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1 results

  • Diffusion-reaction in one dimension

  • David Balding
  • Journal:
    Journal of Applied Probability / Volume 25 / Issue 4 / December 1988
    Published online by Cambridge University Press:
    14 July 2016, pp. 733-743
    Print publication:
    December 1988

  • One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.