$0<\varepsilon<\gamma$}.$$ We define the ideal $\mathcal N({\mathbb{R}}^n)$ of $\mathcal E({\mathbb{R}}^n)$ to be the set of families $(f_\varepsilon)_{0<\varepsilon<1}$ such that for all compact sets $K$, for all multi-indices $I$, and for all $q\in {\mathbb{N}}$ there exist $c>0$ and $\gamma>0$ such that $$\sup_{x\in K}\left|\partial^If_\varepsilon(x)\right|{\leqslant}c\varepsilon^q\quad\textrm{if $0<\varepsilon<\gamma$.}$$ This means that the elements of $\mathcal N({\mathbb{R}}^n)$ have a fast decay (faster than any power of $\varepsilon$) as $\varepsilon \searrow0$. The *Colombeau algebra* is the quotient $$\mathcal G({\mathbb{R}}^n)=\mathcal E({\mathbb{R}}^n)/\mathcal N({\mathbb{R}}^n).$$ All distributions $\eta$ are represented in $\mathcal G({\mathbb{R}}^n)$ because the families of smoothings by convolution with a mollifier are contained there. We denote by $[\eta]$ the set of elements of of $\mathcal G({\mathbb{R}}^n)$ that are associated to the distribution $\eta$. That is, $(f_\varepsilon)_\varepsilon$ belongs to the subset $[\eta]$ of $\mathcal G({\mathbb{R}}^n)$ if, for all $\phi\in C^\infty_c({\mathbb{R}}^n)$, $$\lim_{\varepsilon\to0+}\int\phi(x)f_\varepsilon(x)\,dx=\langle\eta,\phi\rangle.$$

We now proceed to use the Colombeau algebra $\mathcal G({\mathbb{R}}^n)$ to define the object corresponding to the direction of the motion. Let $\mu$ be a probability measure on ${\mathbb{R}}^n$ and let $\eta$ be a distribution satisfying the conditions in Corollary \[cor:two-sided\] including the last part for families of probability measures, so that there exists a family of probabilities $(\mu_t)_{t\in{\mathbb{R}}}$ with derivative $\eta$ at 0 and $\mu_0=\mu$.

Let $[\mu]$ be 