Indexing metadata

Branching Random Walk with Catalysts


 
Dublin Core PKP Metadata Items Metadata for this Document
 
1. Title Title of document Branching Random Walk with Catalysts
 
2. Creator Author's name, affiliation, country Harry Kesten; Cornell University
 
2. Creator Author's name, affiliation, country Vladas Sidoravicius; IMPA
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Branching random walk, survival, extinction.
 
3. Subject Subject classification Primary 60K35; Secondary 60J80.
 
4. Description Abstract

Shnerb et al. (2000), (2001) studied the following system of interacting particles on $\Bbb Z^d$: There are two kinds of particles, called $A$-particles and $B$-particles. The $A$-particles perform continuous time simple random walks, independently of each other. The jumprate of each $A$-particle is $D_A$. The $B$-particles perform continuous time simple random walks with jumprate $D_B$, but in addition they die at rate $\delta$ and a $B$-particle at $x$ at time $s$ splits into two particles at $x$ during the next $ds$ time units with a probability $\beta N_A(x,s)ds +o(ds)$, where $N_A(x,s)\; (N_B(x,s))$ denotes the number of $A$-particles (respectively $B$-particles) at $x$ at time $s$. Conditionally on the $A$-system, the jumps, deaths and splittings of different $B$-particles are independent. Thus the $B$-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of $A$-particles which coincide with the appropriate $B$-particles. One starts the process with all the $N_A(x,0),\, x \in \Bbb Z^d$, as independent Poisson variables with mean $\mu_A$, and the $N_B(x,0),\, x \in \Bbb Z^d$, independent of the $A$-system, translation invariant and with mean $\mu_B$.

Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation $\Bbb E\{N_B(x,t)\}$ tends to infinity, no matter what the values of $\delta, \beta, D_A$, $D_B, \mu_A,\mu_B \in (0, \infty)$ are. We shall show here that nevertheless there is a phase transition in all dimensions, that is, the system becomes (locally) extinct for large $\delta$ but it survives for $\beta$ large and $\delta$ small.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2003-03-24
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/127
 
10. Identifier Digital Object Identifier 10.1214/EJP.v8-127
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 8
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
15. Rights Copyright and permissions The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Summary of the Creative Commons Attribution License

You are free
  • to copy, distribute, display, and perform the work
  • to make derivative works
  • to make commercial use of the work
under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.