], the focus of this paper is on asymptotic properties
of Fokker-Planck equations on graphs, especially when the free energy contains nonlinear
interactive potentials, a case that has not been considered in discrete settings in the past.
We shall show that the resulting nonlinear Fokker-Planck equations converge to one of the
Gibbs distributions exponentially fast as time goes to infinity, which can be viewed as the
discrete analogue of entropy dissipation [6, 11]. We further provide an explicit formula
that bounds the convergence rate. It is worth to point out that there might be multiple
equilibria (Gibbs measures) due to lack of convexity for general interactive potentials.
This is a scenario that can cause technical difficulties for many existing methodologies.
In contrast, to achieve the goal, we borrow strategies from dynamical systems to study
the asymptotic behavior of the nonlinear Fokker-Planck equations. In addition, the de-
duced convergence rate formula seems to suggest, by formal calculations, an interesting
continuous analog that connects to an important identity, known as Yano’s formula, in
Riemanian geometry [28, 29].
The structure of this paper is arranged as follows. We review the related results on
discrete 2-Wasserstein metric and Fokker-Planck equations on graphs in the next section.
Then study its asymptotic convergence in Section 3. In Section 4, we discuss some prop-
erties stemmed from the convergence rate formula, including an interesting connection
with Yano’s formula. We provide several numerical examples to illustrate the convergence
properties in Section 5.
2. Optimal transport on finite graphs
In this section, we review the constructions of 2-Wasserstein metric in discrete states
andFokker-Planck equationsongraphs. Wemainlyfollowtheapproachesgivenin[10,17].
To account for the nonlinear interactive potentials in free energy, our presentation might
be different, although most results can be directly generalized from those ideas reported
in cited references. For this reason, we skip some proofs if similar ones can befoundthere,
unless the modifications are not trivial.
Let us consider a simple weighted finite graph G = (V,E,d), where V = 1,2, ,n
{ ··· }
is the vertex set, E is the edge set, and d d is the weight of edge (i,j) E with
ij
∈ ∈
d = d > 0. We assume that G is undirected and contains no self loops or multiple
ij ji
NonlinearFokker-Planckequations ongraphs 3
edges. The adjacency set, or neighborhood, of vertex i V is denoted by
∈
N(i) = j V (i,j) E .
{ ∈ | ∈ }
The probability set (simplex) supported on all vertices of G is defined by
n
(G) = (p )n p = 1 , p 0 , for any i V ,
P { i i=1 | i i ≥ ∈ }
i=1
X
wherep is thediscreteprobability functionat nodei. Itsinterior is denotedby (G). To
i o
P
assist our presentation, we adopt the following notations and operations on G and (G).
P
A vector field v on G refers to a skew-symmetric matrix on the edges set E:
v := (v ) , with v = v .
ij (i,j)∈E ij ji
−
Given a function Φ= (Φ )n on V, it induces a potential vector field Φ on G as
i i=1 ∇G
1
Φ := ( (Φ Φ )) .
G i j (i,j)∈E
∇ d −
ij
For a given probability function p (G) and a vector field v, we define the product
∈ P
pv, called flux function on G, by
pv := (v g (p)) ,
ij ij (i,j)∈E
where g (p) are specially chosen functions. For example, g (p) can be the logarithmic
ij ij
mean or an upwind function of p which are used in [10, 17]. In this paper, we select the
average of p and p , i.e.
i j
p +p
i j
g (p) = , for any (i,j) E ,
ij
2 ∈
for the simplicity of illustration.
We define the divergence of pv on G by
n
1
div (pv) := v g (p) ,
G ij ij
− d
(cid:18)j∈N(i) ij (cid:19)i=1
X
which returns a value at each node of V. Given two vector fields v = (v ) , w =
ij (i,j)∈E
(w ) on a graph and p (G). The discrete inner product is defined by,
ij (i,j)∈E
∈ P
1
(v,w) := v w g (p) .
p ij ij ij
2
(i,j)∈E
X
The coefficient 1/2 in front of the summation accounts for the fact that every edge in G
is counted twice. In particular, we have
1
(v,v) := v2g (p) .
p 2 ij ij
(i,j)∈E
X
It is worth to remark that we prefer not to replace g by its explicit formula, as done in
ij
[10,17], toemphasize thefreedomof usingdifferent g , whichleads to differentdefinitions
ij
for the flux function, divergence operator and inner product, and hence results in different
formulas for the discrete 2-Wasserstein metric. More details are given in the Appendix.
