统计代写|随机分析作业代写stochastic analysis代写|Homogenization of Diffusions on the Lattice

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统计代写|随机分析作业代写stochastic analysis代写|Periodic Drift Coefficients

In this paper we treat limit theorems for diffusions on the lattice $\mathbf{Z}^{d}$ of the form of those constituting the solution of the homogenization problem of diffusions. For finite dimensional diffusion processes, various models of homogenization (generalized in several directions) have been studied in detail (cf. eg. $[\mathrm{F} 2, \mathrm{FNT}$, FunU, O, PapV, Par] and references therein). On the other hand, for corresponding prohlems of infinite dimensional diffusions only fow results are known (cf. [FunU, ABRY1,2,3]). In this paper we consider a homogenization problem of infinite dimensional diffusion processes indexed by $\mathbf{Z}^{d}$ having periodic drift coefficients with the period $2 \pi$ (cf. (2.1)), by applying an $L^{2}$ type ergodic theorem for the corresponding quotient processes taking values in $[0,2 \pi)^{\mathbf{z}^{d}}$ (cf. Prop. 1). The ergodic theorem which is based on a (weak) Poincaré inequality.

In [ABRY3] the same problem has been discussed by applying the uniform ergodic theorem for the corresponding quotient process, that is available by assuming that the Markov semi-group of the quotient process of the original process satisfies a logarithmic Sobolev inequality. In the same paper it has also

been shown that a homogenization property of the processes starting from an almost every arbitrary point in the state space with respect to an invariant measure of the quotient process holds (cf. also [ABRY1, ABRY2]). In this occasion, the main purpose of the present paper is the comparison between the results derived under the assumption of logarithmic Sobolev inequality and the corresponding results proven by assuming $L^{2}$ ergodic theorem based on (weak) Poincaré inequality, which is strictly weaker than the one for logarithmic Sobolev inequality (cf. [AKR, G]). This paper is a series of works on the considerations of several types of homogenization models for infinite dimensional diffusion processes.

For an adequate understanding of crucial differences between homogenization problems in finite and infinite dimensional situations, we first brietly review a simple case of the homogenization problem for finite dimensional diffusions.

On some complete probability space, suppose that we are given a one dimensional standard Brownian motion process $\left{B_{t}\right}_{t \in \mathbf{R}{+}}$and consider the stochastic differential equation for each initial state $x \in \mathbf{R}$ and each scaling parameter $\epsilon>0$ given by \begin{aligned} X^{\epsilon}(t, x)=& x+\frac{1}{\epsilon} \int{0}^{t} b\left(\frac{X^{\epsilon}(s, x)}{\epsilon}\right) d s \ &+\sqrt{2} \int_{0}^{t} a\left(\frac{X^{\epsilon}(s, x)}{\epsilon}\right) d B_{s}, \quad t \in \mathbf{R}_{+}, \end{aligned}
where $a \in C^{\infty}(\mathbf{R} \rightarrow \mathbf{R})$ is a periodic function with period $2 \pi$ which satisfies
$$\lambda \leq a(x) \leq \lambda^{-1}, \quad \forall x \in \mathbf{R},$$
for some constant $\lambda>0$ and $b(x) \equiv \frac{d}{d x} a^{2}(x)$.

