Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations (Q678091)

The densities of distributions (or transition densities) of some special compositions of stochastic processes are investigated. These densities are shown to satisfy certain higher order parabolic and hyperbolic equations. A typical simple example (which is the first result of the paper) is this: let \(B_1\), \(B_2\) be independent Brownian motions and put \[ X(t)= \begin{cases} B_1(iB_2(t))&\text{if \(B_2(t)>0\),}\\ iB_1(-iB_2(t))&\text{if \(B_2(t)<0\).} \end{cases} \] Then the density function of \(X(t)\) satisfies the equation \( \frac{\partial u}{\partial t} = -\frac 18\frac{\partial^4 u}{\partial x^4} \) with the initial condition \(u(x,0)=\delta(x)\).