Glauber 过程的拌和时间（第二篇）

Chapter 5

5.1　More functional analysis

We will perform more functional analysis. Here are the definitions:

The entropy \(\text{Ent}_\mu(\cdot)\) (in nats, rather than bits) is defined by

\[\text{Ent}_\mu(f)=\mathbb{E}_\mu[f\ln f]-\mathbb{E}_\mu[f]\ln\mathbb{E}_\mu[f],\qquad\forall f:\Omega\to\mathbb{R}_{\geq 0}, \]

with the convention \(0\ln 0=0\). This is consistent with \(\lim_{x\to0^+}x\ln x=0\).

The Dirichlet energy \(\mathfrak{D}(\cdot,\cdot)\) (between two functions) is defined by

\[\mathfrak{D}(f,g)=\langle f,(\mathsf{Id}-\mathsf{P})g\rangle_\mu=\langle f,g\rangle_\mu-\langle f,\mathsf{P}g\rangle_\mu,\qquad\forall f,g:\Omega\to\mathbb{R}. \]

The following computation

\[\begin{aligned} \mathfrak{D}(f,g) &=\sum_{x\in\Omega}\mu(x)f(x)g(x)-\sum_{x,y\in\Omega}w(x,y)f(x)g(y)\\[0.3em] &=\frac{1}{2}\left(\sum_{x,y\in\Omega}w(x,y)(f(x)-f(y))(g(x)-g(y))\right) \end{aligned} \]

helps view \(\mathfrak{D}(f,g)\) from a more local standpoint. In particular,

\[\mathfrak{D}(f,\ln f)=\frac{1}{2}\left(\sum_{x,y\in\Omega}w(x,y)(f(x)-f(y))(\ln f(x)-\ln f(y))\right). \]

Note that \(\mathfrak{D}(f,\ln f)\) is defined only for \(f:\Omega\to\mathbb{R}_{>0}\). To extend the definition to \(f:\Omega\to\mathbb{R}_{\geq 0}\), compute the contribution of edges with at least one endpoint mapped to zero under \(f\) as follows:

1. If two states \(x,y\) with \(w(x,y)\neq0\) satisfy \(f(x)=f(y)=0\), then the edge \((x,y)\) contributes \(0\).

2. If two states \(x,y\) with \(w(x,y)\neq0\) satisfy \(f(x)=0,f(y)\neq0\) or \(f(x)\neq0,f(y)=0\), then the edge \((x,y)\) contributes \(+\infty\). In other words, \(\mathfrak{D}(f,\ln f)=+\infty\) as soon as even a single such edge exists.

Of course, if two states \(x,y\) with \(w(x,y)\neq0\) satisfy \(f(x),f(y)\neq0\), then the edge \((x,y)\) contributes \(w(x,y)(f(x)-f(y))(\ln f(x)-\ln f(y))\), as in the original formula.

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5.2　Modified log-Sobolev inequality

The modified log-Sobolev inequality is parallel to the Poincaré inequality.

For a nonnegative real number \(\varrho\geq 0\), we say the modified log-Sobolev inequality holds with parameter \(\varrho\) if and only if

\[\varrho\cdot\text{Ent}_\mu(f)\leq\mathfrak{D}(f,\ln f),\qquad\forall f:\Omega\to\mathbb{R}_{\geq 0}. \]

A larger \(\varrho\) corresponds to a strictly stronger statement.

Let \(\varrho(\mathsf{P})\) and \(\gamma(\mathsf{P})\) denote the optimal parameters for the modified log-Sobolev inequality and the Poincaré inequality, respectively.

In Section 2.2 we proved that \(\gamma(\mathsf{P})=1-\lambda_2\). Moreover, the function \(f=d+\varphi_2\) attains the minimum value \(\frac{\mathfrak{D}(f,f)}{\text{Var}_\mu(f)}=1-\lambda_2\). Set \(d=-\min \varphi_2(x)\) so that \(f(x)\geq 0\) for all \(x\in\Omega\). In this Section 5.2, when we write \(f\), we refer to this specific function \(f=d+\varphi_2\), not an arbitrary function.

In this section we prove that \(\varrho(\mathsf{P})\leq 2\gamma(\mathsf{P})\).

For \(c>0\), define the function \(f_c=1+\frac{f}{c}\).

We need Taylor expansions with a third-order Peano remainder:

1. The function \(h_+(x)=(1+x)\ln(1+x)\) has \(h_+(x)=x+\tfrac{1}{2}x^2+O(x^3)\) around \(0\).
2. The function \(h_-(x)=\ln(1+x)\) has \(h_-(x)=x-\tfrac{1}{2}x^2+O(x^3)\) around \(0\).

Expanding pointwise and taking expectations gives

\[\mathbb{E}_\mu[f_c\ln f_c]=\tfrac{\mathbb{E}_\mu[f]}{c}+\tfrac{\mathbb{E}_\mu[f^2]}{2c^2}+O(c^{-3}). \]

Expanding \(\mathbb{E}_\mu[f_c]\ln\mathbb{E}_\mu[f_c]\) as a whole gives

\[\mathbb{E}_\mu[f_c]\ln\mathbb{E}_\mu[f_c] =\left(1+\tfrac{\mathbb{E}_\mu[f]}{c}\right)\!\left(\tfrac{\mathbb{E}_\mu[f]}{c}-\tfrac{\mathbb{E}_\mu[f]^2}{2c^2}+O(c^{-3})\right) =\tfrac{\mathbb{E}_\mu[f]}{c}+\tfrac{\mathbb{E}_\mu[f]^2}{2c^2}+O(c^{-3}). \]

Therefore

\[\text{Ent}_\mu(f_c) =\mathbb{E}_\mu[f_c\ln f_c]-\mathbb{E}_\mu[f_c]\ln\mathbb{E}_\mu[f_c] =\tfrac{\mathbb{E}_\mu[f^2]-\mathbb{E}_\mu[f]^2}{2c^2}+O(c^{-3}) =\tfrac{\text{Var}_\mu(f)}{2c^2}+O(c^{-3}).\ \mathbf{(1)} \]

At the same time,

\[(f_c(x)-f_c(y))(\ln f_c(x)-\ln f_c(y)) =\left(\tfrac{f(x)-f(y)}{c}\right)\!\left(\tfrac{f(x)-f(y)}{c}+O(c^{-2})\right) =\tfrac{(f(x)-f(y))^2}{c^2}+O(c^{-3}). \]

Therefore

\[\mathfrak{D}(f_c,\ln f_c) =\frac12\left(\sum_{x,y\in\Omega}w(x,y)(f_c(x)-f_c(y))(\ln f_c(x)-\ln f_c(y))\right) =\tfrac{\mathfrak{D}(f,f)}{c^2}+O(c^{-3}).\ \mathbf{(2)} \]

Combining the two equalities \(\mathbf{(1)}\) and \(\mathbf{(2)}\) gives

\[\frac{\mathfrak{D}(f_c,\ln f_c)}{\text{Ent}_\mu(f_c)} =\frac{2\>\!\mathfrak{D}(f,f)+O(c^{-1})}{\text{Var}_\mu(f)+O(c^{-1})} =2\gamma(\mathsf{P})+O(c^{-1}). \]

Letting \(c\to+\infty\) yields \(\varrho(\mathsf{P})\leq 2\gamma(\mathsf{P})\).

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