Limit of 1-Dimensional Mixed-Mechanism Interacting Particle System Model

TL;DR

This paper studies a 1D mixed-mechanism interacting particle system, deriving its limit as a stochastic partial differential equation driven by space-time white noise, incorporating new mechanisms like random switch and local homogenization.

Contribution

It introduces new mechanisms such as random switch and unbiased local homogenization into the voter process model and establishes the SPDE limit with high-frequency duplication effects.

Findings

Limit described by SPDE driven by space-time white noise

Inclusion of new mechanisms like random switch and local homogenization

Identification of high-frequency duplication as key to the diffusion term

Abstract

Elaborating on the model from voter process with mixed-mechanism under suitable scaling, I have two new mechanisms which are random switch and unbiased local Homogenization and subtly biased advantage but with state dependent coefficient involved. The most crucial one, the existence of high-frequency duplication generating the diffusion term and noise term in each case identifies the limit equation as SPDE driven by space time white noise.

Equations

$${\Delta }_n(f)(x):={\mathcal{H}}_n\sum _{y\sim x}\frac{1}{{\mathcal{N}}_n}(f(y)-f(x)).$$

$$\begin{array}[]{ccc}-1&0&1\\ (1-d_{n})d_{n}&1-2d_{n}(1-d_{n})&d_{n}(1-d_{n})\end{array}$$

$${\partial }_t(d_n{\rho }_n)=\sqrt{2d_n(1-d_n)({\mathcal{H}}_n{\rho }_n)}\dot{W},$$

$${\partial }_t{u}_n=\sqrt{2u_n(1-u_n{\mathcal{S}}_n){\mathcal{H}}_n/({\mathcal{S}}_n{\rho }_n)}\dot{W}.$$

$${\partial }_t(d_n{\rho }_n)=(d_n{\rho }_n){\mathcal{L}}_n/{\mathcal{N}}_n\cdot {\mathcal{N}}_n(1-d_n).$$

$${\partial }_t{u}_n={\mathcal{L}}_n{u}_n(1-u_n{\mathcal{S}}_n).$$

$${\partial }_t{u}_n={\mathcal{L}}_n{u}_n(1-u_n{\mathcal{S}}_n)-{\mathcal{L}}_n(1-u_n{\mathcal{S}}_n)u_n=0.$$

$${\partial }_t(d_n{\rho }_n)=(-d_n{\rho }_n){\mathcal{L}}_n/{\mathcal{N}}_n^2\cdot {\mathcal{N}}_n(1-d_n)\cdot {\mathcal{N}}_n(1-d_n).$$

$${\partial }_t{u}_n=-{\mathcal{L}}_n{u}_n(1-u_n{\mathcal{S}}_n{)}^2.$$

$${\partial }_t{u}_n={\mathcal{L}}_n\left[((1-u_n)u_n^2-u_n(1-u_n{)}^2\right]=2{\mathcal{L}}_n{u}_n(u_n-\frac{1}{2})(1-u_n)$$

$${\partial }_t{u}_n=P(u_n),$$

$${\partial }_t{u}_n{\mathcal{S}}_n=-u_n{\mathcal{S}}_n{\mathcal{L}}_n(u_n)$$

$${\partial }_t(d_n{\rho }_n)=(d_n{\rho }_n){\mathcal{L}}_n(u_n)/{\mathcal{N}}_n\cdot {\mathcal{N}}_n(1-d_n),$$

$${\partial }_t{u}_n=-u_n{\mathcal{L}}_n(u_n)$$

$${\partial }_t{u}_n={\mathcal{L}}_n(u_n)u_n(1-u_n{\mathcal{S}}_n).$$

$${\partial }_{t}u=-uP(u)$$

$${\partial }_{t}u=P(u)u(1-u).$$

$${\partial }_{t}u=\frac{1}{3}\text{Δ}u+4u(u-\frac{1}{2})(1-u)-uP(u)+2\sqrt{u(1-u)}\dot{W},u(0,x)=f_0.$$

$${\partial }_{t}u=\frac{1}{3}\text{Δ}u-uP(u)+2\sqrt{u(1-u)}\dot{W},u(0,x)=f_0.$$