4 Chow,Li,Zhou
Proposition: For any vector field v and potential function Φ on a graph, the following
properties hold
n
(1) div (pv) Φ = (v, Φ) .
− i=1 G |i i ∇G p
Pn
(2) div (pv) = 0 .
i=1 G |i
P
Proof. Following the definitions and v +v = 0, we write
ij ji
n n
1
div (pv) Φ = v g Φ
G i i ij ij i
− | d
ij
i=1 i=1j∈N(i)
X X X
1 1 1
= ( v Φ g + v Φ g )
ij i ij ji j ji
2 d d
ij ij
(i,j)∈E (j,i)∈E
X X
1 1
= v (Φ Φ )g
ij i j ij
2 d −
ij
(i,j)∈E
X
=( Φ,v) .
G p
∇
By taking Φ = 1 for any i V, we obtain
i
∈
n n
div (pv) = div (pv) Φ = (v, Φ) = 0 .
G i G i i G p
| | − ∇
i=1 i=1
X X
(cid:3)
The first identity in above proposition can be viewed as an analogue of integration by
part formula on graphs.
2.1. 2-Wasserstein metric on a graph. Following theapproachin[17], wenowpresent
the discrete analogue of 2-Wasserstein metric on probability set (G).
2 o
W P
Definition 1. For any p0, p1 (G), define
o
∈ P
1 dp
2(p0,p1) := inf (v,v) dt : +div (pv) = 0 , p(0) = p0 , p(1) = p1 , (1)
W2 v { p dt G }
Z0
where the infimum is taken over all vector fields v on a graph, and p is a continuously
differentiable curve p : [0,1] (G).
o
→ P
One can view the above definition as the discrete Briener-Benamou formula [3]. It has
been proved in [10, 17] that possesses properties that are similar to those for the 2-
2
W
Wassersteindistanceinthecontinuousspace. Forexample,whenequippedwith , (G)
2 o
W P
becomes a Riemannian manifold. In addition, Maas used the following decomposition on
discrete sets in his study.
Theorem 2. Given a vector field on a graph v = (v ) with v = v , and a
ij (i,j)∈E ij ji
−
measure p (G), there exists a unique decomposition, such that
o
∈ P
v = Φ+u , and div (pu) = 0 ,
G G
∇
NonlinearFokker-Planckequations ongraphs 5
where Φ is a function defined on V. In addition, the following property holds,
(v,v) = ( Φ, Φ) +(u,u) .
p G G p p
∇ ∇
One may view this decomposition as a discrete analogue of the well-known Hodge de-
composition. Following the idea used in Maas [17], an equivalent definition of is given
2
W
in the next theorem.
Theorem 3. If p0, p1 (G), then
o
∈ P
1 dp
(p0,p1) 2 = inf ( Φ, Φ) dt : +div (p Φ) = 0 , p(0) = p0 , p(1) =p1 ,
2 G G p G G
W ∇GΦ{Z0 ∇ ∇ dt ∇ }
(cid:0) (cid:1) (2)
where infimum is taken over all potential vector fields Φ, and p is continuously differ-
G
∇
entiable curves p :[0,1] (G).
o
→ P
Anotherequivalentdefinition, given in[10]independently,istolook at fromtangent
2
W
space viewpoint. Consider the tangent space at a point p (G),
o
∈P
n
T (G) = (σ )n Rn σ = 0 ,
pPo { i i=1 ∈ | i }
i=1
X
it has been proved that for any given σ T (G), there exists a unique function Φ, up
p o
∈ P
to a constant shift, satisfying σ = div (p Φ), see details in [10]. This one to one map
G G
− ∇
endows an inner product on T (G).
p o
P
Definition 4. For any two tangent vectors σ1,σ2 T (G), define the inner product
p o
g : T (G) T (G) R by ∈ P
p o p o
P × P →
n n
1 1
g(σ1,σ2) := σ1Φ2 = σ2Φ1 = g (p)(Φ1 Φ1)(Φ2 Φ2) ,
i i i i 2 d2 ij i − j i − j
i=1 i=1 (i,j)∈E ij
X X X
where σ1 = div (p Φ1) and σ2 = div (p Φ2).
