Proof. Take any Borel set $A\subset I^k$ . By the definition of the measures ${\mathbf {P}}^{(\beta ,k)}$ , we have

thus proving the claim. In the third line we applied the Markov property of ${\mathbf {P}}_{\mathrm {RW}^{(k)}}.$ The condition $|\beta |\leq \eta /k$ ensures that all moment generating functions actually exist, that is, the integrals are convergent for all ${\mathbf {x}}\in I^k$ , for example, by Hölder’s inequality which says that $\int _{I^k}e^{\beta \sum _{i=1}^k (a_i-x_i)} \boldsymbol p^{(k)}({\mathbf {x}},{\mathrm {d}}{\mathbf {a}}) \leq \prod _{i=1}^k \big (\int _{I^k}e^{k\beta (a_i-x_i)} \boldsymbol p^{(k)}({\mathbf {x}},{\mathrm {d}}{\mathbf {a}})\big )^{1/k} = M(\beta k)$ , where M as always denotes the moment generating function of the annealed one-step law as in (1.3).

Proof. The second bound follows immediately from the first, since by Hölder’s inequality we have that $\int _{I^k} e^{\lambda \sum _{i=1}^k |y_i-x_i|} q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}}) \le \prod _{i=1}^k \big ( \int _{I^k} e^{k\lambda |y_i-x_i|} q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}}) \big )^{1/k}.$

Thus we will prove the first bound. We find for $\lambda \ge 0$ and $\beta \in [-\eta /(2k),\eta /2k]$ and $1\le i \le k$ that

$$ \begin{align*} \int_{I^k} e^{\lambda |y_i-x_i|} \boldsymbol q^{(k)}_\beta ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) &= \frac{\int_{I^k} e^{\lambda |y_i-x_i|} e^{\beta \sum_{j=1}^k (y_j-x_j)}\boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) }{\int_{I^k} e^{\beta \sum_{j=1}^k (y_j-x_j)} \boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) } \\ &\leq \frac{\big(\int_{I^k} e^{2\lambda |y_i-x_i|} \boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) \big)^{1/2}\prod_{j=1}^k \big(\int_{I^k} e^{2k|\beta| |y_j-x_j|} \boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) \big)^{1/(2k)}}{e^{\int_{I^k}\beta \sum_{j=1}^k (y_j-x_j) \boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}})} } \\ &= \frac{\big(\int_I e^{2\lambda |a|} \mu(\mathrm da)\big)^{1/2}\big(\int_I e^{2k |\beta||a|} \mu(\mathrm da)\big)^{1/2}}{e^{\beta k m_1}}. \end{align*} $$

In the first line we simply applied the definition of the measures $\boldsymbol q^{(k)}_\beta $ . In the second line we used Hölder’s inequality for the numerator and Jensen for the denominator. In the last line we used the fact that the marginal law of each $y_j-x_j$ under $\boldsymbol p^{(k)} ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}})$ is just $\mu $ , and $m_1:= \int _I a\mu (\mathrm da).$

By Assumption 1.2 Item (3) the last expression is clearly finite if $\lambda =\eta /2$ and $\beta \in [-\eta /(2k),\eta /(2k)]$ , thus completing the proof.

Proof. The proof of martingality is immediate from the fact that $\mathbf E_{\mathbf {x}}^{(\beta ,k)} [f(\mathbf R_{r+1})|\mathcal F_r] = \mathscr Q_k^\beta f(\mathbf R_r)$ by Proposition 2.7. The growth condition on f ensures that all integrals and expectations are finite, for example, by Lemma 2.9.

Proof. Let us define

(2.5) $$ \begin{align}\mathcal K({\mathbf{x}},\boldsymbol\beta):= \log \int_{I^k} e^{\sum_{i=1}^k \beta_i (a_i-x_i)} \boldsymbol p^{(k)}({\mathbf{x}},{\mathrm{d}}{\mathbf{a}})\;\;\;-\;\;\;\sum_{i=1}^k\log M(\beta_i), \end{align} $$

so that by Definition 2.3 the $(4p)^{th}$ order Taylor expansion of $\mathcal K$ in the $\beta _i$ variables is given by

$$ \begin{align*}\mathcal K_{4p}({\mathbf{x}},\boldsymbol\beta)&:= \sum_{m=1}^{4p} \frac1{m!} \bigg(\sum_{j_1,...,j_m=1}^k \beta_{j_1}\cdots \beta_{j_m}\kappa^{(m;k)} (\mathbf j,{\mathbf{x}})\;\;\;-\;\;\; \sum_{\substack{1\leq j_1,...,j_m\leq k\\ j_i \text{ all same}}} \beta_{j_1}\cdots \beta_{j_m} \kappa^{(m;k)} (\mathbf j,{\mathbf{x}})\bigg) \\ &= \sum_{m=1}^{4p} \frac1{m!}\sum_{\substack{1\leq j_1,...,j_m\leq k\\ j_i \text{ not all same}}}\beta_{j_1}\cdots \beta_{j_m} \kappa^{(m,k)} (\mathbf j; {\mathbf{x}})\\&=\sum_{m=2p}^{4p} \frac1{m!}\sum_{\substack{1\leq j_1,...,j_m\leq k\\ j_i \text{ not all same}}}\beta_{j_1}\cdots \beta_{j_m} \kappa^{(m,k)} (\mathbf j; {\mathbf{x}})\end{align*} $$

In the first equality, the terms with all $j_i$ being the same comes from the $\log M(\beta _i)$ contribution to $\mathcal K$ , and uses the fact that the marginal law of $a_i-x_i$ under $\boldsymbol p^{(k)}({\mathbf {x}},{\mathrm {d}}{\mathbf {a}})$ is precisely $\mu $ for every ${\mathbf {x}}\in I^k$ . In the third line, we used Item (1) in Lemma 2.4 which guarantees that all of the joint cumulants up to order $2p-1$ vanish.

By Taylor’s theorem and the uniform exponential moment bounds in Lemma 2.9, for any fixed indices $\ell _1,...,\ell _m,$ the difference $\mathcal K-\mathcal K_{4p}$ satisfies the bound

$$ \begin{align*}\sup_{{\mathbf{x}}\in I^k} \bigg|\frac{\partial^m}{\partial \beta_{\ell_1}\cdots\partial\beta_{\ell_m}}\bigg( \mathcal K({\mathbf{x}},\boldsymbol\beta)-\mathcal K_{4p}({\mathbf{x}},\boldsymbol\beta) \bigg)\bigg|\le C\bigg|\sum_{i=1}^k|\beta_i|\bigg|^{4p+1-m} ,\end{align*} $$

uniformly over $\beta _1,...,\beta _m \in [-\eta /(2k),\eta /(2k)].$ Furthermore by Item (3) in Lemma 2.4 we have that

$$ \begin{align*}\sup_{{\mathbf{x}}\in I^k} \bigg|\frac{\partial^m}{\partial \beta_{\ell_1}\cdots\partial\beta_{\ell_m}}\bigg( \sum_{m=2p+1 }^{4p} \frac1{m!}&\sum_{\substack{1\leq j_1,...,j_m\leq k\\ j_i \text{ not all same}}}\beta_{j_1}\cdots \beta_{j_m} \kappa^{(m,k)} (\mathbf j; {\mathbf{x}})\bigg)\bigg| \\&\le C\bigg|\sum_{i=1}^k|\beta_i|\bigg|^{2p+1-m} \mathrm{F_{decay}}\big( \min_{1\le i<j\le k}|x_i-x_j|\big),\end{align*} $$

uniformly over $\beta _1,...,\beta _m \in [-\eta /(2k),\eta /(2k)].$ On the other hand, since the joint cumulant of two random variables is just their covariance, the second bullet point in Lemma 2.4 (together with Definition 2.11) implies that when $m=2p$ we have that

$$ \begin{align*}\frac1{m!}\sum_{\substack{1\leq j_1,...,j_m\leq k\\ j_i \text{ not all same}}}\beta_{j_1}\cdots \beta_{j_m} \kappa^{(m,k)} (\mathbf j; {\mathbf{x}}) = \sum_{1\le i<j\le k} \beta_i^p\beta_j^p\boldsymbol{\zeta} (x_i-x_j),\end{align*} $$

which is enough to complete the proof.

Proof. Let $\mathcal K({\mathbf {x}},\boldsymbol \beta )$ be as defined in (2.5) so that

Proposition 2.12 with $m=1$ clearly implies that this quantity is bounded above by

$$ \begin{align*}C|\beta|^{2p-1}\mathrm{F_{decay}}\big(\min_{1\le i<j\le k}|x_i-x_j|\big) +C|\beta|^{4p},\end{align*} $$

thus proving the first bound (actually Proposition 2.12 seems to give a better bound with factor $|\beta |^{2p}$ as opposed to $|\beta |^{2p-1}$ , but this would ignore the term of order $2p$ that had already been subtracted there, which we must take into account here). Next, note that

Proposition 2.12 with $m=2$ clearly implies that this quantity is bounded above by

$$ \begin{align*}C|\beta|^{2p-2}\mathrm{F_{decay}}\big( \min_{1\le i<j\le k}|x_i-x_j|\big) +C|\beta|^{4p-1}.\end{align*} $$

But the first bound readily implies that

with C independent of ${\mathbf {x}}\in I^k$ and $\beta \in [-\eta /(2k),\eta /(2k)]$ . Together with the previous expression, this is enough to imply the second and third bounds.

