l}{\partial y_j}=0,$$ and a similar equation with respect to the variables $x$. Then we get the equations for $l^2$ and $d^2$, i. e. the equations and below.

Recall that a minimal geodesic is a global distance minimizing geodesic. This minimal geodesic which connects $x$ and $y$ characterizes the Riemannian distance $d(x,y)$ in an obvious way. Moreover smoothness of $(x,y)\rightarrow d(x,y)$ for smooth diffusion and drift coefficients $a_{ij},b_i$ follows from the following fact about ordinary differential equations.

Let $F:{\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}$ be a smooth map. Consider the differential system $$\frac{d^2x}{dt^2}=F\left( x,\frac{dx}{dt}\right),$$ where $x$ is a map $I\subset {\mathbb R}\rightarrow {\mathbb R}^n$. Then for each point $(x_0,y_0)$ there exists a neighborhood $U\times V$ of this point and $\epsilon >0$ such that for $(x,v)\in U\times V$ equation (2.69) has a unique solution $x_v:]-\epsilon,\epsilon[\rightarrow {\mathbb R}^n$ with initial conditions $x_v(0)=x$ and $x_v'(0)=v$. Moreover, the map $X:U\times V\times ]-\epsilon,\epsilon[\rightarrow {\mathbb R}^n$ defined by $(t,x,v)\rightarrow X(t,x,v):=x_v(t)$ is smooth.

Finally we get

Let $\Omega \subseteq {\mathbb R}^n$ be some domain. The function $d^2:\Omega\times \Omega \subseteq {\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}_+$ (the leading order term of the WKB-expansion of a parabolic equation with diffusion coefficients $a_{ij}$) is the unique function which satisfies the equations 