$${\partial }_{t}u=\frac{1}{6}\text{Δ}u+\sqrt{2u}\dot{W},u(0,x)=f_0.$$

$${\partial }_{t}u=\dot{W,}u(0,x)=f_0=\frac{1}{2}.$$

$$(P_s(x,y):x,y\in \mathbb{Z}/{\rho }_n,x\sim y)withrate{\mathcal{H}}_n/{\mathcal{N}}_n=n^{1/2},$$

$$({\hat{P}}_s(x):x\in \mathbb{Z}/{\rho }_n)withrateP(u_n),$$

$$({\tilde{P}}_s(x,y,z):x,y,z\in \mathbb{Z}/{\rho }_n,y\sim x,z\sim x)withrate{\mathcal{L}}_n^c/{\mathcal{N}}_n^2=2\theta n^{-1}.$$

$${\xi }_t^n(x)$$

$$+\sum _{y\sim x}\sum _{z\sim x}\underset{0}{\overset{t}{\int }}(1-{\xi }_{s-}^n(x)){\xi }_{s-}^n(y){\xi }_{s-}^n(z)d{\tilde{P}}_s(x,y,z)$$

$$-\sum _{y\sim x}\sum _{z\sim x}\underset{0}{\overset{t}{\int }}{\xi }_{s-}^n(x)(1-{\xi }_{s-}^n(y))(1-{\xi }_{s-}^n(z))d{\tilde{P}}_s(x,y,z)$$

$$={\xi }_0^n(x)+\sum _{y\sim x}\underset{0}{\overset{t}{\int }}({\xi }_{s-}^n(y)-{\xi }_{s-}^n(x))d{P}_s(x,y)-\underset{0}{\overset{t}{\int }}{\xi }_{s-}^n(x)d{\hat{P}}_s(x)$$

$$-\sum _{y\sim x}\sum _{z\sim x}\underset{0}{\overset{t}{\int }}{\xi }_{s-}^n(x)d{\tilde{P}}_s(x,y,z)$$

$$+\sum _{y\sim x}\sum _{z\sim x}\underset{0}{\overset{t}{\int }}(\sum _{p(x,y,z)}{\xi }_{s-}^n(x){\xi }_{s-}^n(y))d{\tilde{P}}_s(x,y,z)$$

$$-2\sum _{y\sim x}\sum _{z\sim x}\underset{0}{\overset{t}{\int }}{\xi }_{s-}^n(x){\xi }_{s-}^n(y){\xi }_{s-}^n(z)d{\tilde{P}}_s(x,y,z).$$

$$\underset{0}{\overset{T}{\int }}\left(\left|{\varphi }_s\right|+{\varphi }_s^2+\left|{\partial }_s{\varphi }_s\right|,1\right)ds<\infty .$$

$$(v_t,{\varphi }_t)$$

Full text

Limit of 1-Dimensional Mixed-Mechanism Interacting Particle System

Model

Tong Zhao

Introduction

Elaborating on the model from voter process with mixed-mechanism under suitable scaling, I have two new mechanisms which are random switch and unbiased local Homogenization and subtly biased advantage but with state dependent coefficient involved. The most crucial one, the existence of high-frequency duplication generating the diffusion term and noise term in each case identifies the limit equation as SPDE driven by space time white noise.

At the beginning, presenting the notations (parameters in the model) used in this paper is necessary for convenience. $\mathcal{N}_{n}$ is the number of neighbors of a voter, $\mathcal{D}_{n}$ is the distance between two voters who neighbor each other at most, $\mathcal{H}_{n}$ is the rate of high-frequency mechanism, $\mathcal{L}_{n}$ is the rate of low-frequency mechanism, $\mathcal{S}_{n}$ is the scale of density and $\mathcal{\rho}_{n}:=\mathcal{N}_{n}/(2\mathcal{D}_{n})$ is the density of voters in the nth mode.

For any fixed $n\in\mathbb{Z}$, a classic 1-dimensional model is the lattice scale is determined by $\mathbb{Z}/\mathcal{\rho}_{n}$ in its nth model. There is a voter on each lattice who is an advocate of either $\boldsymbol{\mathbf{A}}\mathit{}$ or $\boldsymbol{a}$ . If $x,y\in\mathbb{Z}/\mathcal{\rho}_{n}$ and $|x-y|\leqslant\mathcal{D}_{n}$, we regard $x,y$ as neighbors denoted by $x\sim y$.

We will use $d_{n}(t,x)$, $u_{n}(t,x)$ and $u(t,x)$ to denote the density, the $\mathcal{S}_{n}$-scaled density in the nth model and its limit of advocates of $\boldsymbol{\mathbf{A}}$ at $x$ point and time $t$respectively. We can also consider this model on a ring $(\mathbb{Z}/n)/\mathbb{Z}$ as its nth model. Under this circumstance, we need $u$ and $\dot{W}$ of period $1$ in final equation.

In a particular model: $\mathcal{N}_{n}=2n^{1/2},\mathcal{D}_{n}=n^{-1/2},\mathcal{H}_{n}=2n,\mathcal{L}_{n}=2\theta,\mathcal{S}_{n}=1,\mathcal{\rho}_{n}=n,d_{n}=\mathcal{S}_{n}u_{n}$.

Mechanisms

The initiative, 1-dimensional voter process can converge to a SPDE driven by space time white noise with various drift terms—especially the bistable drift term— in our model in vague sense, was motivated by Allee effect. In time, we find it can be generalized to voter process and then go a step further to a more general form.

1. High-Frequency Unbiased Oscillation Mechanism

1.1. Symmetric Duplication

The reason we call it high-frequency unbiased oscillation mechanism is its rate must be high enough to generate the Laplacian term and the white noise term and is not related to the number of neighbors of each individual on $\mathbb{Z}/n$. For anyone of two individuals $x,y$ neighboring one another adopting the view of the other in a period of time (i.e. $[0,t]$) indefinitely times subject to Poisson process at rate $\mathcal{H}_{n}/\mathcal{N}_{n}$ in the nth model, these Poisson processes labeled by $P_{t}(x,y)$ are independent of each other with ordered pairs $(x,y)$.

Let consider how this mechanism gives birth to laplacian term and white noise term. We define:

[TABLE]

Whence \text{\Delta}_{n} is a generator of a random walk at rate $\mathcal{H}_{n}$ , which is the totally rate of diffusion of density if substituting approximate local density $u_{n}$ for $f$. The transition probability of this random walk is uniformly distributed on the lattices in the neighborhood of a specific point. So the variance of the distribution of the transition probability is $\mathcal{D}_{n}^{2}/3$. Since each jump of the random walk is independent of one another and the expectation of the number of jumps in unit time is $\mathcal{H}_{n}$, the variance of the random walk in unit time is $\mathcal{D}_{n}^{2}\cdot\mathcal{H}_{n}/3$. Therefore, we require the convergence of $\mathcal{D}_{n}^{2}\cdot\mathcal{H}_{n}/3$ upon $n$ tending to infinity, assuming this limit is $\sigma^{2}$. According to local central limit theorem, we have the laplacian term \frac{\sigma^{2}}{2}\text{\Delta}u .

As to the white noise term, so $d_{n}\rho_{n}$ is the density of $\boldsymbol{\mathbf{A}}$ in the neighborhood of a certain point x (i.e. the number of $\boldsymbol{\mathbf{A}}$ on a unit interval of space). In a unit interval of time, the expectation of the number of times of occurrence of adopting view of others on a unit interval whose density is uniform and identified with $\mathcal{\rho}_{n}$ is $\mathcal{H}_{n}\mathcal{\rho}_{n}$. The distribution of increment of the density upon the next adopting occurring is:

[TABLE]

Hence we have:

[TABLE]

which leads to

[TABLE]

There we require some convergence property to get a non-trivial term.

2. Low-Frequency Drift Mechanism

Now, we will pay attention to the significant part, low-frequency drift mechanism. The following are various types of them, whose occurrence related to a lattice point $x$ is of the same order as $\mathcal{L}_{n}$.

2.1. Mutation (random switching)

This mechanism in voter process model means every voter switches their views or changes their minds somehow, maybe on a whim or something irrational. Assuming the rate of the mutation of a kind of voter, $\mathcal{L}_{n}$, we are handy to conclude that if $\boldsymbol{\mathbf{A}}$ mutates into $\boldsymbol{a}$at rate $\mathcal{L}_{n}$, we obtain the term $-u_{n}\mathcal{L}_{n}$, and in turn if $\boldsymbol{a}$mutates into $\boldsymbol{\mathbf{A}}$ at rate $\mathcal{L}_{n}$, we obtain the term $(\mathcal{S}_{n}^{-1}-u_{n})\mathcal{L}_{n}$ by observing the quantity of voter $\boldsymbol{\mathbf{A}}$ in density sense. Specially, if the mutation is unbiased, we have a term $(\mathcal{S}_{n}^{-1}-2u_{n})\mathcal{L}_{n}$ .

