Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
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Fokker-Planck Equation on Graph with Finite
Vertices
Yao Li
January 13, 2011
Jointly with S.-N. Chow (Georgia Tech)
Wen Huang (USTC)
Hao-min Zhou(Georgia Tech)
Functional Inequalities and Discrete Spaces
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Outline
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. .1 Introduction
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. 2. Main Results
Main Results
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. .3 Proof of the Theorems
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
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. .4 Further Discussions
Geometry of the Metric Space
Some Remarks
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. .5 Examples
Fokker-Planck equation on a Graph
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Preliminary
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State space: X
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measure on state space X : µ
continuous state space: X - manifold with dimension N, dµ is
absolutely continuous with the
∫ Legesgue measure
density function ρ such that ρdx = 1
.
discrete state space: X - finite set {a1 , · · · , aN }, dµδ-functions
N
{ρi }N
i=1 = {ρ(ai )}i=1 ; i = 1, · · · , N denotes the probability
density function on X = {a1 , · · · , aN }
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Free Energy
Given a probability density function ρ on state space X , the free
energy is
F (ρ) = U(ρ) − βS(ρ)
U(ρ) is the potential energy
{∑
N
i=1 Ψi ρi
∫
U(ρ) =
RN Ψ(x)ρ(x)dx
discrete
continuous
S(ρ) is the entropy,
{ ∑
− N
ρi log ρi
∫ i=1
S(ρ) =
− RN ρ(x) log ρ(x)dx
β is the temperature.
discrete
continuous
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Discrete Free Energy and Gibbs Distribution
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Free energy of probability density function ρ is defined on
X = {a1 , · · · , aN } as
F (ρ) = U(ρ) − βS(ρ) =
N
∑
Ψi ρi + β
N
∑
ρi log ρi
.
.
Gibbs distribution is a unique global minimum of F (ρ), denoted as
{ρ∗i }N
i=1 , with
1
ρ∗i = e −Ψi /β
K
∑N −Ψ /β
Where
K = i=1 e i is a normalizer.
.
i=1
i=1
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Free Energy and Gibbs Density on RN
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Free energy of probability density function ρ is defined on X = RN
as
∫
∫
F (ρ) =
Ψ(x)ρ(x)dx + β
ρ(x) log ρ(x)dx
RN
RN
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And the Gibbs distribution is a unique minimum of F (ρ), denoted
as
1
ρ∗ (x) = e −Ψ(x)/β
K
∫
where
K = RN e −Ψ(x)/β dx is the normalizer.
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Remark
..
The free energy is a constant plus the relative entropy with respect
to
. the Gibbs measure.
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Fokker-Planck Equation (RN )
Given free energy F (ρ) of the density function ρ on RN , we have
.
The Fokker-Planck equation is generated by the following
differential operator
P(·) = ∇ · (∇Ψ·) + β∆·
The semigroup generated by P is called Fokker-Planck equation
∂ρ
= ∇ · (∇Ψρ) + β∆ρ
∂t
(1)
which provides the time evolution of the probability density
N
.function ρ(x, t) on X = R
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Two simple facts
free energy decreases along the solution of Fokker-Planck
equation
Gibbs density is a stable stationary solution of Fokker-Planck
equation
.
The
decay rate of the free energy functional is of exponential.
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Remark 2
..
If we replace ∇Psi by some other vector field (under some
assmuption), free energy still decreases along the solution of
.Fokker-Planck equation.
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Remark 3
..
Fokker-Planck equation is also called forward Kolmogorov
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equation.
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Yao Li
Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Gradient Flow
Given a gradient flow
dx
= −∇Ψ(x),
dt
x ∈ RN
subject to white noise
√
dx
= −∇Ψ(x) + 2βdWt ,
dt
x ∈ RN
The time evolution of its probability density function satisfies
equation (1), which can be vertified by Ito Calculus.
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
2-Wasserstein Metric (RN )
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X = RN , and µ1 , µ2 are Borel probability measures, the 2-Wasserstein distance
is
Z
d 2 (µ1 , µ2 ) =
inf
|x − y |2 p(dxdy )
p∈P(µ1 ,µ2 )
RN ×RN
where P(µ1 , µ2 ) is the collection of Borel probability measures on R n × R n with
.marginals µ1 and µ2 .
