Basic Estimates of Stability Rate for One-Dimensional Diffusions

Abstract

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincaré-type (or Hardy-type) inequalities. The derivations of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.

Appendix

The next result is a generalization of [9; Proposition 1.2].

Proposition 6.3

Let $P_t(x, \cdot)$ be symmetric and have density $p_t(x, y)$ with respect to $\mu.$ Suppose that the diagonal elements $p_{s}(\cdot, \cdot)\in L_{{\rm loc}}^{1/2}(\mu)$ for some $s>0$ and a set ${\fancyscript{K}}$ of bounded functions with compact support is dense in $L^2(\mu).$ Then $\lambda_0 = \varepsilon_{\rm max}.$

Proof

The proof is similar to the ergodic case (cf. [6; Sect. 8.3] and 9; proof of Theorem 7.4]), and is included here for completeness. (a) Certainly, the inner product and norm here are taken with respect to $\mu.$ First, we have

$$ \begin{aligned} P_t(x, K)&= P_s P_{t-s} {\bf 1}_K (x)\\ &=\int \mu(\hbox{d} y) {{\hbox{d} P_s(x, \cdot)}\over {{\hbox{d}} \mu}}(y) P_{t-s} {\bf 1}_K (y)\quad(\hbox{since} P_s\ll \mu)\\ &=\mu\bigg({{\hbox{d} P_s(x, \cdot)}\over {\hbox{d} \mu}} P_{t-s} {\bf 1}_K\bigg)\\ &=\mu\bigg({\bf 1}_K P_{t-s}{{\hbox{d} P_s(x, \cdot)}\over {{\hbox{d}} \mu}} \bigg)\quad(\hbox{by symmetry of} P_t) \\ &\le \sqrt{\mu(K)} \bigg\|P_{t-s}{{\hbox{d} P_s(x, \cdot)}\over {\hbox{d} \mu}}\bigg\|\quad\hbox {(by Cauchy--Schwarz inequality)}\\ &\le \sqrt{\mu(K)} \bigg\|{{\hbox{d} P_s(x, \cdot)}\over {\hbox{d} \mu}}\bigg\| e^{-\lambda_0 (t-s)} \;(\hbox{by}\, L^2-\hbox{exponential convergence})\\ &=\Big(\sqrt{\mu(K) p_{2s}(x, x)} e^{\lambda_0 s}\Big) e^{-\lambda_0 t}\quad(by [6 (8.3)]). \end{aligned} $$

By assumption, the coefficient on the right-hand side is locally $\mu$-integrable. This proves that $\varepsilon_{\rm max}\ge \lambda_0.$

(b) Next, for each $f\in{{\fancyscript{K}}}$ with $\|f\|=1,$ we have

$$ \begin{aligned} \|P_t f\|^2&= (f, P_{2t} f)\quad(\hbox{by symmetry of }P_t)\\ &\le \|f\|_{\infty}\int\nolimits_{{\rm supp} (f)} \mu(\hbox{d} x) P_{2t} |f|(x) \\ &\le \|f\|_{\infty}^2 \int\nolimits_{{\rm supp} (f)} \mu (\hbox{d} x) P_{2t} (x, \hbox{supp}(f)) \\ &\le \|f\|_{\infty}^2 \int\nolimits_{{\rm supp} (f)} \mu (\hbox{d} x) c(x, \hbox{supp} (f)) e^{-2\varepsilon_{\rm max} t} \\ &=: C_f e^{-2\varepsilon_{\rm max} t}. \end{aligned} $$

The technique used here goes back to [17].

(c) The constant $C_f$ in the last line can be removed. Following Lemma 2.2 in [24], by the spectral representation theorem and the fact that $\|f\|=1,$ we have

$$ \begin{aligned} \|P_t f\|^2&=\int\nolimits_0^\infty e^{-2 \lambda t}\hbox{d} (E_\lambda f, f) \\ &\ge \bigg[\int\nolimits_0^\infty e^{-2 \lambda s}\hbox{d} (E_\lambda f, f)\bigg]^{t/s}\quad \hbox {(by Jensen's inequality)}\\ &=\|P_s f\|^{2t/s},\qquad \;t\ge s. \end{aligned} $$

Note that here the semigroup is allowed to be subMarkovian. Combining this with (b), we have $\|P_s f\|^2\le C_f^{s/t} e^{-2 \varepsilon_{\rm max} s}.$ Letting $t\to \infty,$ we obtain

$$ \|P_s f\|^2\le e^{-2\varepsilon_{\rm max} s}, $$

first for all $f\in {{\fancyscript{K}}}$ and then for all $f\in L^2(\mu)$ with $\|f\|=1$ because of the denseness of ${{\fancyscript{K}}}$ in $L^2(\mu).$ Therefore, $\lambda_0\ge \varepsilon_{\rm max}.$ Combining this with (a), we complete the proof. $\square$

The main result (Theorem 6.2) of this paper is presented in the last section (Sect. 10) of the paper [9], as an analog of birth–death processes. Paper [9], as well as [8] for $\varphi^4$-model, is available on arXiv.org.