Opuscula Math. 31, no. 2 (2011), 195-207
http://dx.doi.org/10.7494/OpMath.2011.31.2.195

 
Opuscula Mathematica

On the Bochner subordination of exit laws

Mohamed Hmissi
Wajdi Maaouia

Abstract. Let \(\mathbb{P}=(P_t)_{t\ge 0}\) be a sub-Markovian semigroup on \(L^2(m)\), let \(\beta=(\beta_t)_{t\ge 0}\) be a Bochner subordinator and let \(\mathbb{P}^{\beta}=(P_t^{\beta})_{t\ge 0}\) be the subordinated semigroup of \(\mathbb{P}\) by means of \(\beta\), i.e. \(P^{\beta}_s:=\int_0^{\infty} P_r\,\beta_s(dr)\). Let \(\varphi:=(\varphi_t)_{t\gt 0}\) be a \(\mathbb{P}\)-exit law, i.e. \[ P_t\varphi_s= \varphi_{s+t}, \qquad s,t\gt 0\] and let \(\varphi^{\beta}_t:=\int_0^{\infty} \varphi_s\,\beta_t(ds)\). Then \(\varphi^{\beta}:=(\varphi_t^{\beta})_{t\gt 0}\) is a \(\mathbb{P}^{\beta}\)-exit law whenever it lies in \(L^2(m)\). This paper is devoted to the converse problem when \(\beta\) is without drift. We prove that a \(\mathbb{P}^{\beta}\)-exit law \(\psi:=(\psi_t)_{t\gt 0}\) is subordinated to a (unique) \(\mathbb{P}\)-exit law \(\varphi\) (i.e. \(\psi=\varphi^{\beta}\)) if and only if \((P_tu)_{t\gt 0}\subset D(A^{\beta})\), where \(u=\int_0^{\infty} e^{-s} \psi_s ds\) and \(A^{\beta}\) is the \(L^2(m)\)-generator of \(\mathbb{P}^{\beta}\).

Keywords: sub-Markovian semigroup, exit law, subordinator, Bernstein function, Bochner subordination.

Mathematics Subject Classification: 47A50, 47D03, 39B42.

Full text (pdf)

  • Mohamed Hmissi
  • Département de Mathématiques, Faculté des Sciences de Tunis, TN-2092 El Manar Tunis, Tunisia
  • Wajdi Maaouia
  • Département de Mathématiques, Faculté des Sciences de Tunis, TN-2092 El Manar Tunis, Tunisia
  • Received: 2010-05-12.
  • Revised: 2010-08-17.
  • Accepted: 2010-09-09.
Opuscula Mathematica - cover

Cite this article as:
Mohamed Hmissi, Wajdi Maaouia, On the Bochner subordination of exit laws, Opuscula Math. 31, no. 2 (2011), 195-207, http://dx.doi.org/10.7494/OpMath.2011.31.2.195

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.