Proof. For the upper bound, we have that

$$ \begin{align*} (\mathscr Q^\beta_k-\mathrm{Id}) u_{\mathbf{x}}^{ij}({\mathbf{y}})&= \int_{I^k} \bigg(\big|a_i-a_j-(x_i-x_j)\big|-\big|y_i-y_j-(x_i-x_j)\big|\bigg) \boldsymbol q^{(k)}_\beta ({\mathbf{y}},{\mathrm{d}}{\mathbf{a}}) \\& \leq \int_{I^k} |a_i-a_j - (y_i-y_j)| \boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}} {\mathbf{a}}). \end{align*} $$

From here the uniform bound follows from the uniform exponential moment bounds on the increments, see Lemma 2.9.

For the lower bound, use Jensen’s inequality to say

$$ \begin{align*} \mathscr Q^\beta_k u_{{\mathbf{x}}}({\mathbf{y}}) & = \int_{I^k} |a_i-a_j-(x_i-x_j)| \boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}}{\mathbf{a}}) \\&\ge \bigg| \int_{I^k} (a_i-a_j)\boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}}{\mathbf{a}}) - (x_i-x_j)\bigg| \\&\ge |y_i-y_j - (x_i-x_j)| - \bigg| y_i-y_j - \int_{I^k} (a_i-a_j)\boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}}{\mathbf{a}})\bigg| \\&= u_{{\mathbf{x}}}({\mathbf{y}}) - \bigg| y_i-y_j - \int_{I^k} (a_i-a_j)\boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}}{\mathbf{a}})\bigg|. \end{align*} $$

Thus we just need to show that

(2.7) $$ \begin{align}\bigg| y_i-y_j - \int_{I^k} (a_i-a_j)\boldsymbol q^{(k)}_\beta({\mathbf{y}},{\mathrm{d}}{\mathbf{a}})\bigg|\leq C|\beta|^{4p} +C|\beta|^{2p-1}\mathrm{F_{decay}}\big(\min_{1\le i'<j'\le k}|y_{i'}-y_{j'}|\big). \end{align} $$

For this, first note that $y_i-y_j = \int _{I^k} (a_i-a_j) \boldsymbol p^{(k)}({\mathbf {y}},{\mathrm {d}}{\mathbf {a}})$ where we recall that $\boldsymbol p^{(k)} = \boldsymbol q_0^{(k)}$ . Thus use the definition (2.4) of $\boldsymbol q^{(k)}_\beta $ and Taylor expand the exponentials to order $4p-1$ around $\beta =0$ , and note that all terms of order less than $2p-1$ vanish identically thanks to the deterministic moments condition in Assumption 1.2 Item (4), and the fact that $\mu ^{\otimes k}$ is a permutation-invariant measure. Then note that all terms of order between $2p-1$ and $4p-1$ contribute (at worst) a term of the form $\mathrm {F_{decay}}\big ( \min _{1\le i<j\le k} |x_i-x_j|\big )$ thanks to the moment mixing condition in Assumption 1.2 Item (5). Finally note that the remainder in the Taylor expansion is at worst of order $C|\beta |^{4p},$ which when combined with the exponential moment bounds of Lemma 2.9 completes the proof of (2.7).

Proof. Note by the definition of $\boldsymbol {p}_{\mathbf {dif}}$ that

$$ \begin{align*} \int_I a^2 \boldsymbol{p}_{\mathbf{dif}}(r,\mathrm da)-r^2 &=\int_I (a-r)^2 \boldsymbol{p}_{\mathbf{dif}}(r,\mathrm da) \\&= \int_{I^2} (x-r-y)^2 \boldsymbol p^{(2)}((r,0),({\mathrm{d}} x,{\mathrm{d}} y))\\&=2m_2-2\int_{I^2} (x-r)y\;\boldsymbol p^{(2)}((r,0),({\mathrm{d}} x,{\mathrm{d}} y)), \end{align*} $$

therefore we have that

$$ \begin{align*}\bigg| 2(m_2-m_1^2)- \bigg(\int_I a^2 \boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da) -x^2\bigg) \bigg| = 2\bigg|\int_{I^2} (x-r)y\;\boldsymbol p^{(2)}((r,0),({\mathrm{d}} x,{\mathrm{d}} y))-m_1^2\bigg| \leq 2\mathrm{F_{decay}}(|x|), \end{align*} $$

where we used Assumption 1.2 Item (5). But $m_2-m_1^2 =1$ by Assumption 1.2 Item (3), thus proving (2.8). To prove the second statement, note by the first bullet point of Assumption 1.2 Item (6) and the uniform exponential moment bounds of Lemma 2.9 that $y\mapsto \int _I (y-a)^2 \boldsymbol {p}_{\mathbf {dif}}(y,{\mathrm {d}} a)$ is a continuous function of y. In particular it achieves its minimum on any compact interval. If the minimum was $0$ , then $\int _I (y_0-a)^2 \boldsymbol {p}_{\mathbf {dif}}(y_0,{\mathrm {d}} a)=0$ for some $y\in I$ . Thus using the fact that $\boldsymbol {p}_{\mathbf {dif}}(y_0,\cdot )$ has mean $y_0$ , it would follow that $y_0$ is an absorbing state for the chain. But this would contradict the second bullet point of Assumption 1.2 Item (6).

Proof. We may apply Burkholder-Davis-Gundy, then Jensen, then the definition of $\mathcal S_q(M;P)$ in that order to obtain uniformly over $r\ge s\ge 0$

$$ \begin{align*} E [ |M_r-M_s|^q ] &\leq C_q E \bigg[ \bigg( \sum_{n=s}^{r-1} (M_{n+1}-M_n)^2\bigg)^{q/2}\bigg] \\ &\leq C_q E \bigg[ (r-s)^{\frac{q}2 - 1} \sum_{n=s}^{r-1} |M_{n+1}-M_n|^q \bigg] \\ &\leq C_q (r-s)^{\frac{q}2} \mathcal S_q(M;P). \end{align*} $$

Since the constant $C_q$ in Burkholder-Davis-Gundy is known to be univeral, the claim follows.

Proof. By the Markov property and the fact that there is a supremum over ${\mathbf {x}}$ , it suffices to prove the claim when $r'=0$ , so that $R^i_{r'}-R^j_{r'} = x_i-x_j$ where ${\mathbf {x}}=(x_1,...,x_k)$ in coordinates. We can also assume without any loss of generality that $F\geq \mathrm {F_{decay}}$ because otherwise one may replace F with $\tilde F:=\max \{F,\mathrm {F_{decay}}\}$ , note that $\tilde F$ is still decreasing (for large x) and summable, then prove the claim for $\tilde F$ , and note that $V^{ij}(\tilde F;r)- V^{ij}(\tilde F;r')\ge V^{ij}(F;r)-V^{ij}( F;r')$ to immediately obtain the claim for F.

We now break the proof into five steps.

Step 1. In this step we show that the Markov kernels $\boldsymbol q^{(k)}_\beta $ satisfy a certain coercivity condition on test functions given by absolute values of differences of coordinates, specifically (2.11) below. This will be extremely useful later. For ${\mathbf {x}},{\mathbf {y}}\in I^k$ , let

$$ \begin{align*}u_{\mathbf{x}} ({\mathbf{y}}):= |y_i-y_j-(x_i-x_j)| \;\;\;\;\text{and} \;\;\;\;v_{\mathbf{x}}^\beta:= \mathscr Q^\beta_k u_{\mathbf{x}}-u_{\mathbf{x}},\end{align*} $$

so by Lemma 2.15 we have that $v_{\mathbf {x}}^\beta ({\mathbf {y}})\ge -C|\beta |^{4p} -C|\beta |^{2p-1}\mathrm {F_{decay}}\big (\min _{1\le i'<j'\le k}|y_{i'}-y_{j'}|\big )$ .

We now show that $\inf _{\beta \in [-\eta /(2k),\eta /(2k)]} \inf _{{\mathbf {x}}\in I^k} \mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {x}})>0.$ To prove this, first note that the measure $\boldsymbol q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}})$ is never supported on the set $\{{\mathbf {y}}\in I^k:y_i-y_j=x_i-x_j\}$ , because this would then imply by mutual absolute continuity that $\boldsymbol p^{(k)}({\mathbf {x}},{\mathrm {d}}{\mathbf {y}})$ is also supported on that set. By Lemma 2.2, this would mean that $\boldsymbol {p}_{\mathbf {dif}}(x_i-x_j,\cdot )$ would be a Dirac mass. But $\boldsymbol {p}_{\mathbf {dif}}(x,\cdot )$ is never a Dirac mass, simply because it has mean x and Assumption 1.2 Item (6) implies that there cannot be absorbing states. Furthermore $\mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {x}})$ is a continuous function of ${\mathbf {x}}$ , simply by the assumption of weak continuity of all of the kernels in the paper. So far, this argument shows that $\inf _{\beta \in [-\eta /(2k),\eta /(2k)]} \inf _{{\mathbf {x}}\in I^k\cap K}\mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {x}})>0$ for all compact $K\subset I^k$ . Now if we had that $\mathscr Q^{\beta _n}_k u_{{\mathbf {x}}_n}({\mathbf {x}}_n)\to 0$ for some sequence of values $|{\mathbf {x}}_n|\to \infty $ and $\beta _n\in [-\eta /(2k),\eta /(2k)]$ , this would imply (by mutual absolute continuity and the uniform moment bounds of Lemma 2.9) that the measures $\boldsymbol {p}_{\mathbf {dif}}(x^i_n-x^j_n,x^i_n-x^j_n+\cdot )$ would be converging weakly to a Dirac mass at $0$ . But the exponential moments of these measures are uniformly bounded by Lemma 2.9 (applied with $\beta =0$ ). Thus the variances of $\boldsymbol {p}_{\mathbf {dif}}(x^i_n-x^j_n,\cdot )$ would be converging to zero, which is impossible since Lemma 2.16 shows that they converge to $2>0$ if $|x^i_n-x^j_n|\to \infty $ . Thus we have shown that indeed $\inf _{\beta \in [-\eta /(2k),\eta /(2k)]} \inf _{{\mathbf {x}}\in I^k} \mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {x}})>0.$