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Facts about 2-Wasserstein Metric (RN )
...
...
1
Wasserstein distance comes from the optimal transport.
2
Fokker-Planck equation (1), are the gradient flow of free
energy on 2-Wasserstein space. F.Otto(1998), R.Jordan, D.Kinderlehrer, and
F.Otto(1998)
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.3. Fokker-Planck equation contracts the 2-Wasserstein distance
F. Otto, M. Westdickenberg
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.. Nonnegative Ricci curvature is equivalent to convexity of
4
entropy on 2-Wasserstein space J. Lott, C. Villani, J. McCann, Erausquin, Cordero,
Schmuckenschlager, von Renesse, Sturm, etc.
...
5
Logarithmic Sobolev inequality implies Talagrand inequality
F.
Otto, C. Villani
...
6
We can continue this list on and on ...
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Our Motivation
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Remark
..
All
. the previous remarkable results require X to be a length space.
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Extend?
..
.Can we extend a series of remarkable facts to discrete setting?
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Fact!
..
The connection between Fokker-Planck equation and free energy is
no longer true if we replace Fokker-Planck equation by discrete
Laplace
operator.
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Two Different Approach
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We can discretize the manifold – rough curvature
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We can also discretize the Otto’s calculus
– .our work
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Our Motivation
To analysis the Fokker-Planck equation on a graph, one should
answer the following two questions
what’s 2-Wasserstein distance on graph
how to interpret noise in a Markov process
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Actually, because of the topology on a graph is different from that
on the length space, the previous two questions lead to two
different
Fokker-Planck equations!
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Theorem 1
∑
Given free energy F (ρ) = N
i=1 Ψi ρi + βρi log ρi on a graph
G = (V , E ).
we have a Fokker-Planck equation
dρi
dt
∑
=
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρj
(2)
j∈N(i);Ψj >Ψi
+
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρi
j∈N(i);Ψi >Ψj
+
∑
β(ρj − ρi )
j∈N(i);Ψi =Ψj
equation (2) is the gradient flow of free energy on a
Riemannian manifold (M, g ) with distance dΨ
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Theorem 1
Gibbs density ρ∗ is a stable stationary solution of equation (2)
and is the global minimum of free energy
Given any initial data ρ0 ∈ M, we have a unique solution
ρ(t) : [0, ∞) → M with initial value ρ0 , and
limt→∞ ρ(t) = ρ∗
The boundary of M is a repeller of equation (2)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Theorem 2
Given a graph G = (V , E ), a “gradient Markov process” on graph
G generated by potential {Ψi }N
i=1 , suppose the process is
subjected to “white noise” with strength β > 0
we have a Fokker-Planck equation (different from the
equation in Theorem 1)
dρi
dt
=
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρj
j∈N(i);Ψ̄j >Ψ̄i
+
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρi
(3)
j∈N(i);Ψ̄j >Ψ̄i
where Ψ̄i = Ψi + β log ρi
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Theorem 2
Equation (3) is the “gradient flow” of free energy for metric
space (M, d)
Gibbs density ρ∗ is a stable stationary solution of equation (3)
and is the global minimum of free energy
Given any initial data ρ0 ∈ M, we have a unique solution
ρ(t) : [0, ∞) → M with initial value ρ0 , and
limt→∞ ρ(t) = ρ∗
The boundary of M is a repeller of equation (3)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Remark 1
Both equations (2) and (3) are spatial discretizations of
Fokker-Planck equation (RN ) based on upwind scheme
Both equations are gradient flows of free energy, but on
different spaces
Gibbs density is a stable stationary solution of both equations
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
Remark 1
In both cases, the metrics are bounded by two different
metrics dm and dM , which are independent of the potential
All these metrics has some similarity with 2-Wassersyein
distance.
Near the Gibbs distribution, the difference of the two
equations are very “small”
The reason why we have two different equations may be
related to the fact that the graph is not a length space (See
the lecture notes by J.Lott)
Thanks to Prof. W.Gangbo for the discussion!