统计代写|随机分析作业代写stochastic analysis代写|Fundamental Notations

Let $\mathbf{N}$ and $\mathbf{Z}$ be the set of natural numbers and integers respectively. For $d \in \mathbf{N}$ let $\mathbf{Z}^{d}$ be the $d$-dimensional lattice. We consider the problem for the diffusions taking values in $\mathbf{R}^{\mathbf{Z}^{d}}$. We use the following notions and notations:
By $\mathbf{k}$ we denote $\mathbf{k}=\left(k^{1}, \ldots, k^{d}\right) \in \mathbf{Z}^{d}$. For a subset $A \subseteq \mathbf{Z}^{d}$, we define $|A| \equiv \operatorname{card} A$. For $\mathbf{k} \in \mathbf{Z}^{d}$ and $A \subseteq \mathbf{Z}^{d}$ let
$$A+\mathbf{k} \equiv{\mathbf{l}+\mathbf{k} \mid \mathbf{l} \in A}$$
For any non-empty $A \subseteq \mathbf{Z}^{d}$, we assume that $\mathbf{R}^{A}$ is the topological space equipped with the direct product topology. For each non-empty $A \subseteq Z^{d}$, by $\mathbf{x}{A}$ we denote the image of the projection onto $\mathrm{R}^{A}$ : $$\mathbf{R}^{\mathbf{Z}^{d}} \ni \mathbf{x} \longmapsto \mathbf{x}{A} \in \mathbf{R}^{A}$$
For each $p \in N \cup{0} \cup{\infty}$ we define the set of $p$-times continuously differentiable functions with support $A: C_{A}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi\left(\mathbf{x}{A}\right) \mid \varphi \in C P\left(\mathbf{R}^{A}\right)\right}$, where $C^{P}\left(\mathbf{R}^{A}\right)$ is the set of real valued $p$-times continuously differentiable functions on $\mathbf{R}^{A}$. For $p=0$, we simply denote $C{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ by $C_{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) .$ Also we set
$$C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)|| A \mid<\infty\right}$$
$\mathcal{B}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ is the Borel $\sigma$-field of $\mathbf{R}^{\mathbf{Z}^{d}}$ and $\mathcal{B}{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ is the sub $\sigma$-field of $\mathcal{B}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ that is generated by the family $C{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$. For each $\mathbf{k} \in \mathbf{Z}^{d}$, let $\vartheta^{\mathbf{k}}$ be the shift operator on $\mathbf{R}^{\mathbf{Z}^{d}}$ such that

$$\left(v^{\mathbf{k}} \mathbf{x}\right){{\mathbf{j}}} \equiv \mathbf{x}{{\mathbf{k}+\mathbf{j}}}, \mathbf{x} \in \mathbf{R}^{\mathbf{Z}^{d}}, \mathbf{j} \in \mathbf{Z}^{d},$$
where $\mathbf{x}_{{\mathbf{k}+\mathbf{j}}}$ is the $\mathbf{k}+\mathbf{j}$-th component of the vector $\mathbf{x}$.

统计代写|随机分析作业代写stochastic analysis代写|Theorems

In [ABRY3] we have considered the homogenization problem of the sequence of the diffusions $\left{\left{\mathbb{X}^{c}(t, \mathbf{x})\right}_{t \in \mathbf{R}}\right}_{\epsilon>0}$ in the case where the the following uniform ergodicity (3.1) holds for the quotient process $\left(\left{\eta_{t}\right}_{t \geq 0}, Q_{\mathbf{y}}: \mathbf{y} \in T^{\mathbf{z}^{d}}\right)$. Here we consider the same problem for $\left{\left{\mathbb{X}^{c}(t, \mathbf{x})\right}_{t \in \mathbb{R}{+}}\right}{e>0}$ in the case where the $L^{2}$-type ergodicity holds for $\left(\eta_{t}, Q_{\mathbf{y}}: \mathbf{y} \in T^{\mathbf{Z}^{d}}\right)$, and compare the results available under these two different assumptions of (3.1) and (3.2). Each comparison will be given as a Remark following each Theorem resp. Lemma.

In the sequel we denote the uniform ergodicity (3.1) as $(\mathrm{LS})$ and the $L^{2}$ type ergodicity $(3.2)$ as (WP) respectively. We have to remark that if the

potential $\mathcal{J}$, that satisfies J-1), J-2) and J-3), satisfies in addition DobrushinShlosman mixing condition, then (3.1) holds, more precisely in this case the logarithmic Sobolev inequality (LS) holds for the Dirichlet form $\mathcal{E}(u(\cdot), v(\cdot))$ defined in Remark 2, then the stronger inequality such that the term $(c+t)^{-\alpha}$ in (3.1) is replaced by $e^{-\alpha t}$ for some $\alpha>0$ holds (cf. [S]).

