MULTIVARIATE PARTIAL DIFFERENTIAL EQUATION DESCRIBING THE EVOLUTION OF A GAUSSIAN PROCESS

If {X(t), t ≥ 0} is a Gaussian process, the diffusion equation characterizes its marginal probability density function. How about finite-dimensional distributions? For each n ≥ 1, we derive a system of partial differential equations which are satisfied by the probability density function of the vector (X(t 1 ), …, X(t n )). We then show that these differential equations determine uniquely the finite-dimensional distributions of Gaussian processes. We also discuss situations where the system can be replaced by a single equation, which is either one member of the system, or an aggregate equation obtained by summing all the equations in the system.