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
2-Wasserstein Distance on Graph
The distance dΨ could be seen as a discrete version of
2-Wasserstein distance.
...
From the derivation, dΨ is the upwind discretization of the
Otto calculus
...
The geodesic on (M, dΨ ) is the upwind discretization of the
geodesic on (Rd , w2 )
1
2
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Remark
..
In the interior of M, all the distances dΨ , d, dm and dM are
bounded
by the Euclidean distance. (see the next section)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
A toy example
Consider the following potential
function:
.
Discretization
..
We make a discretization
at five points
{a1 = 1, a2 = 2, a3 =
3,
. a4 = 4, a5 = 5}.
.
Noise
..
We set the noise strength
β
. = 0.8
Figure: Potential Energy
Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
compare - stationary distribution
Fokker-Planck equations.
Central difference scheme
Figure: Central Difference
Figure: Fokker-Planck Equations
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
compare - free energy
The free energy decreases in time if applying both of
Fokker-Planck equations
Figure: Equation in Theorem 1
Figure: Equation in Theorem 2
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Main Results
compare - free energy
The free energy does not always decrease when evolving the
equation obtained from the central difference scheme
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Basic Notation - The Graph
The state space is a finite set V = {a1 , a2 · · · , aN }
On each ai ∈ V , we have the potential energy Ψi
We consider a graph G = (V , E ), where V is the set of
vertices, and E denotes the edges set. Moveover, we always
assume this is a connected simple graph (No multiple edges,
no self-loop)
N(i) denotes the collection of neighborhood o i:
N(i) = {aj ∈ V |{ai , aj } ∈ E }
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Space of Probability Distributions
.
M denotes the space of probability distributions on V .
{
}
N
∑
N
N
M = {ρi }i=1 ∈ R |
ρi = 1; ρi > 0
i=1
.
The tangent space of M at ρ is
N
Tρ M = {{σi }N
i=1 ∈ R |
N
∑
σi = 0}
i=1
.
∂M denotes the boundary of M
{
∂M =
.
N
{ρi }N
i=1 ∈ R |
N
∑
ρi = 1; ρi ≥ 0;
i=1
}
ρi = 0
i=1
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Yao Li
N
∏
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
The Identification
Let W be the vector space:
N
W = {{pi=1 }N
i=1 ∈ R }/ ∼
2 N
1
2
where {pi1 }N
i=1 ∼ {pi }i=1 if and only if pi = pi + c for each i.
1 N
An identification is given from Tρ M to W: {pi1 }N
i=1 ≃ {σi }i=1 .
∑
∑
(pi − pj )ρi
(pi − pj )ρj +
σi =
j∈N(i);Ψi >Ψj
j∈N(i);Ψi <Ψj
+
∑
j∈N(i);Ψi =Ψj
ρi − ρj
(pi − pj )
log ρi − log ρj
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Yao Li
(4)
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Discrete Otto’s Calculus
Recall that 2-Wasserstein distance on R n (or manifold) is
equivalent to the following Otto’s formula
σ = −∇(ρ∇p)
for σ ∈ Tρ M
∫
g (σ1 , σ2 ) =
M
ρ∇p1 ∇p2
The fundamental idea is to discretize the Otto’s formula
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
The Norm
The following inner product is well defined on Tρ M
1
2
g (σ , σ ) =
N
∑
pi1 σi2
i=1
.
If σ 1 = σ 2 = σ, the norm is defined as:
g (σ, σ) =
∑
ρi (pi −pj )2 +
{ai ,aj }∈E ,Ψi >Ψj
∑
{ai ,aj }∈E ;Ψi =Ψj
(pi −pj )2
ρi − ρj
log ρi − log ρj
. Clearly, g (σ, σ) ≥ 0, and g induces a distance dΨ on M
.
