{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T04:10:28Z","timestamp":1682568628391},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p>We consider a system of independent branching random walks on <jats:bold>R<\/jats:bold> which start from a Poisson point process with intensity of the form <jats:italic>e<\/jats:italic>\n               <jats:sub>\u03bb<\/jats:sub>(d<jats:italic>u<\/jats:italic>) = e<jats:sup>-\u03bb<jats:italic>u<\/jats:italic>\n               <\/jats:sup>d<jats:italic>u<\/jats:italic>, where \u03bb \u2208 <jats:bold>R<\/jats:bold> is chosen in such a way that the overall intensity of particles is preserved. Denote by \u03c7 the cluster distribution, and let \u03c6 be the log-Laplace transform of the intensity of \u03c7. If \u03bb\u03c6'(\u03bb) &amp;gt; 0, we show that the system is persistent, meaning that the point process formed by the particles in the <jats:italic>n<\/jats:italic>th generation converges as <jats:italic>n<\/jats:italic> \u2192 \u221e to a non-trivial point process \u03a0<jats:sub>e<jats:sub>\u03bb<\/jats:sub>\n               <\/jats:sub>\n               <jats:sup>\u03c7<\/jats:sup> with intensity <jats:italic>e<\/jats:italic>\n               <jats:sub>\u03bb<\/jats:sub>. If \u03bb\u03c6'(\u03bb) &amp;lt; 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on <jats:bold>R<\/jats:bold> which are cluster invariant with respect to the cluster distribution \u03c7 as mixtures of the point processes \u03a0<jats:sub>\n                  <jats:italic>ce<\/jats:italic>\n                  <jats:sub>\u03bb<\/jats:sub>\n               <\/jats:sub>\n               <jats:sup>\u03c7<\/jats:sup> over <jats:italic>c<\/jats:italic> &amp;gt; 0 and \u03bb \u2208 <jats:italic>K<\/jats:italic>\n               <jats:sub>st<\/jats:sub>, where <jats:italic>K<\/jats:italic>\n               <jats:sub>st<\/jats:sub> = {\u03bb \u2208 <jats:bold>R<\/jats:bold>: \u03c6(\u03bb) = 0, \u03bb\u03c6'(\u03bb) &amp;gt; 0}.<\/jats:p>","DOI":"10.1017\/s0021900200008962","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:50:28Z","timestamp":1459263028000},"page":"226-244","source":"Crossref","is-referenced-by-count":0,"title":["Persistence and Equilibria of Branching Populations with Exponential Intensity"],"prefix":"10.1017","volume":"49","author":[{"given":"Zakhar","family":"Kabluchko","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0021900200008962_ref18","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176988176"},{"key":"S0021900200008962_ref21","volume-title":"Infinitely Divisible Point Processes","year":"1978"},{"key":"S0021900200008962_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176992080"},{"key":"S0021900200008962_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19770770102"},{"key":"S0021900200008962_ref14","doi-asserted-by":"publisher","DOI":"10.1214\/09-AOP455"},{"key":"S0021900200008962_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/10-AAP686"},{"key":"S0021900200008962_ref7","doi-asserted-by":"publisher","DOI":"10.2307\/3213346"},{"key":"S0021900200008962_ref12","doi-asserted-by":"publisher","DOI":"10.2307\/1426743"},{"key":"S0021900200008962_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160310502"},{"key":"S0021900200008962_ref11","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176990544"},{"key":"S0021900200008962_ref5","doi-asserted-by":"publisher","DOI":"10.2307\/3213469"},{"key":"S0021900200008962_ref10","first-page":"11","volume":"4","year":"1960","journal-title":"S\u00e9minaire Brelot\u2013Choquet\u2013Deny Theorie du Potentiel"},{"key":"S0021900200008962_ref4","doi-asserted-by":"publisher","DOI":"10.2307\/1426138"},{"key":"S0021900200008962_ref3","volume-title":"Branching Processes","year":"1972"},{"key":"S0021900200008962_ref24","doi-asserted-by":"publisher","DOI":"10.2307\/3318685"},{"key":"S0021900200008962_ref23","volume-title":"Poblaciones Aleatorias Ramificadas y sus Equilibrios","year":"1994"},{"key":"S0021900200008962_ref22","doi-asserted-by":"publisher","DOI":"10.1214\/009117904000000865"},{"key":"S0021900200008962_ref17","volume-title":"Equilibrium Distributions of Branching Processes","year":"1988"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200008962","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T06:38:50Z","timestamp":1682491130000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200008962\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":18,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["S0021900200008962"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200008962","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}