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3 results

  • 6 - Perturbation Theory

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 116-128

  • Summary

    Chapter 6 gives a brief introduction to perturbation theory. We mainly consider relatively bounded perturbations, with bounded perturbations being a special case. Chernoff’s product formula is established and used to give a probabilistic proof of the Feynman–Kac formula, with the aid of Wiener measure.

  • A diffusion model with loss of particles

  • Lech S. Papież, George A. Sandison
  • Journal:
    Advances in Applied Probability / Volume 22 / Issue 3 / September 1990
    Published online by Cambridge University Press:
    01 July 2016, pp. 533-547
    Print publication:
    September 1990

  • The dynamical behaviour of particles which undergo diffusion with annihilation is modelled by a parabolic (Fokker–Planck) equation. Fundamental, closed-form solutions of this equation, identified with transition densities of the underlying stochastic process, are calculated by utilizing specific methods of probability measures on functional spaces and evolution semigroups.

  • A functional central limit theorem for the Ewens sampling formula

  • Jennie C. Hansen
  • Journal:
    Journal of Applied Probability / Volume 27 / Issue 1 / March 1990
    Published online by Cambridge University Press:
    14 July 2016, pp. 28-43
    Print publication:
    March 1990

  • For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.