N
where
{pi }N
i=1 ∼ {σi }i=1 under the identification (4)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Upper Bound
The upper bound metric dM (·, ·) is induced by inner product
M
1
2
g (σ , σ ) =
N
∑
σi1 pi2
i=1
when σ 1 = σ 2 = σ on Tρ M, we have
∑
g M (σ, σ) =
(pi − pj )2 min(ρi , ρj )
{ai ,aj }∈E
N
where [{pi }N
i=1 ] ≃ {σi }i=1 by the identification
∑
σiM =
(pi − pj ) min(ρi , ρj )
{ai ,aj }∈E
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Lower Bound
The lower bound metric dm (·, ·) is induced by inner product
m
1
2
g (σ , σ ) =
N
∑
σi1 pi2
i=1
when σ 1 = σ 2 = σ on Tρ M, we have
∑
g m (σ, σ) =
(pi − pj )2 max(ρi , ρj )
{ai ,aj }∈E
N
where [{pi }N
i=1 ] ≃ {σi }i=1 by the identification
∑
σim =
(pi − pj ) max(ρi , ρj )
{ai ,aj }∈E
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
The Gradient Flow
∑
∑N
F = N
i=1 Ψi ρi + β
i=1 ρi log ρi is the free energy, its gradient
flow is given by
dρ
= −gradF |ρ
dt
on the Riemannian manifold (M, g ), can be rewitten as
gρ (
dρ
, σ) = −dF |ρ (σ)
dt
∀σ ∈ Tρ M
where dF is the differential of F
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Fokker-Planck Equation
By a direct computation, we obtain the Fokker-Planck equation in
Theorem 1:
dρi
dt
∑
=
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρj
j∈N(i),Ψi <Ψj
∑
+
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρi
j∈N(i),Ψi >Ψj
+
∑
β(ρj − ρi )
j∈E(i)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Boundary is a Repeller
Consider M̄ = M ∪ ∂M
∀0 < ϵ << 1, ∃B(ϵ) which is neighborhood of ∂M such that
∀{ρ0i }N
i=1 ∈ B(ϵ) ∩ M, ∃T > 0 such that
{ρi (t)}N
/ B(ϵ), ∀t > T
i=1 ∈
where {ρi (t)}N
i=1 means the solution of equation (2) with initial
value {ρ0i }N
i=1
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Theorem 2
Given a graph G = (V , E ), a “gradient Markov process” on graph
G generated by potential {Ψi }N
i=1 , suppose the process is
subjected to “white noise” with strength β > 0
we have a Fokker-Planck equation (different from the
equation in Theorem 1)
dρi
dt
=
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρj
j∈N(i);Ψ̄j >Ψ̄i
+
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρi
j∈N(i);Ψ̄j >Ψ̄i
where Ψ̄i = Ψi + β log ρi
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Theorem 2
Equation (3) is the “gradient flow” of free energy for the
metric space (M, d)
Gibbs density ρ∗ is a stable stationary solution of equation (3)
and is the global minimum of free energy
Given any initial data ρ0 ∈ M, we have a unique solution
ρ(t) : [0, ∞) → M with initial value ρ0 , and
limt→∞ ρ(t) = ρ∗
The boundary of M is a repeller of equation (3)
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Idea of Proof
“gradient flow” on the graph
dx
= −∇Ψ(x)
dt
√
dx
= −∇Ψ(x) + 2βdWt
dt
“gradient flow” subject to
“white noise” perturbation
∂ρ
= ∇ · (∇Ψρ) + β∆ρ
∂t
Fokker-Planck equation in
Theorem 2
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Gradient Flow on the Graph
We call the following Markov process X (t) a gradient Markov
process generated by potential {Ψi }N
i=1 :
For {ai , aj } ∈ E , if Ψi > Ψj , the transition rate qij from i to j is
Ψi − Ψj .
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
Main Results
Proof of the Theorems
Further Discussions
Examples
Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Remark: Markov process
Consider a time-homogeneous continuous time Markov process
X (t) with transition graph G . For the transition rate qij = k
(i ̸= j), we mean
P(X (t + h) = aj |X (t) = ai ) = qij h + o(h)
and its Kolmogorov forward equation is
dρi
=
dt
∑
(Ψj − Ψi )ρi +
j∈N(i),Ψi >Ψj
∑
(Ψj − Ψi )ρj
(5)
j∈N(i),Ψj >Ψi
where ρi (t) denotes the probability of {X (t) = ai }
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Two Observations
.
Free Energy
..
