Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in \mathcal{P}$ is thought of a point in $\mathbb{R}^{\mathcal{X}}$.
Let $\mathcal{P}=\{p_{\xi}\}$ be a parameterization. An affine connection $\nabla^{(e)}$, called $e$-connection, is defined on $\mathcal{P}$, given by the Christoffel symbols \begin{eqnarray} \Gamma_{ij,k} & := & \sum_x\partial_i(\partial_j\log p_{\xi}(x))\cdot \partial_k p_{\xi}(x). \end{eqnarray} According to Wikipedia: In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it.
Now suppose that $\gamma(t)$ is a geodesic on $\mathcal{P}$, connecting two points $p$ and $q$. Then, does the above statement mean $$\frac{d}{dt}(\log\gamma(t))=\alpha(t)\cdot\textbf{v},$$ where $\alpha(t)$ is scalar depends on $t$ and $\textbf{v}$ is a fixed vector?