We now claim that there exists $\delta>0$ so that for all ${\mathbf {x}},{\mathbf {y}}\in I^k$ and $\beta \in [-\eta /(2k),\eta /(2k)]$ one has that

(2.11) $$ \begin{align}v_{\mathbf{x}}^\beta ({\mathbf{y}})>\delta \mathbf{1}_{(-\delta,\delta)}\big(y_i-y_j-(x_i-x_j)\big)-C|\beta|^{4p} -C|\beta|^{2p-1}\mathrm{F_{decay}}\big(\min_{1\le i'<j'\le k}|y_{i'}-y_{j'}|\big).\end{align} $$

To prove this, note that

$$ \begin{align*} |\mathscr Q^\beta_k u_{\mathbf{x}}({\mathbf{y}}) - \mathscr Q^\beta_k u_{\mathbf{y}}({\mathbf{y}})| &= \bigg| \int_{I^k} |a_i-a_j-(x_i-x_j)|\boldsymbol q^{(k)}_\beta ({\mathbf{y}}, {\mathrm{d}}{\mathbf{a}})-\int_{I^k} |a_i-a_j-(y_i-y_j)|\boldsymbol q^{(k)}_\beta ({\mathbf{y}}, {\mathrm{d}}{\mathbf{a}})\bigg| \\ &\le \int_{I^k} \big| |a_i-a_j-(x_i-x_j)|-|a_i-a_j-(y_i-y_j)| \big| \boldsymbol q^{(k)}_\beta ({\mathbf{y}}, {\mathrm{d}}{\mathbf{a}}) \\ &\leq \int_{I^k}|x_i-x_j-(y_i-y_j)| \boldsymbol q^{(k)}_\beta ({\mathbf{y}}, {\mathrm{d}}{\mathbf{a}})=|x_i-x_j-(y_i-y_j)|. \end{align*} $$

Recall that $c:=\inf _{\beta \in [-\eta /(2k),\eta /(2k)]} \inf _{{\mathbf {x}}\in I^k} \mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {x}})$ has already been shown to be strictly positive. Thus for all ${\mathbf {x}},{\mathbf {y}}\in I^k$ we have $\mathscr Q^\beta _k u_{\mathbf {y}}({\mathbf {y}})\ge c$ , and from the above bound it then follows that $\mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {y}})> 2c/3$ if $|x_i-x_j-(y_i-y_j)|<c/3.$ Thus we see that $v_{\mathbf {x}}^\beta ({\mathbf {y}}) = \mathscr Q^\beta _k u_{\mathbf {x}}({\mathbf {y}}) - |x_i-x_j-(y_i-y_j)|> c/3$ if $|x_i-x_j-(y_i-y_j)|<c/3$ . Together with Lemma 2.15, this implies (2.11) with $\delta :=c/3.$

Step 2. In this step we are going to find a useful family of martingales and derive a discrete “Tanaka’s formula,” see (2.13) below. That formula will be instrumental in the later analysis. We will also derive bounds related to these martingales, specifically (2.12), (2.14), and (2.15) below.

If $I=c {\mathbb {Z}}$ we will henceforth assume without loss of generality that $\delta =c$ in (2.11), otherwise it may be a smaller value. Let $\mathbf e_i$ denote the standard basis of ${\mathbb {R}}^k$ . Equation (2.11) easily implies that for all ${\mathbf {x}}=(x_1,...,x_k)\in I^k$ one has

(2.12) $$ \begin{align}F(|x_i-x_j|) &\leq \inf_{\beta\in[-\eta/(2k),\eta/(2k)]} \delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|)\big[ v^\beta_{\delta m (\mathbf e_i-\mathbf e_j)}({\mathbf{x}}) + C|\beta|^{4p}\nonumber\\ &\quad +C|\beta|^{2p-1}\mathrm{F_{decay}}\big(\min_{1\le i'<j'\le k}|x_{i'}-x_{j'}|\big)\big].\end{align} $$

For every ${\mathbf {x}},{\mathbf {y}}\in I^k$ , note that $u_{\mathbf {x}}(\mathbf R_r)-\sum _{s=0}^{r-1} v_{\mathbf {x}}^\beta (\mathbf R_r)$ is a ${\mathbf {P}}^{(\beta ,k)}_{\mathbf {y}}$ -martingale by Corollary 2.10. Defining $f_F,g_{\beta ,F}:I^k\to \Bbb R$ by

$$ \begin{align*}f_F:= \delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|) u_{\delta m(\mathbf e_i-\mathbf e_j)},\;\;\;\;\;\;\;\;\; g_{\beta,F}:= \delta^{-1}\sum_{m\in {\mathbb{Z}}}F(\delta |m|) v^\beta_{\delta m(\mathbf e_i-\mathbf e_j)},\end{align*} $$

we see that the process

(2.13) $$ \begin{align}\mathcal M_{\beta,F} (r):=-(f_F(\mathbf R_r)- f_F ({\mathbf{y}}))+\sum_{s=0}^{r-1}g_{\beta,F} (\mathbf R_s)\end{align} $$

is a ${\mathbf {P}}^{(\beta ,k)}_{\mathbf {y}}$ -martingale (for all ${\mathbf {y}}\in I^k)$ with $\mathcal M_{\beta ,F} (0)=0$ . Note that f is globally Lipschitz, in fact

(2.14) $$ \begin{align} |f_F ({\mathbf{x}})-f_F ({\mathbf{y}})| &\leq \bigg(\delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|) \bigg)|x_i-x_j-(y_i-y_j)|\leq C|x_i-x_j-(y_i-y_j)|,\end{align} $$

where we absorbed all constants in the last bound. We also claim that $g_{\beta ,F}$ is a globally bounded function. Indeed, Lemma 2.15 gives that $\Gamma := \sup _{\beta \in [-\eta /(2k),\eta /(2k)]} \sup _{{\mathbf {x}},{\mathbf {y}}\in I^k} v^\beta _{\mathbf {x}}({\mathbf {y}}) <\infty $ , thus we see that

(2.15) $$ \begin{align}\sup_{\beta\in [-\eta/(2k),\eta/(2k)]}\sup_{{\mathbf{x}}\in I^k}|g_{\beta,F}({\mathbf{x}})| \leq \bigg(\delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|) \bigg) \cdot \Gamma <\infty. \end{align} $$

From here and (2.14) one sees that the increments of the martingales $\mathcal M_{\beta ,F}$ have $q^{th}$ moments bounded uniformly over $\beta \in [-\eta /(2k),\eta /(2k)]$ and all times $r\in {\mathbb {Z}}_{\ge 0}$ , thus by Lemma 2.17 one obtains that

(2.16) $$ \begin{align}\sup_{\beta\in [-\eta/(2k),\eta/(2k)]} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |\mathcal M_{\beta,F}(r)|^q]^{1/q} \leq r^{1/2}, \end{align} $$

where C does not depend on q.

Step 3. In this step we will show an a priori bound that the left side of (2.10) can actually be bounded above by some multiple of the left side of (2.9) plus $C(r-r')^{1/2} +C|\beta |^{4p}r,$ which will effectively reduce the problem to showing the first bound. This bound is (2.18) below.

In (2.12), note that $\mathrm {F_{decay}}\big (\min _{1\le i'<j'\le k}|x_{i'}-x_{j'}|\big ) \leq \sum _{1\le i'<j'\le k} \mathrm {F_{decay}}(|x_{i'}-x_{j'}|)$ . Furthermore, recall from the beginning of the proof our assumption (without loss of generality) that $F\ge \mathrm {F_{decay}}$ , thus $\mathrm {F_{decay}}(|x_{i'}-x_{j'}|) \leq F(|x_{i'}-x_{j'}|)$ . Thus from (2.12) we have the preliminary bound

(2.17) $$ \begin{align}\notag \mathbf E_{\mathbf{x}}^{(\beta,k)} &[ V^{ij}(F;r)^q ]^{1/q} \\& \leq \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \bigg( \sum_{s=0}^{r-1} g_{\beta,F}(\mathbf R_s) \bigg)^q\bigg]^{1/q}+C|\beta|^{4p}r + C|\beta|^{2p-1} \sum_{1\le i'<j' \le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ V^{i'j'}(F;r)^q]^{1/q}. \end{align} $$

Using (2.13), (2.14), (2.15), and (2.16), we furthermore have

$$ \begin{align*} \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \bigg( \sum_{s=0}^{r-1} g_{\beta,F}(\mathbf R_s) \bigg)^q\bigg]^{1/q} &\leq \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |\mathcal M_{\beta,F}(r)|^q]^{1/q}+ \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |f_F (\mathbf R_r) - f_F({\mathbf{x}})|^q ] ^{1/q} \notag \\& \leq Cr^{1/2} + C \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r-(x_i-x_j)|^q]^{1/q}. \end{align*} $$

Combining the last two expressions and summing over all indices $(i,j)$ with $1\le i<j\le k$ , then moving all instances of $ \mathbf E_{\mathbf {x}}^{(\beta ,k)} [ V^{ij}(F;r)^q ]^{1/q}$ to the left side of the inequality, yields

$$ \begin{align*}\big(1- C|\beta|^{2p-1} \tfrac12k(k-1)\big)& \sum_{1\le i<j \le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ V^{ij}(F;r)^q ]^{1/q} \\ &\leq C r^{1/2} + C|\beta|^{4p}r + C \sum_{1\le i<j\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r-(x_i-x_j)|^q]^{1/q}. \end{align*} $$

Now choose $\beta _0$ small enough so that $C\beta _0^{2p-1} \tfrac 12k(k-1) \leq \frac 12.$ Then after absorbing all constants into some larger one, the last expression yields for $|\beta |\leq \beta _0$ that

(2.18) $$ \begin{align} \sum_{1\le i<j \le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ V^{ij}(F;r)^q ]^{1/q} \leq C r^{1/2} +C |\beta|^{4p}r + C \sum_{1\le i<j\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r-(x_i-x_j)|^q]^{1/q}. \end{align} $$

This bound will be a key step in proving the theorem shortly.