∑
∑
∑
F =
Ψ i ρi + β
ρi log ρi =
(Ψi + β log ρi )ρi
∫
∫
∫
F =
Ψρ + β
ρ log ρ =
(Ψ + β log ρ)ρ
RN
RN
RN
.
.
Fokker-Planck equation
..
ρt
.
= ∇ · (∇Ψρ) + β∆ρ = ∇ · (∇Ψρ + β∇ρ)
= ∇ · [(∇Ψ + β∇ρ/ρ)ρ] = ∇ · [∇(Ψ + β log ρ)ρ]
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(6)
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
New Potential
.
From above observations, we know that in a system with noise, the
motion depends on both probability distribution and potential. In
some
sense, the noise alters the potential.
.
.
New Potential
..
With noise, the potential Ψ becomes another potential depending
on ρ:
Ψ̄ = Ψ + β log ρ
.
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Markov Process with Noise
Then we have a new Markov process Xβ (t) which is time
inhomogeneous, and could be considered as a “white noise”
perturbation from the original Markov process X (t).
.
If {ai , aj } ∈ E , and Ψ̄i > Ψ̄j , then the transition rate of Xβ (t)
from ai to aj is
.
Ψ̄i − Ψ̄j = (Ψi + β log ρi ) − (Ψj + β log ρj )
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Fokker-Planck Equation
Set ρi = P(Xβ (t) = ai ), the Kolmogorov forward equation is:
.
dρi
dt
=
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρi
j∈N(i),Ψ̄i >Ψ̄j
+
.
∑
∑
((Ψj + β log ρj ) − (Ψi + β log ρi ))ρj
j∈N(i),Ψ̄i <Ψ̄j
This is the Fokker-Planck equation (3) in Theorem 2.
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
The Metric
Recall in the Theorem 2, Equation (3) is the “gradient flow” of
free energy for a metric space (M, d)
.
the metric
..
By the metric space (M, d), we mean the distance induced by the inner
product on T ρM
N
∑
ḡ (σ 1 , σ 2 ) =
σi1 pi2
i=1
where
[{pi }N
i=1 ]
σi =
≃
{σi }N
i=1
∑
by the identification
∑
(pi − pj )ρj +
j∈N(i);Ψ̄i <Ψ̄j
+
∑
(pi − pj )
j∈N(i);Ψ̄i =Ψ̄j
.
(pi − pj )ρi
j∈N(i);Ψ̄i >Ψ̄j
ρi − ρj
log ρi − log ρj
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(7)
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Division of the Space
The space M is divided into segments by
N 2 −N
2
submanifolds Sij :
Sij = {{ρi }N
i=1 |ρi > 0; Ψi + β log ρi = Ψj + β log ρj }
ḡ (·, ·) is smooth on each component divided by {Sij }, and gives a
smooth Riemmannian distance on every component.
.
Remark
..
The
submanifolds Sij intersect at one point – Gibbs distribution
.
.
For every ρ ∈ M that is not on any Sij , there exists a small
neighborhood, such that the solution of (3) in Theorem 2 passing
though
ρ is the gradient flow of the free energy with respect to ḡ .
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
Picture
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Theorem 1 - discrete Otto Calculus
Theorem 2 - intepret noise
The other terms
The remaining proof for Theorem 2 is similar to that for Theorem
1
.1. Gibbs density ρ∗ is a stable stationary solution of equation (3)
.2. Given any initial data ρ0 ∈ M, we have a unique solution
ρ(t) : [0, ∞) → M with initial value ρ0 , and
limt→∞ ρ(t) = ρ∗
.. The boundary of M is a repeller of equation (3)
3
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Geometry of the Metric Space
Some Remarks
Weighted Graph
Recall the identification (4) of distance dΨ , we can rewrite it as
⃗σ = L(G , ρ)⃗p
where L(G , ρ) is the Laplacian matrix of the weighted graph
G = (V , E , w ). V and E are given, and w is the weight on edges:


if eij ∈ E , Ψi > Ψj
 ρi
ρj
if eij ∈ E , Ψi < Ψj
wij =

ρi −ρj

if eij ∈ E , Ψi = Ψj
log ρi −log ρj
the weight depends on the current probability distribution.