2.2. Asymmetric Duplication

We always come into a situation where advocates of the party, $\boldsymbol{\mathbf{A}}$, are more active and aggressive than the other $\boldsymbol{a}$. That means $\boldsymbol{\mathbf{A}}$ has an extra frequency to change $\boldsymbol{a}$’s mind or $\boldsymbol{\mathbf{A}}$’s view is more easily adopted by $\boldsymbol{a}$. Similar to high-frequency mechanism, we use $\widetilde{P}_{t}(x,y)$ to represent the process that account the number of times of $x$trying adopting $y$’s opinion in time interval $[0,t]$, but only if y is an advocate of $\boldsymbol{\mathbf{A}}$ when $\widetilde{P}_{t}(x,y)$ jumps, x succeeds in adopting y’s opinion $\boldsymbol{\mathbf{A}}$. $\widetilde{P}_{t}(x,y)$ is subject to Poisson process with expectation $\mathcal{L}_{n}/\mathcal{N}_{n}$. Same as the above considering a neighborhood of a point x, we have:

[TABLE]

Hence,

[TABLE]

Remark*.*

This mechanism could not be symmetric, otherwise

[TABLE]

This implies compensation, hence no effect.

2.3. Local Homogenization ( multi-consulting )

Some voters may be very stubborn and discreet or even scrupulous, their points of views are more stable. For example, every stubborn individual $x$ inquiring two neighbors $y,z$ several times subject to Poisson process with a fixed rate $\mathcal{L}_{n}/\mathcal{N}_{n}^{2}=\widetilde{P}_{t}(x,y,z)$ about their views in the time interval $\left[0,t\right]$ will alter his or hers at a jumping moment of $\widetilde{P}_{t}(x,y,z)$ only if theirs all differ from his or hers at that time. If voters for $\boldsymbol{\mathbf{A}}$ are stubborn, we considering the neighborhood of a fixed point on lattice obtain:

[TABLE]

Thereby:

[TABLE]

However, if voters for $\boldsymbol{a}$ are also stubborn, we obtain

[TABLE]

similarly without loss of the highest order term, which leads to a symmetric bistable structure if $\mathcal{L}_{n}\rightarrow constant,\mathcal{S}_{n}\rightarrow 1$.

2.4. Polynomial

Following the above suit, we consider the case every stubborn individual $x$ inquiring $m$ neighbors $y_{1},y_{2},\cdots,y_{m}$ several times subject to Poisson process with a fixed rate $\mathcal{L}_{n}/\mathcal{N}_{n}^{m}=\widetilde{P}_{t}(x,y_{1},y_{2},\cdots,y_{m})$ about their views in the time interval $\left[0,t\right]$ will alter his or hers at a jumping moment of $\widetilde{P}_{t}(x,y_{1},y_{2},\cdots,y_{m})$ only if theirs all differ from his or hers at that time. Similarly, we obtain an m-order polynomial in drift term.

If $\mathcal{L}_{n}\rightarrow constant,\mathcal{S}_{n}\rightarrow 1$:

1. (1)

As in the above case, we know $\frac{1}{2}$ is the only zero point between 0 and 1 whatever $m$ is, which also leads to a symmetric bistable structure. 2. (2)

and if we change m and modify the relative intensity to make $\boldsymbol{\mathbf{A}}$ and $\boldsymbol{a}$ is not symmetric, we can have a general formula:

[TABLE]

where $P(\cdot)$ is a polynomial.

2.5. State dependence

In some cases, the rate $\mathcal{L}_{n}$ of a point $x$ may depend on the state of its neighborhood. Considering $\mathcal{L}_{n}(u_{n})$ as a function of $u_{n}$, we apply this assumption to 2.1, 2.2 (others can be prove in the same way)and capture the equations below:

[TABLE]

[TABLE]

i.e.

[TABLE]

[TABLE]

If $\mathcal{S}_{n}\rightarrow constant,\mathcal{L}_{n}(u_{n})=P(u_{n})$ :

[TABLE]

[TABLE]

Preliminaries and Description of the Theorem

Choosing different mechanism and modifying parameters at the beginning appropriately to warrant the convergence, we would get a non-trivial SPDE.