Step 4. In this step we will obtain a bound that is in a sense converse to (2.18). Recall from (2.7) that

(2.19) $$ \begin{align} \bigg| \int_{I^k} \big( y_i-y_j - (x_i-x_j)\big) \boldsymbol q^{(k)}_\beta({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) \bigg| \leq C|\beta|^{2p-1} \mathrm{F_{decay}}\big(\min_{1\le i'<j'\le k} |x_{i'}-x_{j'}|\big) + |\beta|^{4p}\end{align} $$

uniformly over $\beta \in [-\eta /(2k),\eta /(2k)]$ and ${\mathbf {x}}\in I^k.$ Let $\pi _i:I^k\to I$ be the coordinate function ${\mathbf {x}}\mapsto x_i$ , and define the process

$$ \begin{align*}M^{i,j,\beta}_r:= R^i_r-R^j_r - (x_i-x_j)-\sum_{s=0}^{r-1} \big((\mathscr Q^\beta_k-\mathrm{Id}) (\pi_i-\pi_j)\big)(\mathbf R_s).\end{align*} $$

By Corollary 2.10, this is a $ {\mathbf {P}}_{\mathbf {x}}^{(\beta ,k)}$ -martingale for all ${\mathbf {x}}\in I^k$ , which should not be confused with the Tanaka martingales appearing in (2.13), as they are unrelated. Now we can use Lemma 2.17, because (2.19) and Lemma 2.9 show that the increments of this martingale have uniformly bounded $q^{th}$ moments. Thus we see that $ \mathbf E_{\mathbf {x}}^{(\beta ,k)} [ |M^{i,j,\beta }_r|^q]^{1/q} \leq C r^{1/2},$ where C does not depend on integers $r\ge 0$ . Thus we find that

(2.20) $$ \begin{align} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q} \leq C r^{1/2} + \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \bigg( \sum_{s=0}^{r-1} \big((\mathscr Q^\beta_k-\mathrm{Id}) (\pi_i-\pi_j)\big)(\mathbf R_s)\bigg)^q\bigg]^{1/q}. \end{align} $$

Equation (2.19) precisely says $|(\mathscr Q^\beta _k-\mathrm {Id}) (\pi _i-\pi _j)\big )(\mathbf R_s)| \leq C|\beta |^{2p-1} \mathrm {F_{decay}}\big ( \min _{1\le i<j\le k} |R^i_s-R^j_s|\big ) + |\beta |^{4p},$ which can be further bounded above by $C|\beta |^{2p-1} \sum _{1\le i'<j'\le k}\mathrm {F_{decay}}\big ( |R^{i'}_s-R^{j'}_s|\big ) + |\beta |^{4p}$ because $\mathrm {F_{decay}}$ is nonincreasing. Using the definition (2.6) of the processes $V^{ij}$ , this precisely means that

$$ \begin{align*}\sum_{s=0}^{r-1} \big|\big(\mathscr Q^\beta_k-\mathrm{Id}) (\pi_i-\pi_j)\big)(\mathbf R_s) \big| \leq Cr|\beta|^{4p}+ C|\beta|^{2p-1} \sum_{1\le i'<j'\le k}V^{i'j'}(\mathrm{F_{decay}};r).\end{align*} $$

Plugging that bound into (2.20), and recalling $F\ge \mathrm {F_{decay}}$ , we obtain

(2.21) $$ \begin{align} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q} \leq C r^{1/2} + C|\beta|^{4p} r + C|\beta|^{2p-1} \sum_{1\le i'<j'\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ V^{i'j'}(F,r)^q]^{1/q}. \end{align} $$

uniformly over $\beta \in [-\eta /(2k),\eta /(2k)]$ and ${\mathbf {x}}\in I^k.$

Step 5. In this step, we will establish the theorem. Apply (2.21) and then (2.18) in that order, and one finds that

$$ \begin{align*} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q} \leq C r^{1/2} + C|\beta|^{4p} r +C|\beta|^{2p-1} \sum_{1\le i'<j'\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^{i'}_r-R^{j'}_r - (x_i-x_j)|^q]^{1/q}. \end{align*} $$

Now sum the left side over all indices $(i,j)$ with $1\le i<j\le k$ , and move all instances of $ \mathbf E_{\mathbf {x}}^{(\beta ,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q}$ to the left side of the expression, and we obtain

$$ \begin{align*}\big(1-\tfrac12 k(k-1) C |\beta|^{2p-1} \big) \sum_{1\le i<j\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q} \leq Cr^{1/2} + C|\beta|^{4p}r.\end{align*} $$

Now make $\beta _0>0$ smaller (if needed) so that $C \cdot \beta _0^{2p-1} \cdot \frac 12 k(k-1) \leq 1/2$ . Then for $|\beta |\leq \beta _0$ one obtains from the previous expression (after absorbing constants on the right side into some larger constant) that

$$ \begin{align*}\sum_{1\le i<j\le k} \mathbf E_{\mathbf{x}}^{(\beta,k)} [ |R^i_r-R^j_r - (x_i-x_j)|^q]^{1/q} \leq Cr^{1/2} + C|\beta|^{4p}r.\end{align*} $$

This proves (2.9), and then (2.10) follows immediately from (2.9) and (2.18).

Proof. This is immediate from (2.9) and (2.10). The extra term of the form $C|\beta |^{4p}(r-r')$ on the right side of (2.9) and (2.10) will contribute $C|t-s|$ which can be bounded above by $C_T |t-s|^{1/2}$ on the bounded time interval $[0,T].$

Proof. Note that (2.10) easily implies the bound that

(2.23) $$ \begin{align}\sup_{{\mathbf{x}}\in I^k} \mathbf E_{\mathbf{x}}^{(\beta_N,k)} [ V^{ij}(F,r)] \leq Cr^{1/2}, \end{align} $$

uniformly over $N\ge 1$ and $r\le Nt.$ Since $V^{ij}$ is an additive functional, one may iteratively apply the Markov property to automatically obtain higher moment bounds just from this first moment bound, specifically we will show that

(2.24) $$ \begin{align} \sup_{{\mathbf{x}}\in I^k} \mathbf E_{\mathbf{x}}^{(\beta_N,k)} [ V^{ij}(F;r)^m ] \leq C^m \sqrt{m!}\cdot r^{m/2}, \end{align} $$

where C is independent of $m\in \mathbb N$ and $r\le Nt$ . If we can prove (2.24), the claim will follow immediately simply by Taylor expanding the exponential, and setting $r=Nt$ .

The proof of (2.24) follows from a straightforward induction on m. The $m=1$ case is simply (2.23). For the inductive step, let us define $\Omega _m$ to be the optimal constant so that the left side of (2.24) is bounded above by $\Omega _m r^{m/2}$ uniformly over all $r\ge 0$ . We then have

$$ \begin{align*} h_m(r;{\mathbf{x}})&:= \sum_{s_1,s_2,...,s_m=0}^{r-1} \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \prod_{j=1}^{m} F(|R^i_{s_j}-R^j_{s_j}|) \bigg] \\&= \sum_{\substack{0\le s_1,s_2,...,s_m<r\\ \mathrm{no\;repeated\;index}}} \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \prod_{j=1}^{m} F(|R^i_{s_j}-R^j_{s_j}|) \bigg] +\sum_{\substack{0\le s_1,s_2,...,s_m<r\\\mathrm{repeated\;index}}} \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \prod_{j=1}^{m} F(|R^i_{s_j}-R^j_{s_j}|) \bigg] \\&\leq m! \cdot \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \sum_{0\leq s_1 < ...<s_m< r} \prod_{j=1}^{m} F(|R^i_{s_j}-R^j_{s_j}|) \bigg] + \|F\|_{\infty} h_{m-1}(r;{\mathbf{x}}) \\&= m! \cdot \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \sum_{s_1=0}^{r-1} F(|R^i_{s_1}-R^j_{s_1}|)\cdot \frac1{(m-1)!} h_{m-1}(r-s_1;\mathbf R_{s_1})] \bigg] + \|F\|_{\infty} h_{m-1}(r;{\mathbf{x}}) \\ &\le m\Omega_{m-1} \mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \sum_{s_1=0}^{r-1} F(|R^i_{s_1}-R^j_{s_1}|) \cdot (r-s_1)^{(m-1)/2}\bigg] + \|F\|_{\infty} h_{m-1}(r;{\mathbf{x}}) \end{align*} $$

We used the Markov property in the fourth line. Now for the first term on the right side, use summation by parts and we obtain that

$$ \begin{align*}\sum_{s=0}^{r-1} F(|R^i_{s}-R^j_{s}|)\cdot (r-s)^{(m-1)/2} = \sum_{s=0}^{r-1} \big[ (r-s)^{(m-1)/2}-(r-1-s)^{(m-1)/2}\big]\cdot \bigg(\sum_{t=0}^{s-1} F(|R^i_{t}-R^j_{t}|)\bigg).\end{align*} $$

Now note that $(r-s)^{(m-1)/2}-(r-1-s)^{(m-1)/2}$ can be bounded above by $C (\frac {m-1}{2}) (r-s)^{(m-3)/2} $ , then apply the expectation $ \mathbf E_{\mathbf {x}}^{(\beta ,k)} $ over the last expression, then use the $m=1$ case once again, and we see that the expectation is bounded above by $C (\frac {m-1}{2})\cdot \sum _{s=0}^{r-1} (r-s)^{(m-3)/2}s^{1/2} $ which is equal to $r^{m/2} $ multiplied by a Riemann sum approximation for $\int _0^1 (1-u)^{(m-3)/2}u^{1/2}du $ . Notice that the latter is $O(m^{-3/2})$ as $m\to \infty .$ This whole discussion yields

$$ \begin{align*}\mathbf E_{\mathbf{x}}^{(\beta,k)} \bigg[ \sum_{s_1=0}^{r-1} F(|R^i_{s_1}-R^j_{s_1}|) \cdot (r-s_1)^{(m-1)/2}\bigg] \leq Cm^{-1/2}r^{m/2}\end{align*} $$

for some absolute constant C.

