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3044 results in Probability theory and stochastic processes

Page 29 of 153

  • 9 - Varopoulos Semigroups

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 166-178

  • Summary

    Fractional calculus and the Riemann–Liouville integral are used to motivate the Hardy–Littlewood–Sobolev inequality. These are then generalised to the Riesz potential, and via heat kernels, toVaropoulos’s class of ultracontractive semigroups, which we show also satisfy this key inequality. As an application, we establish the Nash inequality for uniformly elliptic second-order diffusion operators and show that these are in the Varopoulos class.

  • 7 - Markov and Feller Semigroups

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 129-148

  • Summary

    Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.

  • 4 - Self-Adjoint Semigroups and Unitary Groups

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 83-101

  • Summary

    We study self-adjoint semigroups and give necessary and sufficient conditions for these to be induced by a convolution semigroup (through the procedure of Chapter 3). Stone’s theorem is proved and applied to Schrödinger’s equation and the Heisenberg equation of quantum mechanics. We then introduce quantum dynamical semigroups and the Lindblad generator.

  • 3 - Convolution Semigroups of Measures

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 46-82

  • Summary

    Chapter 3 studies semigroups on function spaces obtained via convolution semigroups of probability measures. Motivating examples that are studied in detail are the heat kernel (Brownian motion) and the Poisson kernel (Cauchy process). The characteristic functional (Fourier transform) is used to establish the Levy–Khinchine formula, and applications are given to stable laws. The generator and the semigroup are written as pseudo-differential operators.

  • 1 - Semigroups and Generators

  • David Applebaum, University of Sheffield
  • Book:
    Semigroups of Linear Operators
    Published online:
    27 July 2019
    Print publication:
    15 August 2019, pp 9-30
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  • Summary

    We define semigroups of linear operators in Banach spaces and introduce their generators (which may be unbounded operators) and their resolvents. A number of examples are given, which will be referred back to throughout the volume.

  • 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.

Page 29 of 153