  For nonlinear functions f of a random vector Y, E[f(Y)] and f(E[Y]) usually
differ. Consequently the mathematical expectation of Y is not intrinsic: when
we change coordinate systems, it is not invariant.This article is about a
fundamental and hitherto neglected property of random vectors of the form Y =
f(X(t)), where X(t) is the value at time t of a diffusion process X: namely
that there exists a measure of location, called the "intrinsic location
parameter" (ILP), which coincides with mathematical expectation only in special
cases, and which is invariant under change of coordinate systems. The
construction uses martingales with respect to the intrinsic geometry of
diffusion processes, and the heat flow of harmonic mappings. We compute
formulas which could be useful to statisticians, engineers, and others who use
diffusion process models; these have immediate application, discussed in a
separate article, to the construction of an intrinsic nonlinear analog to the
Kalman Filter. We present here a numerical simulation of a nonlinear SDE,
showing how well the ILP formula tracks the mean of the SDE for a Euclidean
geometry.