As for the second term $ \|F\|_{\infty } h_{m-1}(r;{\mathbf {x}}) $ , we can absorb $\|F\|_{\infty }$ into some absolute constant C since F is just a fixed function. By the inductive hypothesis, $ \|F\|_{\infty } h_{m-1}(r;{\mathbf {x}}) $ may be bounded above by $ C\Omega _{m-1} r^{(m-1)/2} \leq C \Omega _{m-1} r^{m/2}. $

The entire above discussion yields the relation $\Omega _m \leq Cm^{1/2} \Omega _{m-1}$ for some absolute constant C independent of $m\in \Bbb N,$ which easily implies (2.24).

Proof. Let us use the bound

$$ \begin{align*}\sup_{r\leq Nt} \mathcal H^{2k}(N^{-\frac1{4p}},{\mathbf{x}},\mathbf R,r)^q &\leq \exp \bigg( q\sum_{s=1}^{Nt} \bigg| \log \mathbf E_{{\mathbf{x}}}^{(0,2k)} \big[ e^{N^{-\frac1{4p}}\sum_{j=1}^{2k} (R^j_s -R^j_{s-1})} \big| \mathcal F_{s-1}\big] \; - 2k \log M(N^{-\frac1{4p}})\bigg|\bigg)\\&=\exp \bigg( q\sum_{s=1}^{Nt} \bigg| \log \int_{I^k} e^{N^{-\frac1{4p}}(y_j - R^j_{s-1})} \boldsymbol p^{(k)} (\mathbf R_{s-1}, {\mathrm{d}} {\mathbf{y}}) \; - 2k \log M(N^{-\frac1{4p}})\bigg|\bigg), \end{align*} $$

where we applied the Markov property in the second line. Using Proposition 2.12 with $m=0$ and all $\beta _i = N^{-\frac 1{4p}}$ will show that the expression inside the absolute value can be bounded above by

(2.25) $$ \begin{align}CN^{-1/2} \sum_{1\le i<j\le k} \mathrm{F_{decay}}\big( |R^i_r-R^j_r| \big) + CN^{-1-\frac1{4p}},\end{align} $$

where C is a large constant independent of N. An application of Corollary 2.21 will then immediately prove the first bound. The second bound can then be proved by first using the elementary bound $|e^u-1|\leq |u|e^{|u|}$ , then applying Cauchy-Schwarz, and then applying the first bound in conjunction with Corollary 2.20 (noting the termwise bound (2.25) of each summand).

Proof. We set $V(x):= \sqrt {|x|+1}.$ We will show that $P_{\mathbf {dif}} V(x)\leq V(x)$ for all positive and sufficiently large x, and the proof for negative x would be completely symmetric.

For $x>2$ and $a\in (x/2,3x/2)$ we can write

$$ \begin{align*}V(a)-V(x) &= V'(x) (a-x) + \frac12 V"(x) (a-x)^2 +R(a,x) \\ &= \tfrac12 (x+1)^{-1/2} (a-x) -\frac14 (x+1)^{-3/2} (a-x)^2 + R(a,x), \end{align*} $$

where by Taylor’s remainder theorem one has uniformly over all $x>2$ and $a\in (x/2,3x/2)$ the bound

$$ \begin{align*}|R(a,x)| \leq \frac16 |a-x|^3 \sup_{b\in [x/2,3x/2]}|V"'(b)| \le \frac3{48} (x/2)^{-5/2} |a-x|^3 .\end{align*} $$

Now recall that $\int _I (a-x)\boldsymbol {p}_{\mathbf {dif}}(x,{\mathrm {d}} a) = 0$ for all $x\in I$ , which is immediate from the definitions, see, for example, Lemma 2.2. Consequently we can disregard the first-order term when calculating $P_{\mathbf {dif}} V$ and we find that

(3.1) $$ \begin{align}\notag P_{\mathbf{dif}} V(x)-V(x) &= \int_I (V(a)-V(x)) \boldsymbol{p}_{\mathbf{dif}} (x,{\mathrm{d}} a) \\ \notag &\leq -\frac14 (x+1)^{-3/2}\int_I (a-x)^2 \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a) + \frac3{48}(x/2)^{-5/2} \int_I |a-x|^3 \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_I |V(a)-V(x)| \mathbf{1}_{\{|x-a|>x/2\}} \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a).\end{align} $$

Let $\eta $ be as in Assumption 1.2 Item (3), and let us define constants

$$ \begin{align*} \omega_{exp}: &= \sup_{y\in I} \int_I e^{\frac\eta4|a-y|} \boldsymbol{p}_{\mathbf{dif}}(y,{\mathrm{d}} a),\\ \omega_2:&= \sup_{y\in I} \int_I (a-y)^2 \boldsymbol{p}_{\mathbf{dif}}(y,{\mathrm{d}} a),\\ \omega_3 :&= \sup_{y\in I} \int_I |a-y|^3 \boldsymbol{p}_{\mathbf{dif}}(y,{\mathrm{d}} a), \\ \delta_2 :&= \inf_{y\in I} \int_I (a-y)^2 \boldsymbol{p}_{\mathbf{dif}}(y,{\mathrm{d}} a). \end{align*} $$

The first three constants are all finite, for example, by Lemma 2.9, and the last constant is strictly positive as noted in Lemma 2.16. Notice that V is a globally Lipschitz function on I with Lipschitz constant 1/2, that is, $|V(x)-V(a)|\leq \frac 12 |x-a|,$ thus using Cauchy-Schwarz and Markov’s inequality we see that

$$ \begin{align*} \int_I |V(a)-V(x)| &\mathbf{1}_{\{|x-a|>x/2\}} \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a) \\&\leq \bigg(\int_I (V(a)-V(x))^2\boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a)\bigg)^{1/2} \bigg(\int_I \mathbf{1}_{\{|x-a|>x/2\}} \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a)\bigg)^{1/2} \\ &\leq \bigg(\frac14 \int_I (a-x)^2\boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a)\bigg)^{1/2} \bigg(\frac{\int_I e^{\frac\eta4|x-a|} \boldsymbol{p}_{\mathbf{dif}}(x,{\mathrm{d}} a)}{e^{\frac\eta4(x/2)}}\bigg)^{1/2} \\ &\leq \frac12 \omega_2^{1/2} \omega_{exp}^{1/2}\cdot e^{-\frac{\eta x}{16}}. \end{align*} $$

Plugging this bound back into (3.1), we find that for $x>2$ one has

$$ \begin{align*}P_{\mathbf{dif}} V(x) - V(x) \leq -\frac{\delta_2} 4 (x+1)^{-3/2} + \frac{3\omega_3}{48} (x/2)^{-5/2} + \frac12 \omega_2^{1/2} \omega_{exp}^{1/2} \cdot e^{-\frac{\eta x}{16}}.\end{align*} $$

The right side is clearly negative for sufficiently large $x>2$ , thus proving the claim.

Proof. By the first bullet point in Assumption 1.2 Item (6), the Markov kernel $\boldsymbol {p}_{\mathbf {dif}}$ is Strong Feller, that is, the operator $P_{\mathbf {dif}}$ sends bounded measurable functions to bounded continuous functions. In particular $\boldsymbol {p}_{\mathbf {dif}}$ is a “T-chain” in the sense of [Reference Meyn and TweedieMT93, Definition 6.0.0 (iii)].

Furthermore $\boldsymbol {p}_{\mathbf {dif}}$ is “open-set irreducible” in the sense of [Reference Meyn and TweedieMT93, Section 6.1.2] by the second bullet point of Assumption 1.2 Item (6). In particular, by [Reference Meyn and TweedieMT93, Proposition 6.1.5] the chain is “ $\psi $ -irreducible” where the maximal irreduciblilty measure $\psi $ has full support due to the open-set irreducibility and the Strong Feller property.

Furthermore, by Proposition 3.4 and [Reference Meyn and TweedieMT93, Theorem 9.4.1], for all $x\in I$ , there exists a compact set $K\subset I$ such that the Markov chain $(X_r)_{r\ge 0}$ visits K infinitely often ${\mathbf {P}}_x$ -a.s.. In other words, the Markov kernel $\boldsymbol {p}_{\mathbf {dif}}$ is “nonevanescent” in the sense of [Reference Meyn and TweedieMT93, Section 9.2.1]. Now by [Reference Meyn and TweedieMT93, Theorem 9.2.2 (ii)] any nonevanescent $\psi $ -irreducible T-chain is automatically Harris recurrent.

By [Reference Meyn and TweedieMT93, Theorem 17.3.2], Harris recurrent chains can be characterized as exactly those chains for which a unique invariant measure $\pi ^{\mathrm {inv}}$ exists (up to scalar multiple) and satisfies the ratio limit theorem as in the third bullet point of the theorem statement. By [Reference Meyn and TweedieMT93, Theorem 10.4.9], the measure $\pi ^{\mathrm {inv}}$ is equivalent to the maximal irreducibility measure $\psi $ , which has already been explained to have full support. By [Reference Meyn and TweedieMT93, Theorem 12.3.3], the invariant measure $\pi ^{\mathrm {inv}}$ is finite on compact sets as long as the Foster-Lyapunov drift criterion is satisfied as we have already shown in Proposition 3.4.