Set :

1. (1)

$D(f,\delta)(x):=sup\left\{\left|f\left(y\right)-f\left(x\right)\right|:\left|y-x\right|\leqslant\delta,y\in\mathbb{Z}/\mathbb{\rho}_{n}\right\}$, for $x\in\mathbb{Z}/\mathbb{\rho}_{n},\delta>0$. 2. (2)

$\xi_{t}^{n}(x)\in\left\{0,1\right\}$ is an identifier of state of voter at $x\in\mathbb{Z}/\mathbb{\rho}_{n}$ and time $t$, without lose of generality, $\boldsymbol{\mathbf{A}}$, $\boldsymbol{a}$ corresponding to 1 and 0 respectively. Then the dynamics of $\xi_{t}^{n}(x)$ are noted according to various mechanism. 3. (3)

$A(f)(x):=\left(\mathcal{N}_{n}\mathcal{S}_{n}\right)^{-1}\underset{y\sim x}{\sum}f(y)$ for $x\in\mathbb{Z}/\mathbb{\rho}_{n}$ , then linearly interpolated. 4. (4)

$u_{n}(t,x):=A(\xi_{t}^{n})(x)$ . 5. (5)

$\hat{u}_{n}(t,x):=u_{t}(x)-(v_{0},\psi_{t}^{x}).$ 6. (6)

$\tilde{u}_{n}(t,x):=\hat{u}_{n}(t,x)$ on the lattice $z\in\mathbb{Z}/\mathcal{\rho}_{n},\;t\in\mathbb{N}/(n\mathcal{\rho}_{n})$, then linearly interpolate first in $x$ and then in $t$ to obtain a continuous $\mathcal{\mathscr{C}}$ valued process. 7. (7)

v_{t}^{n}(x):=$$\left(\mathcal{\rho}_{n}\mathcal{S}_{n}\right)^{-1}\underset{x}{\sum}\delta_{x}I\text{\left(\xi_{t}^{n}(x)=1\right)} the measure valued process. 8. (8)

$(f,g):=\mathcal{\rho}_{n}^{-1}\underset{x}{\sum}f(x)g(x)$ for $f,g:\mathbb{Z}/\mathcal{\rho}_{n}\longrightarrow\mathfrak{\mathbb{R}}$ and $(v,f)=\int fdv$ for $f:\mathbb{Z}/\mathcal{\rho}_{n}\longrightarrow\mathfrak{\mathbb{R}},v\in\mathcal{M}(\mathbb{Z}/\mathcal{\rho}_{n})$. 9. (9)

$e_{\lambda}(x):=e^{\lambda\left|x\right|}$ for all $x\in\mathcal{\mathbb{R}}$. 10. (10)

$\mathscr{C:=}\left\{f:\mathbb{R\rightarrow}[0,\infty)\;continuous\;with\;\left|f(x)e_{\lambda}(x)\right|\rightarrow 0\;as\;\left|x\right|\rightarrow\infty\;for\;all\;\lambda<0\right\}$ . 11. (11)

$\left\|f\right\|_{\lambda}:=\underset{x}{\sup}\left|f(x)e_{\lambda}(x)\right|$ 12. (12)

$\underset{p(x,y,z)}{\sum}f(x,y,z):=f(x,y,z)+f(z,x,y)+f(y,z,x)$ 13. (13)

$p(\sigma^{2},\cdot)$ is density function of centered normal distribution with variance $\sigma^{2}$. 14. (14)

$C$ , a constant having nothing to do with our interest, is different from line to line. 15. (15)

$\left|P(\cdot)\right|:=\stackrel{{\scriptstyle[}}{{i}}=0]{n}{\sum}\left|p_{i}\right|,\quad if\;P(x)=\stackrel{{\scriptstyle[}}{{i}}=0]{n}{\sum}p_{i}x^{i}.$

In the following argument, we will omit superscript $n$ without ambiguity.

Theorem**.**

Upon $n$ tend to infinity, and $u_{n}(0,x)$ converge to $f_{0}$ in $\mathscr{C}$ sense. Then $u_{n}(t,x)$ converge in distribution sense to a continuous $\mathscr{C}$ valued process $u(t,x)$ which solves the following SPDEs under respective conditions.