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Laplacian Matrix
Let G = (V , E , w ) be a weighted graph, then the Laplacian matrix
L is given by
Lij
= −wij
Lii
=
N
∑
if eij ∈ E
wij
eij ∈ E
j=1
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Weighted Graph
Similarly, the distance dm and dM could be rewitten as matrix
forms.
For dm
{
ρi if ρi > ρj
wij =
ρj if ρi < ρj
For dM
{
wij =
ρi
ρj
if
if
ρi < ρ j
ρi > ρ j
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Relationship with Euclidean Distance
So for ⃗σ ∈ T M, we have
gρ (⃗σ , ⃗σ ) = ⃗p T L(G , ρ)⃗p
the Euclidean norm of ⃗σ is
||⃗σ ||2 = ⃗p T L(G , ρ)T L(G , ρ)⃗p
which means when ρ is not on ∂M, we have
1
1
||⃗σ ||2
||⃗σ ||2 ≤ gρ (⃗σ , ⃗σ ) ≤
λN
λ2
λN is the largest eigenvalue of L(G , ρ), λ2 is the smallest positive
eigenvalue of L(G , ρ).
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Fokker-Planck Equation on Graph with Finite Vertices
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Eigenvalue Bound - Upper Bound
There are classical results about the eigenvalue bounds of the
Laplacian matrices. For example, Oscar Rojo gives the following
upper bounds
N
∑
λN ≤
max wkj
1
λN ≤ max
2 i∼j
{
j=1
k∼j
∑
∑
}
wi + wj + k∼i,k̸∼j wik + k∼j,k̸∼i wjk
∑
+ k∼i,k∼j |wik − wjk |
where i ∼ j means eij ∈ E
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Eigenvalue Bound - Lower Bound
Abraham Berman and Xiao-Dong Zhang gives the following lower
bound
√
λ2 ≥ σ̄ − σ̄ 2 − ic (G )2
where σ̄ is the maximal of the diagonal elements in L(G , ρ), and
ic (G ) is called the iso perimetric number is G :
∑
wij /|X |)
ic (G ) =
min
(
X ⊂V ,|X |≤N/2
i∈X ,j̸∈X
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Fokker-Planck Equation on Graph with Finite Vertices
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The Geodesic
We can compute the geodesic equation on the space M. For
simplicity, we assume the potential function on vertices are all
distinct.
Note that the geodesic is of constant speed so it’s also the
minimum of the energy functonal
∫
E (γ) = g (γ̇, γ̇)dt
So the equation of geodesic satisfies the Euler-Lagrange equation.
However, the Hamiltonian equation reveals more information about
the metric dΨ .
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The Geodesic
After legendre transformation, we obtain the following Hamiltonian
equation
dpi
dt
= −
dρi
dt
=
1
2
∑
(pi − pj )2
j∈N(i),Ψi >Ψj
∑
(pi − pj )ρi +
(8)
∑
(pi − pj )ρj (9)
j∈N(i),Ψi <Ψj
j∈N(i),Ψi >Ψj
.
Remark
..
This is the upwind discrete of geodesic equation on 2-Wasserstein
space
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Fokker-Planck Equation on Graph with Finite Vertices
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The Geodesic
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Some Remarks
One Example
v
u N √
√
u∑
dΨ (ρ1 , ρ2 ) = 2t ( ρ1i − ρ2i )
i=1
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Fokker-Planck Equation on Graph with Finite Vertices
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Some Remarks
1. the Master Equation
.
Master Equation
..
In (3), when the potential function is a constant function, we
obtained the master equation.
dρi
=β
dt
∑
(ρj − ρi )
{ai ,aj }∈E
.
.
Master equation also describes the time evolutionary of probability
distribution of the standard random walk. And the entropy
increases
monotonically along the master equation.
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Fokker-Planck Equation on Graph with Finite Vertices
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2. From Markov Process to Fokker-Planck Equation
Let’s consider the Markov process on a graph, without any
potential function assigned.
Given a time-homogeneous Markov process. In some cases, we can
still give a potential energy based on this Markov process, like this
picture.
