The Isolation Time of Poisson Brownian Motions

Let the nodes of a Poisson point process move independently in R^d according to Brownian motions. We study the isolation time for a target particle that is placed at the origin, namely how long it takes until there is no node of the Poisson point process within distance r of it. In the case when the target particle does not move, we obtain asymptotics for the tail probability which are tight up to constants in the exponent in dimension d ≥ 3 and tight up to logarithmic factors in the exponent for dimensions d = 1, 2. In the case when the target particle is allowed to move independently of the Poisson point process, we show that the best strategy for the target to avoid isolation is to stay put.

[1]  Elliott H. Lieb,et al.  A General Rearrangement Inequality for Multiple Integrals , .

[2]  A phase transition in a model for the spread of an infection , 2004, math/0410371.

[3]  A. M. Berezhkovskii,et al.  Wiener sausage volume moments , 1989 .

[4]  Shashi Phoha,et al.  Surveillance coverage of sensor networks under a random mobility strategy , 2003, Proceedings of IEEE Sensors 2003 (IEEE Cat. No.03CH37498).

[5]  Alistair Sinclair,et al.  Mobile Geometric Graphs, and Detection and Communication Problems in Mobile Wireless Networks , 2010, ArXiv.

[6]  F. Spitzer.  Electrostatic capacity, heat flow, and brownian motion , 1964 .

[7]  Yuval Peres,et al.  Mobile geometric graphs: detection, coverage and percolation , 2010, Probability Theory and Related Fields.

[8]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[9]  G. Parisi.  Brownian motion , 2005, Nature.

[10]  E. Lieb,et al.  A general rearrangement inequality for multiple integrals , 1974 .

[11]  Eli Upfal,et al.  Infectious Random Walks , 2010, ArXiv.

[12]  Takis Konstantopoulos.  Response to Prof. Baccelli's lecture on Modelling of Wireless Communication Networks by Stochastic Geometry , 2010, Comput. J..

[13]  O Bénichou,et al.  Lattice theory of trapping reactions with mobile species. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Andrea E. F. Clementi,et al.  MANETS: High Mobility Can Make Up for Low Transmission Power , 2009, ICALP.

[15]  Rongfeng Sun,et al.  Survival Probability of a Random Walk Among a Poisson System of Moving Traps , 2010, 1010.3958.

[16]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[17]  Zhenming Liu,et al.  Information Dissemination via Random Walks in d-Dimensional Space , 2011, SODA.

[18]  D. J. White,et al.  Dynamic Boolean models , 1997 .

[19]  Harry Kesten,et al.  The spread of a rumor or infection in a moving population , 2003, math/0312496.

[20]  Perla Sousi,et al.  An isoperimetric inequality for the Wiener sausage , 2011, 1103.6059.

[21]  Z. Ciesielski,et al.  First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path , 1962 .