Equations of Mathematical Physics and Compositions of Brownian and Cauchy processes

TL;DR

This paper investigates various processes formed by composing Brownian, fractional Brownian, and Cauchy processes, deriving PDEs for their distributions and linking them to classical equations like wave and heat equations.

Contribution

It introduces new classes of iterated processes and establishes the PDEs they satisfy, connecting stochastic processes with fundamental differential equations.

Findings

Many PDEs, including wave and heat equations, are satisfied by the laws of these iterated processes.

Certain composed processes are governed by fractional diffusion equations.

The work extends the understanding of stochastic processes and their relation to PDEs.

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.