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Fokker-Planck Equation on Graph with Finite Vertices
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2. From Markov Process to Fokker-Planck Equation
More precisely, if the transition probability graph of the given
random walk is a ”gradient like” graph, then the potential energy
function is well defined.
.
For a weighted directed simple graph without, if every directed
path from i to j have the same total weight for any two vertices i
and j, then a potential energy function is well defined on this
graph.
Moreover, the potential function is unique up to a constant.
.
.
example
..
Remark Every connected tree with weighted edges is a ”gradient
like”
graph.
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Fokker-Planck Equation on Graph with Finite Vertices
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2. From Markov Process to Fokker-Planck Equation
Figure: The left graph is gradient like
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Fokker-Planck Equation on Graph with Finite Vertices
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3. Upwind Scheme
On lattices, both equation (2) and (3) has upwind structure. The
drift part of equation (2) is a upwind scheme with respect to
potential {Ψi }N
i=1 ; where the whole equation of (3) is a upwind
scheme with equivalent potential {Ψ̄i }N
i=1
.
Conservation Law
..
In numerics, upwind scheme is a common used conservative
scheme
for the nonlinear hyperbolic equation
.
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Fokker-Planck Equation on Graph with Finite Vertices
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4. The Fundamental Relationship
For the discrete space case, the free energy, Fokker-Planck
equations and stochastic process have the same strong relationship
as in continuous case.
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
A Markov Process on Graph
Then we will show one general example about discrete
Fokker-Planck equation for on a graph.
Suppose G = (X , E ) is a graph, and a Markov process is defined
on this graph. Then we will show:
The Markov process
The graph with potential
The discrete Fokker-Planck equation of the perturbed Markov
process
The Gibbs disribution
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
The Markov Process
This is the graph. The black number is the label, and the red
number is the transition rate.
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
The Transition Matrix
The transition probability rate

0
 0

Q=
 0
 0
0
matrix Q of this Markov process is:

1 0 0 0
0 0 0 0 

2 0 0 1 

0 1 0 2 
1 0 0 0
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
The Graph with Potential
Since the underlying weighted directed graph is gradient like, we
may associate the potential energy on each vertices:
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
The Equation
Suppose the strengthen of noise is β = 0.9, then the
Fokker-Planck equation is:
dρ1 /dt = −ρ1 − 0.9(log ρ1 − log ρ2 )ρ1
dρ2 /dt = ρ1 + 2ρ3 + ρ5 + 0.9((log ρ5 − log ρ2 )ρ5 + (log ρ3 − log ρ2 )ρ3
+(log ρ1 − log ρ2 )ρ1 )
dρ3 /dt = ρ4 − 3ρ3 − 0.9((log ρ3 − log ρ2 )ρ3 + (log ρ3 − log ρ5 )ρ3
−(log ρ4 − log ρ3 )ρ4 )
dρ4 /dt = −3ρ4 − 0.9((log ρ4 − log ρ3 )ρ4 + (log ρ4 − log ρ5 )ρ4 )
dρ5 /dt = −ρ5 + 2ρ4 + ρ3 + 0.9(−(log ρ5 − log ρ2 )ρ5
+(log ρ4 − log ρ5 )ρ4 + (log ρ3 − log ρ5 )ρ3 )
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Yao Li
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
The Gibbs Distribution
Then the discrete Fokker-Planck equation converges to the Gibbs
distribution, which is showed in the picture.( The noise level is
β = 0.9)
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
Free Energy vs Time
And the Free energy decreases with time:
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Fokker-Planck Equation on Graph with Finite Vertices
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Fokker-Planck equation on a Graph
Things to do
This is the beginning of our work, many questions still don’t have
answer
.
.. rate of convergence (Just done by a student of Wen Huang)
1
.. Functional inequalities
2
.
.
...
...
3
Convexity of entropy or relative entropy (free energy)
4
Relationship between convexity of entropy and nonnegative
Ricci curvature
...
..
..
5
Application in the other area
6
......
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Fokker-Planck Equation on Graph with Finite Vertices
Introduction
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Fokker-Planck equation on a Graph
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Fokker-Planck Equation on Graph with Finite Vertices