Spectral condition, hitting times and Nash inequality

Eva Löcherbach; Oleg Loukianov; Dasha Loukianova

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1213-1230
  • ISSN: 0246-0203

Abstract

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Let X be a μ -symmetric Hunt process on a LCCB space 𝙴 . For an open set 𝙶 ⊆ 𝙴 , let τ 𝙶 be the exit time of X from 𝙶 and A 𝙶 be the generator of the process killed when it leaves 𝙶 . Let r : [ 0 , ∞ [ → [ 0 , ∞ [ and R ( t ) = ∫ 0 t r ( s ) d s . We give necessary and sufficient conditions for 𝔼 μ R ( τ 𝙶 ) l t ; ∞ in terms of the behavior near the origin of the spectral measure of - A 𝙶 . When r ( t ) = t l , l ≥ 0 , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order l + 1 for τ 𝙶 implies the Nash inequality of order p = l + 2 l + 1 for the whole process. The associated rate of convergence of the semi-group in 𝕃 2 ( μ ) is bounded by t - ( l + 1 ) . Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order t - ( l + 1 ) of the semi-group implies the existence of moments of order l + 1 - ε for τ 𝙶 , for all ε g t ; 0 .

How to cite

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Löcherbach, Eva, Loukianov, Oleg, and Loukianova, Dasha. "Spectral condition, hitting times and Nash inequality." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1213-1230. <http://eudml.org/doc/271947>.

@article{Löcherbach2014,
abstract = {Let $X$ be a $\mu $-symmetric Hunt process on a LCCB space $\mathtt \{E\}$. For an open set $\mathtt \{G\}\subseteq \mathtt \{E\}$, let $\tau _\{\mathtt \{G\}\}$ be the exit time of $X$ from $\mathtt \{G\}$ and $A^\{\mathtt \{G\}\}$ be the generator of the process killed when it leaves $\mathtt \{G\}$. Let $r:[0,\infty [\,\rightarrow [0,\infty [$ and $R(t)=\int _\{0\}^\{t\}r(s)\,\mathrm \{d\} s$. We give necessary and sufficient conditions for $\mathbb \{E\}_\{\mu \}R(\tau _\{\mathtt \{G\}\})&lt;\infty $ in terms of the behavior near the origin of the spectral measure of $-A^\{\mathtt \{G\}\}$. When $r(t)=t^\{l\}$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for $\tau _\{\mathtt \{G\}\}$ implies the Nash inequality of order $p=\frac\{l+2\}\{l+1\}$ for the whole process. The associated rate of convergence of the semi-group in $\mathbb \{L\}^\{2\}(\mu )$ is bounded by $t^\{-(l+1)\}$. Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order $t^\{-(l+1)\}$ of the semi-group implies the existence of moments of order $l+1-\varepsilon $ for $\tau _\{\mathtt \{G\}\}$, for all $\varepsilon &gt;0$.},
author = {Löcherbach, Eva, Loukianov, Oleg, Loukianova, Dasha},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {recurrence; hitting times; Dirichlet form; Nash inequality; weak Poincaré inequality; $\alpha $-mixing; continuous time Markov processes; continuous-time Markov processes; hitting times; Dirichlet form; Nash inequality; weak Poincaré inequality},
language = {eng},
number = {4},
pages = {1213-1230},
publisher = {Gauthier-Villars},
title = {Spectral condition, hitting times and Nash inequality},
url = {http://eudml.org/doc/271947},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Löcherbach, Eva
AU - Loukianov, Oleg
AU - Loukianova, Dasha
TI - Spectral condition, hitting times and Nash inequality
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1213
EP - 1230
AB - Let $X$ be a $\mu $-symmetric Hunt process on a LCCB space $\mathtt {E}$. For an open set $\mathtt {G}\subseteq \mathtt {E}$, let $\tau _{\mathtt {G}}$ be the exit time of $X$ from $\mathtt {G}$ and $A^{\mathtt {G}}$ be the generator of the process killed when it leaves $\mathtt {G}$. Let $r:[0,\infty [\,\rightarrow [0,\infty [$ and $R(t)=\int _{0}^{t}r(s)\,\mathrm {d} s$. We give necessary and sufficient conditions for $\mathbb {E}_{\mu }R(\tau _{\mathtt {G}})&lt;\infty $ in terms of the behavior near the origin of the spectral measure of $-A^{\mathtt {G}}$. When $r(t)=t^{l}$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for $\tau _{\mathtt {G}}$ implies the Nash inequality of order $p=\frac{l+2}{l+1}$ for the whole process. The associated rate of convergence of the semi-group in $\mathbb {L}^{2}(\mu )$ is bounded by $t^{-(l+1)}$. Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order $t^{-(l+1)}$ of the semi-group implies the existence of moments of order $l+1-\varepsilon $ for $\tau _{\mathtt {G}}$, for all $\varepsilon &gt;0$.
LA - eng
KW - recurrence; hitting times; Dirichlet form; Nash inequality; weak Poincaré inequality; $\alpha $-mixing; continuous time Markov processes; continuous-time Markov processes; hitting times; Dirichlet form; Nash inequality; weak Poincaré inequality
UR - http://eudml.org/doc/271947
ER -

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