Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds

Abstract

The asymptotic behavior of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than (arXiv:2107.11568), generalize the recent ones in Wang (Stoch Process Appl 144:271–287, 2022) and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works.

Appendix

In appendix, we prove Remark 2.1. The proof may be familiar for experts. However, we present it here for completeness.

Proof

Firstly, for every $x\in M$, $t\mapsto p_t(x,x)$ is decreasing in $(0,\infty )$. Indeed, by the symmetry, the semigroup property and the contraction property, for every $0<s<t<\infty$,

$$\begin{aligned} p_t(x,x)= & {} \Vert p_{\frac{t}{2}}(x,\cdot )\Vert _{L^2(\mu )}^2=\Vert P_{\frac{t-s}{2}}p_{\frac{s}{2}}(\cdot ,x)\Vert _{L^2(\mu )}^2\nonumber \\\le & {} \Vert p_{\frac{s}{2}}(\cdot ,x)\Vert _{L^2(\mu )}^2=p_s(x,x). \end{aligned}$$

(A1)

Secondly, for every $t\ge t_0/2$,

$$\begin{aligned} \lim _{N\rightarrow \infty }\sup _{\Vert f\Vert _{L^2(\mu )}\le 1}\Vert P_tf\mathbbm {1}_{\{|P_tf|\ge N\}}\Vert _{L^2(\mu )}=0. \end{aligned}$$

Indeed, letting $A_{N}=\{|P_tf|> N\}$ for each $N\in \mathbb {N}$, every $f\in L^2(\mu )$ and every $t>0$, by Minkowski’s inequality, the Cauchy–Schwarz inequality, Fubini’s theorem and properties of $(p_t)_{t>0}$, we have, for every $f\in L^2(\mu )$ and every $t>0$,

$$\begin{aligned} \Vert P_tf\mathbbm {1}_{\{|P_tf|\ge N\}}\Vert _{L^2(\mu )}&=\Big \{\int _M\Big (\int _Mf(y)p_t(x,y)\,\mu (d y)\Big )^2\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big \}^{1/2}\\&\le \int _M\Big (\int _M f(y)^2p_t(x,y)^2\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big )^{1/2}\,\mu (d y)\\&\le \Big \{\int _M\Big (\int _M p_t(x,y)^2\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big )\,\mu (d y)\Big \}^{1/2}\Vert f\Vert _{L^2(\mu )}\\&=\Big \{\int _M\Big (\int _M p_t(x,y)^2\,\mu (d y)\Big )\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big \}^{1/2}\Vert f\Vert _{L^2(\mu )}\\&=\Big (\int _Mp_{2t}(x,x)\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big )^{1/2}\Vert f\Vert _{L^2(\mu )}. \end{aligned}$$

Let $A_N^*=\{\sup _{\Vert f\Vert _{L^2(\mu )}\le 1}|P_t f|>N\}$ for every $N\in \mathbb {N}$ and every $t>0$. By the Cauchy–Schwarz inequality,

$$\begin{aligned} \begin{aligned}&\int _M \sup _{\Vert f\Vert _{L^2(\mu )}\le 1}|P_t f|\,d \mu =\int _M \sup _{\Vert f\Vert _{L^2(\mu )}\le 1}\Big |\int _M f(y)p_t(x,y)\,\mu (d y)\Big |\,\mu (d x)\\&\le \int _M \Big (\int _M p_t(x,y)^2\,\mu (d y)\Big )^{1/2}\,\mu (d x)\le \sqrt{\gamma (2t)}<\infty ,\quad t\ge t_0/2, \end{aligned} \end{aligned}$$

which implies that $\sup _{\Vert f\Vert _{L^2(\mu )}\le 1}|P_t f|<\infty$ $\mu$-a.e., $t\ge t_0/2$. Then for every $t\ge t_0/2$, $\mathbbm {1}_{A_N^*}\rightarrow 0$ $\mu$-a.e. as $N\rightarrow \infty$. By Fatou’s lemma, we obtain that

$$\begin{aligned} \begin{aligned}&\lim _{N\rightarrow \infty }\sup _{\Vert f\Vert _{L^2(\mu )}\le 1}\Vert P_tf\mathbbm {1}_{\{|P_tf|\ge N\}}\Vert _{L^2(\mu )}\\&\le \limsup _{N\rightarrow \infty }\sup _{\Vert f\Vert _{L^2(\mu )}\le 1}\Big (\int _Mp_{2t}(x,x)\mathbbm {1}_{A_N}(x)\,\mu (d x)\Big )^{1/2}\\&\le \limsup _{N\rightarrow \infty }\Big (\int _Mp_{2t}(x,x)\mathbbm {1}_{A_N^*}(x)\,\mu (d x)\Big )^{1/2}\le 0,\quad t\ge t_0/2, \end{aligned} \end{aligned}$$

where the first inequality in the last line is due to that $\sup _{\Vert f\Vert _{L^2(\mu )\le 1}}\mathbbm {1}_{A_N}\le \mathbbm {1}_{A_N^*}$ for any $t>0$ since $A_N\subset A_N^*$ for every $f\in L^2(\mu )$ with $\Vert f\Vert _{L^2(\mu )}\le 1$ and every $t>0$.

Finally, due to [28, Corollary 1.6.9 and Corollary 1.6.6], we deduce that the essential spectrum of L is empty, which finishes the proof. $\square$