The following ODE consists of a sum of two time-dependent vector fields $u$ and $v$:
\begin{align} \begin{cases} d\mathbf{x}_{t} & = \left[ u(\mathbf{x}_{t},t) +v(\mathbf{x}_{t},t) \right] dt \\ \mathbf{x}_0 &= \mathbf{x}_{\rm{init}}, \end{cases} \end{align}
I am interested in a desired propery of the solution of that ODE. To derive that property, I lifted up the dimension by one resulting in the following ODE:
\begin{align} \begin{cases} d\breve{\mathbf{x}}_{t} & = \left[ U(\breve{\mathbf{x}}) +V(\breve{\mathbf{x}}) \right] dt \\ \breve{\mathbf{x}}_0 &= \breve{\mathbf{x}}_{\rm{init}}, \end{cases} \end{align}
where $\breve{\mathbf{x}} = (\mathbf{x},t)$, $U(\breve{\mathbf{x}}) = (u(\mathbf{x}_{t},t),\alpha)$ and $V(\breve{\mathbf{x}}) = (v(\mathbf{x}_{t},t),\beta)$ with $\alpha, \beta \in \mathbb{R}$ such that $\alpha + \beta =1$.
Here is the crux: I derived the desired property only if $\alpha \neq 0$. Intuitively, we have different dynamics of the lifted ODE for each pair of $\alpha$ and $\beta$. Therefore, to be really able to claim that the desired property holds, it must be valid for all combinations of $\alpha$ and $\beta$ with $\alpha + \beta =1$.
Question: Is that intuition really the case? Or can I say that the desired property holds in an almost sure sense? If so, how to argue that in a rigorous mathematical sense?
I hope the question is clear/ makes sense. I cannot find any references on that topic and any comment would be very much appreciated.
Thanks for your interest and attention!