Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \subset \mathcal P (\Omega)$ be the sequence of minimizers in the *minimizing movement scheme* of the time step $\tau$.

The following is taken from section 8.3 in Santambrogio's *Optimal transport for applied mathematicians*:

Let us define two interpolations between the measures $\varrho_{(k)}^\tau$. With this time-discretized method, we have obtained, for each $\tau>0$, a sequence $(\varrho_{(k)}^\tau)_k$. We can use it to build at least two interesting curves in the space of measures:

- first we can define some piecewise constant curves, i.e., $\varrho_t^\tau:=\varrho_{(k+1)}^\tau$ for $t \in ] k \tau,(k+1) \tau]$; associated with this curve, we also define the velocities $\mathbf{v}_t^\tau=\mathbf{v}_{(k+1)}^\tau$ for $t \in] k \tau,(k+1) \tau]$, where $\mathbf{v}_{(k+1)}^\tau$ is defined as $\mathbf{v}_{(k+1)}^\tau=(\mathrm{id}-$ $\left.\mathrm{T}_{k+1}^\tau\right) / \tau$ where $\mathrm{T}_{k+1}^\tau$ is the optimal transport map from $\varrho_{(k+1)}^\tau$ to $\varrho_{(k)}^\tau$; we also define the momentum variable $E^\tau=\varrho^\tau \mathbf{v}^\tau$;
- then, we can also consider the densities $\tilde{\varrho}_t^\tau$ that interpolate the discrete values $(\varrho_{(k)}^\tau)_k$ along geodesics: $$ \tilde{\varrho}_t^\tau=\left(\frac{k \tau-t}{\tau} \mathbf{v}_{(k)}^\tau+\mathrm{id}\right)_{\sharp} \varrho_{(k)}^\tau, \quad \text {for} \quad t \in](k-1) \tau, k \tau[; \quad (8.11) $$ the velocities $\tilde{\mathbf{v}}_t^\tau$ are defined so that $\left(\tilde{\varrho}^\tau, \tilde{\mathbf{v}}^\tau\right)$ satisfy the continuity equation and $\left\|\tilde{\mathbf{v}}_t^\tau\right\|_{L^2\left(\tilde{\varrho}_t^\tau\right)}=\left|\left(\tilde{\varrho}^\tau\right)^{\prime}\right|(t)$. To do so, we take $$ \tilde{\mathbf{v}}_t^\tau=\mathbf{v}_t^\tau \circ\left((k \tau-t) \mathbf{v}_{(k)}^\tau+\mathrm{id}\right)^{-1}; $$ as before, we define a momentum variable: $\tilde{E}_\tau=\tilde{\varrho}^\tau \tilde{\mathbf{v}}^\tau$.

When $t^* := (k-1) \tau$ we have by (8.11) that $$ \tilde{\varrho}_{t^*} = \left(\mathbf{v}_{(k)}^\tau+\mathrm{id}\right)_{\sharp} \varrho_{(k)}^\tau. $$

I could not see how $\tilde{\varrho}_{t^*}^\tau = \varrho_{(k-1)}^\tau$. As such, I could not see how $(\tilde{\varrho}_t^\tau)_t$ ''interpolates'' $(\varrho^\tau_{(k)})_n$.

Could you elaborate on my confusion?

Thank you so much for your help!