caa a22 4500 47707698X CHVBK 20180405111441.0 cr unu---uuuuu 170330e19961101xx s 000 0 eng 10.1007/BF00047125 doi (NATIONALLICENCE)springer-10.1007/BF00047125 Intersections of random walks and Wiener sausages in four dimensions [Elektronische Daten] [Sergio Albeverio, Xian Zhou] Let {S 1 (n)} n≥0and {S 2 (n)} n≥0be independent simple random walks in Z 4 starting at the origin, and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaqGPbaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiaadggacaGGSaGaamOyaiab-LcaPiabg2da9i% ab-Tha7Hqbciab+Hha4jabgIGiolab+PfaAnaaCaaaleqabaGaaGin% aaaakiaacQdaieGacaqFtbWaaSbaaSqaaiaabMgaaeqaaOGae8hkaG% Iaa0xBaiab-LcaPiabg2da9iab+Hha4baa!5761!\[\Pi _{\rm{i}} (a,b) = \{ x \in Z^4 :S_{\rm{i}} (m) = x\]for the some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciaa-1gacqGHiiIZtCvAUfKttLearyqr1ngBPrgaiuaacqGF% OaakcaWGHbGaaiilaiaadkgacqGFPaqkcqGF9bqFaaa!4936!\[m \in (a,b)\} \]. Let two integervalued sequences {a n}n≥0and {b n}n≥0be given, such that the limit limn→∞ a nexists and lim n →∞ b n=+∞. In this paper, it is shown that the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaiaacYcaaeqaaOGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqGH% GjsUieaacaGFydaaaa!5904!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_{n,} a_n + b_n ) \ne \O \] is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdaaaGaciiBaiaac+gacaGGNbWe% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIae8xmaeJae83kaSIaam% OyamaaBaaaleaacaWGUbaabeaakiaac+cacaWGHbWaaSbaaSqaaiaa% d6gaaeqaaOGae8xkaKIae83la8IaciiBaiaac+gacaGGNbGaamOyam% aaBaaaleaacaWGUbaabeaaaaa!5364!\[\frac{1}{2}\log (1 + b_n /a_n )/\log b_n \] if it tends to zero as n→∞, and the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaaqabaGccaGGSaGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqWF% 9aqpieaacaGFydaaaa!583C!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \]is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaaabaGaam4yaiaacUfaciGGSbGaai4BaiaacEgatCvAUfKt% tLearyqr1ngBPrgaiuaacqWFOaakcaWGHbWaaSbaaSqaaiaad6gaae% qaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbaabeaakiab-LcaPiab% -9caViab-XgaSjab-9gaVjab-DgaNjab-HcaOiaadggadaWgaaWcba% GaamOBaaqabaGccqGHRaWkcaaIYaGae8xkaKIae8xxa01aaWbaaSqa% beaacqWFTaqlcqWFXaqmcqWFVaWlcqWFYaGmaaaaaaa!5BAC!\[\begin{array}{l} \Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \\ c[\log (a_n + b_n )/log(a_n + 2)]^{ - 1/2} \\ \end{array}\], for some constant c, if it tends to a finite constant (≤1) as n→∞. These results extend some results obtained by G. F. Lawler about the intersection properties of simple random walks in Z 4. By using similar arguments, we also get corresponding results for the intersections of Wiener sausages in four dimensions. In particular, a conjecture suggested by M. Aizenman, which describes nonintersection of independent Wiener sausages in R 4, is proven. Kluwer Academic Publishers, 1996 intersections nationallicence hitting distribution nationallicence range of random walks nationallicence random walks nationallicence Wiener sausages nationallicence Albeverio Sergio Institute of Mathematics, Ruhr-University Bochum, Postfach 102148, D-44780, Bochum 1, Germany aut Zhou Xian Department of Mathematics, Beijing Normal University, 100875, Beijing, P.R. China aut Acta Applicandae Mathematica Kluwer Academic Publishers 45/2(1996-11-01), 195-237 0167-8019 45:2<195 1996 45 10440 https://doi.org/10.1007/BF00047125 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF00047125 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Albeverio Sergio Institute of Mathematics, Ruhr-University Bochum, Postfach 102148, D-44780, Bochum 1, Germany aut NATIONALLICENCE 700 1- Zhou Xian Department of Mathematics, Beijing Normal University, 100875, Beijing, P.R. China aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 45/2(1996-11-01), 195-237 0167-8019 45:2<195 1996 45 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer