Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

Abstract

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ℝ ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.