On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations

Y. Nikitin and E. Orsingher
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Y. Nikitin: State University of St. Petersburg
E. Orsingher: Universit'a di Roma “La Sapienza”

Journal of Theoretical Probability, 2000, vol. 13, issue 4, 997-1012

Abstract: Abstract It is well known that the sojourn time of Brownian motion B(t), t>0 on the positive half-line, during the interval [0, t] and under the condition B(t)=0, is uniformly distributed, while it has the form of the “corrected arc-sine law” when the condition B(t)>0 is assumed. We find the analogues of these laws for “processes” X(t), t>0 governed by signed measures whose densities are the fundamental solutions of third and fourth-order heat-type equations. Surprisingly, both laws hold for the fourth-order “process.” The uniform law is still valid for the third-order “process” but a different law emerges when the condition X(t)>0 is considered.

Keywords: Brownian motion; Feynman–Kac functional; arc-sine law; sojourn time; higher order heat equations; Laplace transform; airy function (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1007861923910

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