G G G G
− ∇ − ∇
Hence (1) is equivalent to
1 dp
(p0,p1) 2 =inf g(σ,σ)dt : = σ , p(0) = p0 , p(1) = p1 , p , (3)
2
W { dt ∈ C}
Z0
(cid:0) (cid:1)
where is the set of all continuously differentiable curves p : [0,1] (G).
o
C → P
We use both definitions given in (2) or (3) in our investigation, dependingon which one
is more convenient for the discussion.
2.2. Fokker-Planck equations on finite graphs. We consider a discrete free energy
: (G) R containing three terms: linear potential, interactive potential, and linear
F P →
entropy given by
n n n n
1
(p) = V p + W p p +β p logp , (4)
i i ij i j i i
F 2
i=1 i=1 j=1 i=1
X XX X
6 Chow,Li,Zhou
where (V )n is a constant vector and (W ) is a symmetric matrix, and β > 0 a
i i=1 ij 1≤i,j≤n
small constant. Following approaches given in [10], we derive the gradient flow of (p) on
F
Riemannian manifold ( (G), ).
o 2
P W
Theorem 5 (Gradient flows). For a finite graph G and a constant β > 0, the gradient
flow of the discrete free energy (p) defined in (4) on manifold ( (G), ) is
o 2
F P W
dp 1 ∂ ∂
i
= g (p) (p) (p) .
dt d2 ij ∂p F − ∂p F (5)
j∈N(i) ij j i
X (cid:0) (cid:1)
For any initial condition p0 (G), (5) has a unique solution p(t) : [0, ) (G).
o o
∈ P ∞ → P
Moreover,
(i) there exists a constant depending on p0, c = c(p0) > 0, such that p (t) c for all
i
≥
i V and t > 0;
∈
(ii) the free energy (p) is a Lyapunov function for (5), i.e.
F
d 1 1 ∂ ∂
(p(t)) = [ (p) (p)]2g (p) 0 .
dtF −2 d2 ∂p F − ∂p F ij ≤
(i,j)∈E ij i j
X
If lim p(t) exists, denoted by p∞, then p∞ is one of the Gibbs measures.
t→∞
Proof. For any σ T (G), there exists Φ, such that σ = div (p Φ). Since dp =
∈ pPo − G ∇G dt
(dpi)n , is in T (G). By Definition 4, we have
dt i=1 pPo
n
dp dp
i
( ,σ)p = Φi . (6)
dt dt
i=1
X
On the hand side, we also have
n n
∂
d (p) σ = (p) σ = F (p)div (p Φ)
i i G G i
F · ∂p F · − ∇
i
i=1 i=1
X X (7)
n
=( F(p), Φ) = Φ div (p F(p)) ,
G G p i G G i
∇ ∇ − ∇
i=1
X
in which we use short handed notation F (p) = ∂ (p), for any i V . Combining (6)
i ∂piF ∈
and (7), and the definition of gradient flow on manifold, we obtain
dp
0 =( ,σ) +d (p) σ
p
dt F ·
n
dp
i
= div (p F(p)) Φ .
G G i
{ dt − ∇ }
i=1
X
Since the above formula is true for all (Φ )n Rn,
i i=1 ∈
dp 1
i
+ g (p) F (p) F (p) = 0
dt d2 ij i − j
j∈N(i) ij
X (cid:0) (cid:1)
holds for all i V, which is (5).
∈
Theproofsof(i)and(ii)arejustslightmodificationsof[10],seemoredetailsin[16]. (cid:3)
NonlinearFokker-Planckequations ongraphs 7
Different from the situations in [10, 17], the nonlinear Fokker-Planck equation (5) may
have multiple Gibbs measures. We will give an example in Section 5. In this case, one can
not construct a time reversible Markov process corresponding to (5). Therefore it does
not fit into the framework of [17]. However, methodologies used in [10] can go through
with nominal modifications.
3. Entropy dissipation on finite graphs
In this section, we study the asymptotic behavior of gradient flow (5). To further
simplify our discussion, we introduce the following notations.
∂2 ∂2 ∂2 ∂2
h = ( + ) (p) , for any i, j, k, l V ,
ij,kl
∂p ∂p ∂p ∂p − ∂p ∂p − ∂p ∂p F ∈
i k j l i l j k
and
1 1
λ (p) = min h (Φ Φ )g (Φ Φ )g , (8)
F (Φi)ni=1∈D 4 (i,j)∈E(k,l)∈E d2ijd2kl ij,kl i − j ij k − l kl
X X
where
1 1
D = (Φ )n Rn (Φ Φ )2g = 1 .
