Entrance laws for annihilating Brownian motions and the continuous-space voter model

Matthias Hammer, Marcel Ortgiese, Florian Vollering

Research output: Contribution to journalArticlepeer-review

29 Downloads (Pure)


Consider a system of particles moving independently as Brownian motions until two of them meet, when the colliding pair annihilates instantly. The construction of such a system of annihilating Brownian motions (aBMs) is straightforward as long as we start with a finite number of particles, but is more involved for infinitely many particles. In particular, if we let the set of starting points become increasingly dense in the real line it is not obvious whether the resulting systems of aBMs converge and what the possible limit points (entrance laws) are. In this paper, we show that aBMs arise as the interface model of the continuous-space voter model. This link allows us to provide a full classification of entrance laws for aBMs. We also give some examples showing how different entrance laws can be obtained via finite approximations. Further, we discuss the relation of the continuous-space voter model to the stepping stone and other related models.
Finally, we obtain an expression for the $n$-point densities of aBMs starting from an arbitrary entrance law.
Original languageEnglish
Pages (from-to)240-264
Number of pages25
JournalStochastic Processes and their Applications
Early online date21 Jan 2021
Publication statusPublished - 30 Apr 2021

Bibliographical note

Funding Information:
This project received financial support by the German Research Foundation (DFG) within the DFG Priority Programme 1590 ‘Probabilistic Structures in Evolution’, grant OR 310/1-1 .

Publisher Copyright:
© 2021 Elsevier B.V.


  • Annihilating Brownian motions
  • Entrance laws
  • Moment duality
  • Stepping stone model
  • Symbiotic branching
  • Voter model

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


Dive into the research topics of 'Entrance laws for annihilating Brownian motions and the continuous-space voter model'. Together they form a unique fingerprint.

Cite this