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 $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ 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.