1. (1)

If you choose symmetric duplication as high-frequency and multiple consulting and asymmetric state dependent mutation as low-frequency mechanism respectively, then tune parameters to $\mathcal{N}_{n}=2n^{1/2},\mathcal{D}_{n}=n^{-1/2},\mathcal{H}_{n}=2n,\mathcal{L}_{n}^{c}=2,\mathcal{L}_{n}^{m}=P(u_{n}),\mathcal{S}_{n}=1,\mathcal{\rho}_{n}=n,d_{n}=\mathcal{S}_{n}u_{n},$ $u(t,x)$ will fit

[TABLE] 2. (2)

If you choose symmetric duplication and asymmetric state dependent mutation as high-frequency and low-frequency mechanism respectively, then tune parameters to $\mathcal{N}_{n}=2n^{1/2},\mathcal{D}_{n}=n^{-1/2},\mathcal{H}_{n}=2n,\mathcal{L}_{n}(u_{n})=P(u_{n}),\mathcal{S}_{n}=1,\mathcal{\rho}_{n}=n,d_{n}=\mathcal{S}_{n}u_{n}$, $u(t,x)$will fit

[TABLE] 3. (3)

If you choose asymmetric duplication and mutation as high-frequency mechanism and no low-frequency mechanism , then tune parameters to $\mathcal{N}_{n}=2n^{3/2},\mathcal{D}_{n}=n^{-1/2},\mathcal{H}_{n}=n,\mathcal{L}_{n}=0,\mathcal{S}_{n}=n^{-1},\mathcal{\rho}_{n}=n^{2},d_{n}=\mathcal{S}_{n}u_{n}$, $u(t,x)$ will fit

[TABLE] 4. (4)

If you only choose symmetric mutation as high-frequency and no low-frequency mechanism, then tune parameters to $\mathcal{N}_{n}=2n^{1/2},\mathcal{D}_{n}=n^{-1/2},\mathcal{H}_{n}=n,\mathcal{L}_{n}=0,\mathcal{S}_{n}=1,\mathcal{\rho}_{n}=n,d_{n}=\mathcal{S}_{n}u_{n}$, $u(t,x)$ will fit

[TABLE]

The conditions and parameters are representative, you can follow my suit to derive similar SPDE with various combinations of mechanisms from following process. Therefore we will only give the proof of case 1.

All mechanisms of one kind are subject to i.i.d Poisson processes, and process between different kinds are independent mutually:

voter $x$ takes value of $y$

[TABLE]

state dependent mutation from $\boldsymbol{\mathbf{A}}\mathit{}$ to $\boldsymbol{a}$

[TABLE]

voter x consults $y$and $z$

[TABLE]

The dynamics of the process in case 1 is described below:

[TABLE]

Then take a test function $\phi:[0,\infty)\times\mathbb{Z}/\mathcal{\rho}_{n}\rightarrow\mathbb{R}$ with $t\rightarrow\phi_{t}(x)$ continuously differentiable and satisfying

[TABLE]

Implementing integration by parts, for $t\leqslant T$, we have

[TABLE]

3. Semi-Martingale Decomposition

3.1. Laplacian Term

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where

[TABLE]

is a martingale with brackets process given by

[TABLE]

Alternatively, we bound it by

[TABLE]

3.2. Drift Term

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where

[TABLE]

is a martingale with brackets process given by

[TABLE]

Alternatively, if $-1\leqslant u_{t}\leqslant 1$,we bound it by

[TABLE]

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where

[TABLE]

is a martingale with brackets process given by

[TABLE]

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where

[TABLE]

is a martingale with brackets process given by

[TABLE]

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where

[TABLE]

is a martingale with brackets process given by

[TABLE]

3.3. White-Noise Term

We break the term into two parts, a fluctuation term and an average term

[TABLE]

where $Z_{t}(\phi)$ is a martingale with brackets process given by

[TABLE]

Collecting all terms above, we get the final semi-martingale decomposition:

[TABLE]

4. Green’s Function Representation

We need to choose a special test function to the second term in the above final semi-martingale decomposition. For each $z\in\mathbb{Z}/\mathcal{\rho}_{n}$, we define:

1. (1)

a function $\psi_{t}^{z}$ to be the unique solution of

[TABLE] 2. (2)

a random walk $X_{t}$ with generator $\Delta_{n}$ jumping at rate $\mathcal{H}_{n}$ with symmetric steps of variance $\mathcal{D}_{n}^{2}/3.$ 3. (3)

$\bar{\psi}_{0}^{z}(x):=\mathcal{\rho}_{n}P(X_{t}=x|X_{0}=z).$

Maybe you note there is a relation between $\psi_{t}^{z}$ and $\bar{\psi}_{t}^{z}$, i.e.

