Abstract
We study the scaling limit for a catalytic branching particle system whose particles perform random walks on ℤ and can branch at 0 only. Varying the initial (finite) number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n β particles and consider the scaled process \( Z^{n}_{t} {\left( \bullet \right)} = Z_{{nt}} {\left( {{\sqrt {n \bullet } }} \right)} \), where Z t is the measure–valued process representing the original particle system. We prove that \( Z^{n}_{t} \) converges to 0 when \( \beta < \frac{1} {4} \) and to a nondegenerate discrete distribution when \( \beta = \frac{1} {4} \) . In addition, if \( \frac{1} {4} < \beta < \frac{1} {2} \) then \( n^{{ - {\left( {2\beta - \frac{1} {2}} \right)}}} Z^{n}_{t} \) converges to a random limit, while if \( \beta > \frac{1} {2} \) then \( n^{{ - \beta }} Z^{n}_{t} \) converges to a deterministic limit.
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* Research supported partially by DFG and grants RFBR 02–01–00266 and Russian Scientific School 1758.2003.1
** Research supported partially by NSA and by Alexander von Humboldt Foundation
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Vatutin*, V., Xiong**, J. Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks. Acta Math Sinica 23, 997–1012 (2007). https://doi.org/10.1007/s10114-005-0757-4
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DOI: https://doi.org/10.1007/s10114-005-0757-4