{ i i=1 ∈ | 2 d2 i− j ij }
(i,j)∈E ij
X
We assume that the initial measure p0 is in a basin of attraction of equilibrium p∞ for
the gradient flow, meaning
(A) lim p(t) =p∞ and p∞ is an isolated equilibrium.
t→∞
Now, we are ready to present the main theorem of this paper.
Theorem 6. Let (A) hold, λ (p∞) > 0. Then there exists a constant C = C(p0,G) > 0
F
such that
(p(t)) (p∞) e−Ct( (p0) (p∞)) . (9)
F −F ≤ F −F
Furthermore, for any sufficiently small ǫ > 0 satisfying (λ (p∞) ǫ) > 0, there exists a
F
−
time T > 0 depending on ǫ and p0, such that when t > T,
(p(t)) (p∞) e−2(λF(p∞)−ǫ)(t−T)( (p0) (p∞)) .
F −F ≤ F −F
Before giving the proof, we would like to comment that the quadratic formula involv-
ing h in (8) relates to the Hessian operator of (p), denoted as Hess (p), at the
ij,kl F W2F
equilibrium p∞ with respect to 2-Wassertein metric on (G). This is different from
2 o
W P
the Hessian operator of (p) with respect to the standard Euclidean metric on (G),
o
F P
HessRn (p). We will discuss their connections in Subsection 3.2. In addition, λF(p∞) is
F
the smallest eigenvalue of Hess (p) at p∞.
W2F
8 Chow,Li,Zhou
3.1. Proof of main results. To prove this theorem, we borrow some techniques from
dynamical systems. If the energy function (p) is λ convex at the equilibrium, meaning
F
F
Hess (p)p∞ λFI, where λF > 0, then the following steps will show the results.
F | (cid:23)
(1) Calculate the second order derivative of (p(t)) with respect to time t to get
F
Lemma 7.
d2 d
(p(t)) 2λ (p(t)) m(p(t)) (p(t)) ,
dt2F ≥ − F − dtF
(cid:0) (cid:1)
where λ isdefined in (8) and m(p(t)) isa functionsatisfying lim m(p(t)) =0.
F t→∞
(2) Apply λ -convexity at p∞ and Gronwall’s inequality to show (9).
F
Proof of Lemma 7. From Theorem 5, the first order derivative of free energy along the
gradient flow (5) gives
n
d 1 1
(p(t)) = (F F )2g .
dtF −2 d2 i − j ij
i=1j∈N(i) ij
X X
By the product rule, we compute the second order derivative,
d2 1 n 1 dg
(p(t)) = (F F )2 ij
dt2F − 2 d2 i − j dt ♣
i=1j∈N(i) ij
X X (10)
n
1 dF dF
i j
( )(F F )g .
− d2 dt − dt i − j ij ♠
i=1j∈N(i) ij
X X
We shall prove
d
m(p(t)) (p(t)) , (11)
♣ ≥ dtF
with lim m(p(t)) = 0, and
t→∞
d
2λ (p(t)) (p(t)) . (12)
F
♠ ≥ − dtF
Let us prove (11) first. Note
n
1 1 d
= (F F )2 g (p(t))
♣ − 2 d2 i − j dt ij
i=1j∈N(i) ij
X X
n
1 1 ∂g dp ∂g dp
= (F F )2( ij i + ij j) .
− 2 d2 i − j ∂p dt ∂p dt
i=1j∈N(i) ij i j
X X
NonlinearFokker-Planckequations ongraphs 9
Since min p ,p g max p ,p , we have
i j ij i j
{ } ≤ ≤ { }
∂g dp ∂g dp ∂g dp
ij i ij j ij i
+ 2max max
∂pi dt ∂pj dt ≤ i∈V |∂pi |· i∈V | dt |
∂g 1
ij
=2max max (F F )g
i∈V |∂pi |· i∈V |j∈N(i) d2ij j − i ij|
X
∂g 1
ij
2max ∆(G) maxp max max F F
≤ i∈V |∂pi |· · i∈V i·(i,j)∈E d2ij ·(i,j)∈E| i − j|
∂g max p 1
ij i∈V i
2max ∆(G) g max max F F .