[TABLE]

and a property that

[TABLE]

Then linearly interpolate $\psi_{t}^{z}$ and $\bar{\psi}_{t}^{z}$, you will get a convergence of them to $p(\mathcal{D}_{n}^{2}\mathcal{H}_{n}t/3,x-z)$.

4.1. Property of $\psi_{t}^{z}$ and $\bar{\psi}_{t}^{z}$

Under different coefficients, we have similar estimations associated with $\psi_{t}^{z}$ and $\bar{\psi}_{t}^{z}$ to [1, Lemma 3]. In our case we can use the conclusion.

Set $\phi_{s}=e^{-\mathcal{L}_{n}^{c}(t-s)}\psi_{t-s}^{x}$ for $s\leqslant t$ and substitute it into the final semi-martingale decomposition equation then the second term vanishes and $(v_{t},\phi_{t})=u_{t}(x)$, i.e. $u_{n}(t,x)$. Hence it turns out that:

[TABLE]

Because we will only use this representation of $u_{n}(t,x)$ to estimate moment required for the proof of tightness ($e^{-\mathcal{L}_{n}^{c}(t-s)}$ is bounded and positive), we reduce it to

[TABLE]

for convenience without loss of generality.

If $\mathcal{L}_{n}^{c},\mathcal{S}_{n}$ are constant, for simplicity, we rewrite the above formula:

[TABLE]

where

[TABLE]

Lemma 1**.**

If $f:\mathbb{Z}/\rho_{n}\longrightarrow[0,\infty)$ with $(f,f)<\infty,\lambda\in\mathbb{R},m\in\mathbb{N^{+}},$ then:

1. (1)

$(v_{t},\psi_{s}^{z})=(u_{t},\bar{\psi}_{s}^{z}),$ 2. (2)

$\left|(v_{t},f)-(u_{t},f)\right|\leqslant\left\|D(f,\mathcal{D}_{n})\right\|_{\lambda}(v_{t},e_{-\lambda}),$ 3. (3)

$(v_{t},u_{t}^{m}e_{-\lambda})\leqslant C(m)(u_{t}^{m+1},e_{-\lambda})$ (if $u_{t}(x)$ is bounded).

Proof:

[TABLE]

[TABLE]

so

[TABLE]

[TABLE]

i.e.

[TABLE]

Then substitute $u_{t}^{m-1}e_{-\lambda}$ for $f$, we obtain:

[TABLE]

From now on, the incoming method is valid for the cases where $\mathcal{S}_{n}=1,0\leqslant u_{n}(0,x)\leqslant 1$, since the method branches according to $\mathcal{S}_{n}$ with different value, however, we still use notation $\mathcal{S}_{n}$ to identify its trace.

Lemma 2**.**

For $T\geqslant 0,\;p\geqslant 2,\lambda>0,$ we have:

[TABLE]

Proof:

For $i\geqslant 2$, we can prove it in an identical way. The greatest jump of these martingale is bounded by $\left(\mathcal{\rho}_{n}\mathcal{S}_{n}\right)^{-1}\underset{s\geqslant 0}{\sup}\left\|\psi_{s}^{z}\right\|_{\infty}\leqslant C\left(\mathcal{\rho}_{n}\mathcal{S}_{n}\right)^{-1}n^{1/2}$. So by implementing Burkholder’s inequality, we get:

[TABLE]

For $i=1$, also use Burkholder’s inequality, we have:

[TABLE]

Let us have a look at the first term:

[TABLE]

then the second term:

[TABLE]

Finally, we get:

[TABLE]

According to our assumption at the beginning, our version of the above bound is:

[TABLE]

5. Tightness

Our objective is to prove the tightness of $u_{n}(t,x)$ by estimating moment of their discrepancy at different times and locations.

