1. Kolmogorov’s Equations

A stochastic process is a parameterized family of random variables. For the most part, in this book the stochastic processes with which we will deal are parameterized by \(t\in [0,\infty )\), to be thought of as “time,” and take values in some Euclidean space \({\mathbb {R}^N}\). When attempting to analyze such a family \(\{X(t): t\ge 0\}\) of random variables on a probability space \((\varOmega ,\mathcal {F},{\mathbb {P}})\), the first step is to understand the distribution \(\mu _t\) of X(t) for each \(t\ge 0\). Thus one is forced to consider paths \(t\rightsquigarrow \mu _t\) in the space \(\mathbf {M}_1({\mathbb {R}^N})\) of probability measures on \({\mathbb {R}^N}\), and, as usual, this leads one to consider the “tangent field” along the path. With this in mind, the goal of this chapter is to first make precise exactly what the tangent to a \(\mathbf {M}_1({\mathbb {R}^N})\)-valued path is and to then develop a procedure for recovering the path from its tangent field.