Proof. For simplicity of notation we will prove the claim with $b=0$ but the proof for general b is similar, recentering around b rather than $0$ everywhere in the argument below.

By the triangle inequality, we can write

$$ \begin{align*}|P_{\mathbf{dif}} f(x)-f(x)| &= |\int_I (f(a)-f(x)) \boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da)| \\ &\leq \int_I |f(a)-|a|| \boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da) + \bigg|\int_I (|a|-|x|)\boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da)\bigg| + ||x|-|f(x)|| \\ &\leq C \int_I e^{-\theta|a|} \boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da) + \bigg|\int_I (|a|-|x|)\boldsymbol{p}_{\mathbf{dif}}(x,\mathrm da)\bigg| + C e^{-\theta |x|}. \end{align*} $$

Let us call these terms $E_1,E_2,E_3$ . Note that $E_3$ already satisfies the required bound. Since $-|a| \leq |x-a| - |x|,$ we can bound $E_1$ by

$$ \begin{align*}C \bigg( \sup_{x'\in I}\sup_{\theta'\in[0,\frac\eta2 ]}\int_I e^{\theta |x'-a|} \boldsymbol{p}_{\mathbf{dif}}(x',\mathrm da)\bigg)\cdot e^{-\theta|x|},\end{align*} $$

and the supremum is finite by the assumption $\theta <\eta /2.$ This is indeed a bound of the desired form after replacing $C $ by a larger constant.

Finally we need to bound $E_2$ . Without loss of generality we assume $x \ge 0$ , because the case $x\le 0$ is symmetric. Then we have $|x| = x = \int _I a \;\boldsymbol {p}_{\mathbf {dif}}(x,\mathrm da),$ since every $\boldsymbol {p}_{\mathbf {dif}}(x',\cdot )$ is centered at $x'$ by construction. Thus we have that

$$ \begin{align*} E_2 &= \bigg| \int_I (|a|-a) \boldsymbol{p}_{\mathbf{dif}} (x,\mathrm da)\bigg| = 2 \int_I |a|\mathbf{1}_{\{a\le 0\}} \boldsymbol{p}_{\mathbf{dif}} (x,\mathrm da) \end{align*} $$

Note that for $a\le 0$ and $x\ge 0$ one has

$$ \begin{align*}|a|\mathbf{1}_{\{a\le 0\}} \leq \theta^{-1} \sqrt r e^{-\theta a} = \theta^{-1}e^{-\theta (a-x)}e^{-\theta x} =\theta^{-1} e^{\theta |a-x|} e^{-\theta |x|}.\end{align*} $$

We can thus bound $E_2$ by

$$ \begin{align*}C\bigg( \sup_{x'\in I}\sup_{\theta'\in[0,\frac\eta2 ]}\int_I e^{\theta |x'-a|} \boldsymbol{p}_{\mathbf{dif}}^{r'}(x',\mathrm da)\bigg)\theta^{-1} e^{-\theta |x|},\end{align*} $$

which is again finite by the previous lemma, as explained before. This proves the lemma.

Proof. Take $\beta _N=0$ in Proposition 3.1.

Proof. We break the proof into several steps.

Step 1. Let $u_b(y):=|y-b|$ , and $v_b:=P_{\mathbf {dif}} u_b-u_b$ . In this step we will prove that for every fixed $x,b\in I$ , the process $N^{-1/2}\sum _{r=0}^{Nt} v_b(X_r)$ converges in law under ${\mathbf {P}}_x$ to the process $L_0^W$ , where the latter denotes the local time at zero of a Brownian motion W of rate 2 started from $0$ .

Fix $T>0$ and $q\ge 1$ and $b\in I$ . We claim that there exists some $C>0$ such that for $N\ge 1$ and $0\le s\le t \le T$ with $s,t\in N^{-1}{\mathbb {Z}}_{\ge 0}$ and $q\ge 1$ one has

$$ \begin{align*}\sup_{x\in I} \mathbf E_x \bigg[\bigg|N^{-1/2}\sum_{r=Ns}^{Nt} v_b(X_r)\bigg|^q\bigg]^{1/q} \leq C(b)\sqrt{t-s}.\end{align*} $$

Indeed, this is immediate from applying Corollary 2.20 with $\beta =0$ , since we have exponential decay of $v_b$ by Lemma 3.6 with $\theta =\eta /2$ . The tightness of the pair of processes is thus immediate from this bound. Therefore we just need to show how to identify the limit points. To do this we will use martingale problems. We already know from Proposition 3.7 that $(N^{-1/2}X_{Nt})_{t\ge 0}$ is converging to a Brownian motion.

Using Corollary 2.10 with $\beta =0$ , one sees that the process $Z_r:=u_b(X_r)- \sum _{s=0}^{r-1} v_b(X_s)$ is a ${\mathbf {P}}_x$ -martingale for every $x\in I$ . Consider any joint limit point $(\mathcal X,\mathcal U,\mathcal V)$ of the 3-tuple of processes

$$ \begin{align*}\big(N^{-1/2}X_{Nt},N^{-1/2} u_b(X_{Nt}), N^{-1/2} \sum_{s=0}^{Nt-1} v_b(X_s) \big)_{t\ge 0}.\end{align*} $$

Tightness of the first and third coordinates has already been explained, and the second coordinate is tight by Lemma 3.6 and Corollary 2.20, since Lemma 3.6 with $\theta =\eta /2$ guarantees that $|v(x)|\leq C e^{-\frac 2{\eta }|x|}$ . We have already shown that $\mathcal X$ must be distributed as W for a Brownian motion W. For fixed $b\in I$ , the functions $x\mapsto N^{-1/2}u_b(N^{1/2}x)$ are converging uniformly on compacts as $N\to \infty $ to absolute value, thus we must have $\mathcal U=|\mathcal X|=|W|$ . Again using the fact that martingality is preserved by limit points, we see that $\mathcal U- \mathcal V = |W|-\mathcal V$ must be a martingale for the limit point (with respect to the canonical filtration on the space of 3-tuples of continuous paths), which by Tanaka’s formula forces $\mathcal V = L_0^W,$ where $L_0^W$ is the local time.

Step 2. In this step we show that $v_b\in L^1(\pi ^{\mathrm {inv}})$ for every $b\in I$ , and moreover $\int _I v_b \; {\mathrm {d}} \pi ^{\mathrm {inv}}$ does not depend on $b.$

We first show that $v_0\in L^1(\pi ^{\mathrm {inv}})$ , that is, the case $b=0$ . Suppose it was not the case, we will now derive a contradiction. By convexity of $u_0$ and Jensen’s inequality, we know that $v_0$ is nonnegative, thus $\int _I v_0\; {\mathrm {d}}\pi ^{\mathrm {inv}} = +\infty .$ On the other hand by the first bullet point of Theorem 3.5 we have that $\mathbf {1}_{[-J,J]}\in L^1(\pi ^{\mathrm {inv}})$ for all $J>0$ . Then by the third bullet point of Theorem 3.5 one easily shows that for all $J>0$ that

$$ \begin{align*}\lim_{N\to \infty} \frac{\sum_{r=0}^N \mathbf{1}_{[-J,J]}(X_r) }{\sum_{r=0}^N v_0(X_r)} = 0\end{align*} $$

${\mathbf {P}}_x$ -a.s. for all $x\in I$ . By the result of Step 1, this means that for all $J>0$ we have that

$$ \begin{align*}N^{-1/2} \sum_{r=0}^N \mathbf{1}_{[-J,J]} (X_r) = \frac{\sum_{r=0}^N \mathbf{1}_{[-J,J]}(X_r) }{\sum_{r=0}^N v_0(X_r)} \cdot N^{-1/2} \sum_{r=0}^N v_0(X_r) \;\;\;\stackrel{N\to\infty}{\longrightarrow}\;\;\; 0 \cdot L_0^W(1) = 0,\end{align*} $$

in distribution (hence in probability) under every ${\mathbf {P}}_x$ . On the other hand by Corollary 2.20 with $\beta =0$ and $q=2$ we know that for all (fixed) $J>0$

$$ \begin{align*}\sup_{N\ge 1} \sup_{x\in I} \mathbf E_x \bigg[ \bigg(N^{-1/2} \sum_{r=0}^N \mathbf{1}_{[-J,J]} (X_r)\bigg)^2\bigg]<\infty,\end{align*} $$

thus by uniform integrability and the previous convergence statement, for all $x\in I$ one has

$$ \begin{align*}\lim_{N\to \infty} \mathbf E_x \bigg[ N^{-1/2} \sum_{r=0}^N \mathbf{1}_{[-J,J]} (X_r)\bigg]=0, \;\;\; for\;all\;\; J>0.\end{align*} $$

Define $g_N(J):= \mathbf E_0 \big [ N^{-1/2} \sum _{r=0}^N \mathbf {1}_{[-J,J]} (X_r)\big ],$ so we have shown that $\lim _{N\to \infty }g_N(J)=0$ for all $J>0.$ For any $\theta>0 $ note that $\mathbf {1}_{[-J,J]}(x) \leq e^{\theta J/2} e^{-\theta |x|/2},$ consequently we find that

$$ \begin{align*}\sup_{N\ge 1} g_N(J) \leq e^{\theta J/2} \sup_{N\ge 1} \mathbf E_0 \bigg[ N^{-1/2} \sum_{r=0}^N e^{-\theta |X_r|/2} \bigg] = C(\theta) e^{\theta J/2}\end{align*} $$

where we know that the second supremum is finite by Corollary 2.20. Thus by Fubini’s theorem and the Dominated Convergence Theorem we see that for all fixed $\theta>0$

$$ \begin{align*}\lim_{N\to \infty} \mathbf E_0 \bigg[ N^{-1/2}\sum_{r=0}^N \bigg( \sum_{J=1}^\infty e^{-\theta J}\mathbf{1}_{[-J,J]} (X_r)\bigg) \bigg] &= \lim_{N\to\infty} \sum_{J=1}^\infty e^{-\theta J} g_N(J) =\sum_{J=1}^\infty e^{-\theta J}\lim_{N\to\infty} g_N(J) = 0. \end{align*} $$

On the other hand, we know that $v_0$ decays exponentially fast at infinity thanks to Lemma 3.6, which means that there exists some $C,\theta>0$ such that $v_0 \leq C \sum _{J=1}^\infty e^{-\theta J}\mathbf {1}_{[-J,J]}$ . Thus the last expression implies that $N^{-1/2}\sum _{r=0}^N v_0(X_r)$ converges to zero in $L^1({\mathbf {P}}_0)$ , contradicting the result of Step 1 and thus completing the proof that $v_0\in L^1(\pi ^{\mathrm {inv}})$ .

