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 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(...|Bn+1(t) . . .)|)|) are governed by fractional diffusion equations.