≤ i∈V |∂pi |· · mini∈V pi · ij ·(i,j)∈E d2ij ·(i,j)∈E| i − j|
where ∆(G) is the maximal degree of graph G, meaning the largest number of edges
incident to one vertex.
Defining
∂g 1 max p (t)
ij i∈V i
m(p(t)) = 2 max max max F F ∆(G) , (13)
· i∈V |∂pi |·(i,j)∈E d2ij ·(i,j)∈E| i − j|· · mini∈V pi(t)
we obtain
1 1 d
m(p(t)) (F F )2g = m(p(t)) (p(t)) .
♣ ≥ − · 2 d2 i − j ij dtF
(i,j)∈E ij
X
From Theorem 5, we know that lim F (p(t)) F (p(t)) = 0 and inf maxi∈V pi(t) > 0,
t→0| i − j | t>0 mini∈V pi(t)
therefore
lim m(p(t)) = 0 .
t→∞
Next, we estimate to prove (12). Denote f = ∂2 (p), we have
♠ ij ∂pi∂pjF
1 1 1 dF dF
i j
= ( )(F F )g
2♠ − 2 d2 dt − dt i − j ij
(i,j)∈E ij
X
n n
1 1 ∂F dp ∂F dp
i k j k
= (F F )g ( )
− 2 d2 i − j ij ∂p dt − ∂p dt
(i,j)∈E ij k=1 k k=1 k
X X X
n
1 1 dp
k
= (F F )g (f f )( )
2 d2 i − j ij ik − jk − dt
(i,j)∈E ij k=1
X X
n
1 1 1
= (F F )g (f f )[ (F F )g ]
2 d2 i − j ij ik − jk d2 k − l kl
(i,j)∈E ij k=1 l∈N(k) kl
X X X
1 1 1
= (F F )g (f f )(F F )g .
2 d2 i − j ij d2 ik − jk k − l kl
(i,j)∈E ij (k,l)∈E kl
X X
10 Chow,Li,Zhou
And
1
(f f )(F F )g
d2 ik − jk k − l kl
(k,l)∈E kl
X
1 1 1
= [ (f f )(F F )g + (f f )(F F )g ]
2 d2 ik − jk k − l kl d2 ik − jk k − l kl
(k,l)∈E kl (k,l)∈E kl
X X
Relabel k, l in the second formula.
1 1 1
= [ (f f )(F F )g + (f f )(F F )g ]
2 d2 ik − jk k − l kl d2 il− jl l − k lk
(k,l)∈E kl (l,k)∈E lk
X X
1 1 1
= [ (f f )(F F )g (f f )(F F )g ]
2 d2 ik − jk k − l kl − d2 il− jl k − l kl
(k,l)∈E kl (k,l)∈E kl
X X
1 1
= (f +f f f )(F F )g ,
2 d2 ik jl− jk − il k − l kl
(k,l)∈E kl
X
where the second last equality is from g = g and d = d .
kl lk kl lk
Combining above two steps, we have
1 1
= 2 h (F F )g (F F )g . (14)
♠ · 4 d2 d2 ij,kl i − j ij k − l kl
(i,j)∈E(k,l)∈E ij kl
X X
To prove (12), we need to show
1 1 1 1
h (F F )g (F F )g λ (p) (F F )2g . (15)
4 d2 d2 ij,kl i− j ij k− l kl ≥ F ·2 d2 i− j ij
(i,j)∈E(k,l)∈E ij kl (i,j)∈E ij
X X X
The proof of (15) can be divided into two cases. Let
1 1
C(F):= (F F )2g .
2 d2 i − j ij
(i,j)∈E ij
X
If C(F) = 0, then F = F for any (i,j) E, which implies that both sides of (15)
i j
∈
equal to 0, thus (15) holds.
If C(F) > 0, we consider a vector field given by ( Fi )n D. Substituting Φ in
√C(F) i=1 ∈ i
the definition of λ by Fi , we obtain (15).
F √C(F)
Using (11) and (12), we have
d2 d
(p(t)) = + 2λ (p(t)) m(p(t)) (p(t)) ,
dt2F ♣ ♠ ≥ − F − dtF
which finishes the proof of Lemma 7. (cid:0) (cid:1) (cid:3)
Now we are ready to prove Theorem 6. The main idea is to use the asymptotic com-
parison between d (p(t)) and d2 (p) in Lemma 7.
dtF dt2F