It remains to show that $v_b\in L^1(\pi ^{\mathrm {inv}})$ and $\int _I v_b \;{\mathrm {d}} \pi ^{\mathrm {inv}} = \int _I v_0 \;{\mathrm {d}} \pi ^{\mathrm {inv}}$ , for all $b\in I$ . First note that $\int _I v_0 \;{\mathrm {d}} \pi ^{\mathrm {inv}}\neq 0$ . Indeed $v_0\ge 0$ and $\pi ^{\mathrm {inv}}$ has full support by Theorem 3.5. So if the integral vanishes, then $v_0=0$ which would imply that $|X_r|$ is a nonnegative martingale, thus convergent, contradicting the result of Proposition 3.7. From the result of Step 1 we know that for every $b\in I$

$$ \begin{align*}\frac{\sum_{r=0}^{N} v_b(X_r)}{\sum_{r=0}^{N} v_0(X_r)} \;\;\; \stackrel{N\to\infty}{\longrightarrow} \;\;\; \frac{L_0^W(1)}{L_0^W(1)} =1\end{align*} $$

in distribution (hence in probability) under every ${\mathbf {P}}_x$ . Using the third bullet point of Theorem 3.5, it is clear that this would be impossible unless $\int _I v_b \;{\mathrm {d}} \pi ^{\mathrm {inv}}<\infty $ and $\int _I v_b \;{\mathrm {d}} \pi ^{\mathrm {inv}} = \int _I v_0 \;{\mathrm {d}} \pi ^{\mathrm {inv}}$ , for all $b\in I$ .

Step 3. In this step we will finally prove the claim being made in the proposition statement. Let F be as in the theorem statement. The argument will proceed very similarly to the proof of (2.10), but we repeat the details in the present context for additional clarity.

We first show that $\inf _{b\in I} P_{\mathbf {dif}} u_b(b)>0.$ To prove this, note that $\boldsymbol {p}_{\mathbf {dif}}(b,\cdot )$ is never a Dirac mass, simply because it has mean x and Assumption 1.2 Item (6) implies that there cannot be absorbing states. Consequently $P_{\mathbf {dif}} u_b(b)= \int _I |b-a|\boldsymbol {p}_{\mathbf {dif}}(b,\mathrm da)>0$ for all $b\in I$ . Furthermore $P_{\mathbf {dif}} u_b(b)$ is a continuous function of b, simply by the assumption of weak continuity of all of the kernels in the paper. If we had that $u_{b_n}(b_n)\to 0$ for some $|b_n|\to \infty $ , this would mean that the measures $\boldsymbol {p}_{\mathbf {dif}}(b_n,b_n+\cdot )$ would be converging weakly to a Dirac mass at $0$ . But the exponential moments of these measures are uniformly bounded by Lemma 2.9. Thus the variances of $\boldsymbol {p}_{\mathbf {dif}}(x_n,\cdot )$ would be converging to zero, which is impossible since we have already observed in (2.8) that they converge to $2>0$ .

We now claim that there exists $\delta>0$ so that for all $b\in I$ one has that $v_b>\delta \mathbf {1}_{(b-\delta ,b+\delta )}$ . To prove this, note that

$$ \begin{align*} |P_{\mathbf{dif}} u_b(y) - P_{\mathbf{dif}} u_y(y)| &= \bigg| \int_I |a-b|\boldsymbol{p}_{\mathbf{dif}} (y, \mathrm da)-\int_I |a-y|\boldsymbol{p}_{\mathbf{dif}} (y, \mathrm da)\bigg| \\ &\le \int_I \big| |a-b|-|a-y| \big| \boldsymbol{p}_{\mathbf{dif}} (y, \mathrm da) \\ &\leq \int_I|b-y| \boldsymbol{p}_{\mathbf{dif}} (y, \mathrm da)=|b-y|. \end{align*} $$

Recall that $c:= \inf _{y\in I} P_{\mathbf {dif}} u_y(y)$ has already been shown to be strictly positive. Thus for all $b,y\in I$ we have $P_{\mathbf {dif}} u_y(y)>c$ , and from the above bound it then follows that $P_{\mathbf {dif}} u_b(y)> 2c/3$ if $|b-y|<c/3.$ Thus we see that $v_x(y) = P_{\mathbf {dif}} u_b(y) - |b-y|> c/3$ if $|b-y|<c/3$ , thus proving the claim with $\delta :=c/3.$

If $I=c {\mathbb {Z}}$ we will henceforth assume that $\delta =c$ , otherwise it may be a smaller value. Using the fact that $v_b>\delta \mathbf {1}_{(b-\delta ,b+\delta )}$ , we obtain

$$ \begin{align*}F\leq \delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|) v_{\delta m} .\end{align*} $$

Integrating both sides with respect to $\pi ^{\mathrm {inv}}$ and then using the result of Step 2, we find that

$$ \begin{align*}\int_I h\;{\mathrm{d}}\pi^{\mathrm{inv}} \leq \delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta |m|) \int_I v_{\delta m}\;{\mathrm{d}}\pi^{\mathrm{inv}} = \bigg( \delta^{-1}\sum_{m\in {\mathbb{Z}}} F(\delta|m|)\bigg)\int_I v_0\;{\mathrm{d}}\pi^{\mathrm{inv}},\end{align*} $$

and the sum is bounded above by $2\delta ^{-1} \sum _{k=0}^\infty F(k)$ .

Proof. Step 1. We first establish the claim in the very special case that $f=(P_{\mathbf {dif}} - \mathrm {Id})u$ where $u(x) = |x|.$ In this case it is clear that $\gamma (f)=1$ . Letting $\overline u({\mathbf {x}}):= |x_i-x_j|$ and $\overline f_\beta ({\mathbf {x}}):= (\mathscr Q^\beta _k -\mathrm {Id})\overline u$ , both of which are functions on $I^k$ , we claim that

(3.2) $$ \begin{align}|f(x_i-x_j) - \overline f_\beta({\mathbf{x}})|\leq |\beta|^{2p-1} F(\min_{1\le i'<j'\le k} |x_{i'}-x_{j'}| ) + C|\beta|^{4p}. \end{align} $$

for some large enough constant $C>0$ and some decreasing function $F:[0,\infty )\to [0,\infty )$ satisfying $\sum _{k=0}^{\infty } F(k)<\infty $ . To prove (3.2), we first prove an intermediate claim that

(3.3) $$ \begin{align} \bigg| \int_{I^k} |y_i-y_j| \boldsymbol q^{(k)}_\beta({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) -\mathrm{sign}(x_i-x_j) \int_{I^k} (y_i-y_j) \boldsymbol q^{(k)}_\beta({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) \bigg| \leq Ce^{-\frac\eta{8k} |x_i-x_j| }. \end{align} $$

To prove the latter bound, it suffices by symmetry to consider the case $x_i-x_j\ge 0$ . In this case, the integral appearing on the left is bounded above by $\int _{I^k} |y_i-y_j| \mathbf {1}_{\{y_i-y_j\le 0\}} \boldsymbol q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}}), $ which by Cauchy-Schwarz can be bounded from above by $\big (\int _{I^k} (y_i-y_j)^2 \boldsymbol q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}})\big )^{1/2}\big (\int _{I^k} \mathbf {1}_{\{y_i-y_j\le 0\}} \boldsymbol q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}})\big )^{1/2}. $ Using Lemma 2.9, the first term can be bounded above by $(C+C(x_i-x_j)^2)^{1/2},$ with C uniform over $\beta $ and ${\mathbf {x}}$ . For the second term, note that the distance of the point ${\mathbf {x}}$ to the set $\{{\mathbf {y}}:y_i-y_j \le 0\}$ is $x_i-x_j$ , thus by the uniform exponential moment bounds of Lemma 2.9 and Markov’s inequality we have an upper bound of $Ce^{-\frac {\eta }{2k} |x_i-x_j|}.$ Finally note that by making the constant larger, one has $(C+C(x_i-x_j)^2)^{1/2} e^{-\frac {\eta }{2k} |x_i-x_j|} \leq Ce^{-\frac {\eta }{8k} |x_i-x_j|},$ thus establishing (3.3).

Note that (3.3) reduces proving (3.2) to showing that

$$ \begin{align*}\bigg| \int_{I^k} (y_i-y_j) \boldsymbol q^{(k)}_\beta({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) - \int_{I^k} (y_i-y_j) \boldsymbol p^{(k)} ({\mathbf{x}},{\mathrm{d}}{\mathbf{y}}) \bigg| \leq C|\beta|^{2p-1} \mathrm{F_{decay}}\big(\min_{1\le i'<j'\le k} |x_{i'}-x_{j'}| \big) + C|\beta|^{4p}.\end{align*} $$

But this is clear simply by performing a Taylor expansion of $\boldsymbol q^{(k)}_\beta $ from (2.4) of order $4p-1$ in the variable $\beta $ , then applying Items (4) and (5) of Assumption 1.2.

