Probabilistic representation of fundamental solutions to (∂ u)/(∂ t) = κ_m (∂^m u)/(∂ x^m)

2012/03/14 by Enzo Orsingher, Orsingher, Enzo, Mirko D’Ovidio +1
Computer Science · Economics, Econometrics and Finance · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1203.3133

openalex publication_date 2012/03/14 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

For the fundamental solutions of heat-type equations of order n we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process X_n related to the higher-order heat-type equation with positively skewed stable r.v.'s T^j_1/3, j=1,2, ..., n we obtain genuine r.v.'s whose explicit distribution is given for n=3 in terms of Cauchy asymmetric laws. We also prove that X₃(T¹_1/3(...(Tⁿ_(1/3)(t))...)) has a stable asymmetric law.

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