General Fractional Calculus, Evolution Equations, and Renewal Processes

2011/05/06 by Kochubei, Anatoly N. · 7 citations
#26A33 #34A08 #35R11 #60K05 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1105.1239

Abstract

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form (Du)(t)=(d)/(dt)∫₀^tk(t-τ)u(τ) dτ-k(t)u(0) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation Du=-λu, λ>0, proved to be (under some conditions upon k) continuous on [(0,∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.

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