Lemma 3**.**

For $0\leqslant s\leqslant t\leqslant T,\;y,z\in\mathbb{Z}/\mathcal{\rho}_{n},\;\left|t-s\right|,\left|y-z\right|\leqslant 1,\;\lambda>0,\;p\geqslant 2,\;\mathcal{L}_{n}^{c},\mathcal{S}_{n}\;are\;constant$

[TABLE]

Proof:

[TABLE]

[TABLE]

Set

[TABLE]

First, we pay attention to the first term:

[TABLE]

Let us look at the 1st expectation:

[TABLE]

Let us look at the 2nd expectation:

[TABLE]

Let us look at the 3rd expectation:

[TABLE]

Let us look at the 4th expectation:

[TABLE]

Let us look at the 5th expectation:

[TABLE]

Let us look at the 6th expectation:

[TABLE]

Put all the above conclusions together:

[TABLE]

Similarly, we consider the second term:

[TABLE]

Let us look at the 1st expectation:

[TABLE]

Let us look at the 2nd expectation:

[TABLE]

Let us look at the 3rd expectation:

[TABLE]

Let us look at the 4th expectation:

[TABLE]

Let us look at the 5th expectation:

[TABLE]

Let us look at the 6th expectation:

[TABLE]

Let us look at the 7th expectation:

[TABLE]

Let us look at the 8th expectation:

[TABLE]

Let us look at the 9th expectation:

[TABLE]

Put all the above conclusions together:

[TABLE]

Lemma 4**.**

For any $\lambda>0,T<\infty$

1. (1)

$P\left(\underset{t\leqslant T}{\sup}\left\|\hat{u}_{t}(z)-\tilde{u}_{t}(z)\right\|_{-\lambda}\geqslant 7n^{-1/4}\right)\rightarrow 0$ as $n\rightarrow\infty$, 2. (2)

$\underset{t\leqslant T}{\sup}\left\|(v_{0},\psi_{t}^{\cdot})-P_{t/3}f_{0}\right\|_{-\lambda}\rightarrow 0$ as $n\rightarrow\infty$.

The detail of proof is in [1, Lemma 7].

From Kolmogorov’s continuity criterion and the above moment estimate, we can get the tightness of $\tilde{u}_{t}(z)$ as continuous $\mathscr{C}$ valued process. Then the tightness of $\hat{u}_{t}(z)$ follows, also the continuity of all limit points follow from the above lemma.

6. Characterizing limit points

Taking a continuous function $\phi:\mathbb{R}\longrightarrow\mathbb{R}$ with compact support, we define

[TABLE]

where $E(\underset{t\leqslant T}{\sup}\left|E_{t}^{(6)}(\phi)\right|)\leqslant C\left\|D(\phi,\mathcal{D}_{n})\right\|_{\lambda}$. From the tightness of $u_{n}(t)$, we can get the tightness of $v_{t}^{n}$ as cadlag Radon measure valued process with the vague topology once a compact containment condition is checked and all limit points are again continuous.

Because of simultaneous convergence of $u_{n}(t)$ and $v_{t}^{n}$, we write it in pair $(u_{n}(t),v_{t}^{n})$. By Skorokhod theorem, we can find random variables with the same distribution as $\xi_{t}^{n}$, which converges almost sure, and we still label it as $(u_{n}(t),v_{t}^{n})$. Since the limits are continuous, the almost sure convergence holds not only in Skorokhod sense but also in uniform sense on compact sets. Thus, with probability one, for $T<\infty,\;\lambda>0,\phi$ of compact support, we have:

[TABLE]

where $v_{t}(dx)=u_{t}(x)dx$ for all $t\geqslant 0$.

Pick up a $\phi$ three times continuously differentiable and with compact support then substitute it for $\phi_{t}$ into (4.1): we have:

[TABLE]

When n goes to infinity,

$E_{t}^{(i)}(\phi)\;\left(1\leqslant i\leqslant 5\right)$ tend to zero for all t almost surely, from our assumption in case 1.

$\Delta_{n}(\phi)(x)$ tends to $\frac{1}{3}\Delta$ uniformly.

$\mathcal{L}_{n}^{c},\mathcal{S}_{n}$ are constants.

Therefore, $Z_{t}(\phi)$ tends to a continuous local martingale $z_{t}(\phi)$ where

[TABLE]

From (3.1), there exists

[TABLE]

where

[TABLE]

Fortunately, we have a bound

[TABLE]

So

[TABLE]

is a continuous martingale.

Hence we have

[TABLE]