Equations of Mathematical Physics and Compositions of Brownian and\n Cauchy processes

2010/08/05 by Luisa Beghin, Enzo Orsingher, Beghin, Luisa +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #35Q99 #60K99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1008.0928

openalex publication_date 2010/08/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider different types of processes obtained by composing Brownian\nmotion B(t), fractional Brownian motion B_H(t) and Cauchy processes %\nC(t) in different manners.\n We study also multidimensional iterated processes in \ℝ^d, like,\nfor example, \( B₁(|C(t)|),...,B_d(|C(t)|)\) and \(\nC₁(|C(t)|),...,C_d(|C(t)|)\) , deriving the corresponding partial\ndifferential equations satisfied by their joint distribution.\n We show that many important partial differential equations, like wave\nequation, equation of vibration of rods, higher-order heat equation, are\nsatisfied by the laws of the iterated processes considered in the work.\n Similarly we prove that some processes like %\nC(|B₁(|B₂(...|B_n+1(t)|...)|)|) are governed by fractional diffusion\nequations.\n

Citations

• Iterated Brownian motion in an open set

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• Representations of the fractional d'Alembertian and initial conditions in fractional dynamics

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