Correspondingly, if $\mathcal{E}(u(\cdot), v(\cdot))$ satisfies the weak Poincare (WP) inequality, then (3.2) holds. We remark that the logarithmic Sobolev inequality is strictly stronger than the the weak Poincare inequality (cf. [RWang]).
Precisely, we define the ergodicities (LS) and (WP) as follows:
(LS) For some Gibbs state $\mu$, there exists a $c=c(\mathcal{J})>0$ and an $\alpha=$ $\alpha(\mathcal{J})>1$ which depend only on $\mathcal{J}$, such that for each $A \in \mathbf{Z}^{d}$ with $|\Lambda|<\infty$ there exists $K(A) \in(0, \infty)$ and for $\forall t>0, \forall \varphi \in C_{A}^{\infty}\left(T^{\mathbf{Z}^{d}}\right)$ the following holds
$$\left|\int_{T^{\mathbf{z}}} \varphi\left(\mathbf{y}{A}\right) p\left(t,{ }^{,}, d \mathbf{y}\right)-\langle\varphi, \mu)\right|{L^{\infty}} \leq K(\Lambda)(c+t)^{-\alpha}\left(|\nabla \varphi|_{L^{\infty}}+|\varphi|_{L^{\infty}}\right)$$
(WP) There exist $c=c(\mathcal{J})>0, \alpha=\alpha(\mathcal{J})>1$ and $K>0$, that depends only on $\mathcal{J}$, and the following holds
$$\left|\mathcal{P}{t} \varphi-<\varphi, \mu>\right|{L^{2}(\mu)} \leq K(c+t)^{-\alpha}|\varphi|_{L^{2}(\mu)}, \forall t>0, \forall \varphi \in C\left(T^{\mathbf{Z}^{d}}\right)$$
We also remark that (3.1) or (3.2) gives the uniqueness of the Gibbs state, since by (3.1) or (3.2) we see that a Gibbs state $\mu$ that satisfies (3.1) or (3.2) is the only invariant measure for $p\left(t,{ }^{-}, d \mathbf{y}\right)$, but every Gibbs state is an invariant measure. From now on we denote the unique Gibbs measure by $\mu$ (cf. [ABRY3, $\mathrm{AKR}]$ ).

统计代写|随机分析作业代写stochastic analysis代写|Periodic Drift Coefficients

λ≤一种(X)≤λ−1,∀X∈R,

统计代写|随机分析作业代写stochastic analysis代写|Fundamental Notations

：ķ我们表示ķ=(ķ1,…,ķd)∈从d. 对于一个子集一种⊆从d，我们定义|一种|≡卡片⁡一种. 为了ķ∈从d和一种⊆从d让

C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left (\mathbf{R}^{\mathbf{Z}^{d}}\right)|| 一个 \mid<\infty\right}C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left (\mathbf{R}^{\mathbf{Z}^{d}}\right)|| 一个 \mid<\infty\right}

统计代写|随机分析作业代写stochastic analysis代写|Theorems

(LS) 对于某些吉布斯状态μ，存在一个C=C(Ĵ)>0和一种= 一种(Ĵ)>1这仅取决于Ĵ, 这样对于每个一种∈从d和|Λ|<∞那里存在ķ(一种)∈(0,∞)并且对于∀吨>0,∀披∈C一种∞(吨从d)以下成立
|∫吨和披(是一种)p(吨,,,d是)−⟨披,μ)|大号∞≤ķ(Λ)(C+吨)−一种(|∇披|大号∞+|披|大号∞)
(WP) 存在C=C(Ĵ)>0,一种=一种(Ĵ)>1和ķ>0，这仅取决于Ĵ, 并且以下成立
|磷吨披−<披,μ>|大号2(μ)≤ķ(C+吨)−一种|披|大号2(μ),∀吨>0,∀披∈C(吨从d)

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