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Admissibility of transition from the Smolukhovskii integral equation to the Einstein-Fokker-Planck differential equation

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Abstract

For Markov processes we consider limits for the growth rate of the probability density w(to, xo; t, x) of the transition from state xo at time to to states for which x lies between x and x + dx at time t. These limits follow from the condition of admissibility of transition from the Smolukhovskii integral equation to the Einstein-Fokker-Planck differential equation (EFP). We examine the influence of these limits on the form of the coefficients in EFT. The indicated examination is performed for the “prelimit” region as well, that is, the region at the “end” of which we have w(to, xo; t, x) ≡ 0.

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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 99–104, October, 1986.

The authors are deeply grateful to D. M. Sedrakyan and A. O. Melikyan for their interest in this work and useful comments.

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Airapetyan, Y.N., Airapetyan, V.N. Admissibility of transition from the Smolukhovskii integral equation to the Einstein-Fokker-Planck differential equation. Soviet Physics Journal 29, 859–863 (1986). https://doi.org/10.1007/BF00900756

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Keywords

  • Growth Rate
  • Differential Equation
  • Integral Equation
  • Probability Density
  • Markov Process