In Chapter 2, Section 2.2, we showed how one can start with a transition probability function P(s, x; t, ·) and end up with a Markov process. The problem is: where does P(s, x; t, ·) come from? The example we gave there, namely: $$ P(s,x;t,\Gamma ) = \int\limits_\Gamma {g_d } \left( {t - s,y - x} \right)dy $$ (11) is a natural one from the probabilistic point of view because of its connection with independent increments and Gaussian processes. It turns out to be natural from another point of view as well: the theory of second order parabolic partial differential equations. The connection between the P(s, x; t, ·) in (1.1) and partial differential equations is well-known and easy to derive. Namely, if ϕ ∈ C b (R d ) and $$ f(s,\,x)\, = \,\int {g_d (T\, - \,s,\,y\, - \,x)\phi (y)\,dy,\,s < \,T,} $$ then $$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial f}} {{\partial s}} + \frac{1} {2}\Delta f\, = \,0,\,s\, < \,T,} \\ {\mathop {\lim }\limits_{s\dag T} f(s, \cdot \,)\, = \,\phi } \\ \end{array} } \right. $$ (1.2) where Δ is Laplace’s operator $$ \sum\limits_1^d {\frac{{\partial ^2 }} {{\partial x_i^2 }}}. $$