A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process

TL;DR

This paper introduces a novel stochastic process called the LKSP to explicitly solve a linearized Kuramoto-Sivashinsky PDE in multiple dimensions, linking stochastic processes with complex PDEs.

Contribution

It presents the first explicit solution to a linearized Kuramoto-Sivashinsky PDE using a new imaginary-Brownian-time-Brownian-angle process.

Findings

Explicit solution to the PDE via LKSP

Connection between stochastic process and PDE

Extension to multi-dimensional space

Abstract

We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto-Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order PDEs, we give an explicit solution to a linearized Kuramoto-Sivashinsky PDE in $\scriptstyle d$-dimensional space: $\scriptstyle{\eight{u_t=-\frac18\Delta^2u-\frac12\Delta u-\frac12u}}$. The solution is given in terms of a functional of our LKSP.