Math 6410 • ODE

The Ornstein-Uhlenbeck (OU) differential equation is
\[
  \frac{\partial p}{\partial t}
  =\frac{\partial}{\partial x}(kxp)+D\frac{\partial^2p}{\partial x^2}.
\]
\begin{enumerate}
\item[(a)] What type of partial differential equation is this? What calculation is
needed to justify this statement?
\item[(b)] If \(t\) has units of time and \(x\) has units of length, what are the units
of the parameters \(k\) and \(D\)?
\item[(c)] Express this equation in nondimensional variables? How many nondimensional
parameters remain after nondimensionalization?
\item[(d)] Find the solution of the OU equation on the domain \(-\infty<x<\infty\),
\(t\ge0\) with initial data \(p(x,0)=\delta(x-x_0)\) for arbitrary \(x_0\). Give an
interpretation of this solution in terms of a probability.
\end{enumerate}