Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion BH(t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ℝd, like, for example, (B1(|C(t)|),..., Bd(|C(t)|)) and (C1(|C(t)|),..., Cd(|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(|B1(|B2(...|Bn+1(t)|...)|)|) are governed by fractional diffusion equations.