With (3.2) established, use the bound $F\big (\min _{1\le i'<j'\le k} |x_{i'}-x_{j'}| \big ) \leq \sum _{1\le i'<j'\le k} F( |x_{i'}-x_{j'}| ),$ and we find that

$$ \begin{align*}&\mathbf E_{{\mathbf{x}}_N}^{(\beta_N,k)} \bigg[ N^{-1/2} \sum_{r=0}^{Nt-1} |f(R_r^i-R_r^j)-\overline f_{\beta_N} (\mathbf R_r)|\bigg]\\ &\quad \leq \mathbf E_{{\mathbf{x}}_N}^{(\beta_N,k)} \bigg[ N^{-1/2} \sum_{1\le i'<j'\le k} \sum_{r=0}^{Nt-1} \big(|\beta_N|^{2p-1} F(|R^i_r-R^j_r|)+|\beta_N|^{4p}\big)\bigg].\end{align*} $$

Using $|\beta _N|\leq CN^{-\frac 1{4p}}$ , we may use the second bound in Corollary 2.20 to immediately conclude that the above expectation tends to zero. Thus (recalling $\gamma (f)=1$ ) in order to complete this step, it suffices to prove that

(3.4) $$ \begin{align} \lim_{N\to\infty}\mathbf E_{{\mathbf{x}}_N}^{(\beta_N,k)} \bigg[ N^{-1/2} \sum_{r=0}^{Nt-1} \overline f_{\beta_N}(\mathbf R_r)\bigg] = \mathbf E_{\mathrm{BM}}^{x_i-x_j}[L_0^W(t)]. \end{align} $$

But by Corollary 2.10 we know that $|R^i_r-R^j_r| - \sum _{s=0}^{r-1} \overline f_\beta (\mathbf R_s)$ is a $ {\mathbf {P}}_{\mathbf {x}}^{(\beta ,k)}$ -martingale, thus the expectation on the left side is precisely

$$ \begin{align*}\mathbf E_{{\mathbf{x}}_N}^{(\beta_N,k)} \bigg[ N^{-1/2}\big| R^i_{Nt} -R^j_{Nt}|\bigg] - N^{-1/2}|x_N^i-x_N^j|,\end{align*} $$

where ${\mathbf {x}}_N:= (x^1_N,...,x^k_N)$ in coordinates. By Proposition 3.1 and the uniform $L^q$ bounds in Corollary 2.20, we immediately have that the last expectation converges to

$$ \begin{align*}\mathbf E^{x_i-x_j}_{\mathrm{BM}}[|W_t|]- |x_i-x_j|.\end{align*} $$

By Tanaka’s formula, this is precisely $\mathbf E_{\mathrm {BM}}^{x_i-x_j}[L_0^W(t)],$ thus establishing (3.4) and proving the claim for this special case.

Step 2. Now we consider the case of general f. For this we use a Krylov-Bogoliubov type of trick. Fixing $t>0$ henceforth, consider the sequence of measures $\gamma _N$ on I given by

$$ \begin{align*}\int_{I} f \;{\mathrm{d}} \gamma_N:= N^{-1/2} \sum_{s=0}^{Nt-1}\mathbf E^{(\beta_N,k)}_{{\mathbf{x}}_N} [ f(R^i_s-R^j_s)].\end{align*} $$

We will show that any subsequence $\gamma _{N_k}$ of this sequence of measures has a further subsequence $\gamma _{N_{k_j}}$ converging as $N\to \infty $ to an invariant measure $\gamma $ for the Markov kernel $\boldsymbol {p}_{\mathbf {dif}}$ . By convergence, we mean that for all continuous $f:I\to {\mathbb {R}}$ of exponential decay, we have $\int _I f\;{\mathrm {d}}\gamma _N \to \int _I f\;{\mathrm {d}}\gamma $ along this subsequence $\gamma _{N_{k_j}}$ .

If we can prove this, then the lemma would be proved for all functions f of exponential decay, because by the uniqueness in Theorem 3.5, this would mean that the subsequential limit $\gamma $ must be a constant multiple of $\pi ^{\mathrm {inv}}$ . But the result of Step 1 uniquely identifies the constant as the one appearing in the lemma statement (the expectation of the Brownian local time). If every subsequence of $\gamma _N$ has a further subsequence converging to some fixed measure, then the sequence $\gamma _N$ must itself converge to that fixed measure, thus completing the proof.

Thus consider a subsequence $\gamma _{N_k}$ . There is a further convergent subsequence $\gamma _{N_{k_j}}$ simply by the second bound in Corollary 2.20 (with $q=1$ ) combined with, for example, Banach-Alaoglu (to extract subsequential limits on every compact set, then applying a diagonal argument). Let us call this subsequential limit $\gamma $ . We need to show that for all continuous $f:I\to {\mathbb {R}}$ of exponential decay at infinity, one has $\int _I P_{\mathbf {dif}} f \;{\mathrm {d}}\gamma = \int _I f\;{\mathrm {d}}\gamma .$ To prove this, consider such f, say $|f(x)| \leq e^{-\theta |x|}$ where $\theta \leq \eta /(2k)$ , and define $\overline f: I^k\to {\mathbb {R}}$ by $\overline f({\mathbf {x}}) = f(x_i-x_j)$ . To show that $\int _I P_{\mathbf {dif}} f \;{\mathrm {d}}\gamma = \int _I f\;{\mathrm {d}}\gamma ,$ it suffices to show that

(3.5) $$ \begin{align}\lim_{N\to\infty}N^{-1/2} \sum_{s=0}^{Nt-1} \mathbf E^{(\beta_N,k)}_{{\mathbf{x}}_N}[ |(\mathscr Q^0_k-\mathscr Q^{\beta_N}_k) \overline f(\mathbf R_s)|] = 0, \end{align} $$

where $\mathscr Q^0_k$ denotes the operator $\mathscr Q_k^\beta $ with $\beta =0$ (these operators were defined in Corollary 2.10). Indeed, it suffices to show this because $\mathscr Q^0_k\overline f(\mathbf R_s) = P_{\mathbf {dif}} f(R^i_s-R^j_s)$ so that $N^{-1/2} \sum _{s=0}^{Nt-1}\mathbf E^{(\beta _N,k)}_{{\mathbf {x}}_N} [ \mathscr Q^0_k\overline f(\mathbf R_s)] $ converges to $\int _I P_{\mathbf {dif}} f \;{\mathrm {d}}\gamma $ along the subsequence $N_{k_j}$ , and moreover because $N^{-1/2} \sum _{s=0}^{Nt-1}\mathbf E^{(\beta _N,k)}_{{\mathbf {x}}_N} [ \mathscr Q^{\beta _N}_k \overline f(\mathbf R_s)]$ by the Markov property is equal to $\int _I f{\mathrm {d}}\gamma _N + N^{-1/2} \big ( \mathbf E^{(\beta _N,k)}_{{\mathbf {x}}_N}[f(R^i_{Nt} -R^j_{Nt})] - f(x^N_i-x^N_j)\big ),$ which converges to $\int _I f\;{\mathrm {d}}\gamma $ along the subsequence $N_{k_j}$ .

To prove (3.5), we claim by the exponential decay assumption on f and from the definition (2.4) of the measures $\boldsymbol q^{(k)}_\beta $ , one has the bound

(3.6) $$ \begin{align}|(\mathscr Q^0_k-\mathscr Q^{\beta}_k) \overline f({\mathbf{x}})| \leq C\min\{ |\beta|, e^{-\frac\theta{2} |x_i-x_j|} \}\leq C|\beta|^{1/2} e^{-\frac\theta{4} |x_i-x_j|}. \end{align} $$

The second inequality is immediate from the first one using $\min \{u,v\}\leq u^{1/2}v^{1/2}$ . To prove the first inequality, the upper bound of $C|\beta |$ is clear simply from differentiability of the kernels $\boldsymbol q^{(k)}_\beta $ in the variable $\beta $ near $\beta =0$ , see their explicit expression (2.4). For the upper bound of $Ce^{-\frac \theta {2} |x_i-x_j|}$ , we can show that both of the terms $\mathscr Q^{\beta }_k \overline f({\mathbf {x}})$ and $\mathscr Q^{0}_k) \overline f({\mathbf {x}})$ are bounded above by such a quantity. To show this, write the definition $\mathscr Q^\beta _k \overline f({\mathbf {x}}):= \int _{I^k} f(y_i-y_j) \boldsymbol q^{(k)}_\beta ({\mathbf {x}},{\mathrm {d}}{\mathbf {y}}), $ then split the integral into two parts: $\{{\mathbf {y}}: |{\mathbf {y}}-{\mathbf {x}}| \leq \frac 12|x_i-x_j|\} $ and $\{{\mathbf {y}}: |{\mathbf {y}}-{\mathbf {x}}|> \frac 12|x_i-x_j|\} $ , where $|{\mathbf {x}}|=\sum _{j=1}^k |x_j|$ . On the first set, use the exponential decay bound on f to bound $|f({\mathbf {y}}) |\leq Ce^{-\frac \theta 2|x_i-x_j|}$ . On the second set, use the uniform moment bounds of Lemma 2.9 and Markov’s inequality (and $|f|\leq 1$ ) to obtain an upper bound of $Ce^{-\frac \eta {4k} |x_i-x_j|} \leq Ce^{-\frac \theta 2 |x_i-x_j|}.$

With (3.6) proved, the claim (3.5) follows immediately from the second bound of Corollary 2.20. While this proves the claim for all f of exponential decay, the general claim for all f as in the theorem statement can be proved using the uniform bound (2.22) (only $q=1$ is needed there) and an approximation argument of f by some sequence $f_n$ with $|f_n|\leq |f|$ with each $|f_n|$ decaying exponentially.