Theorem 1: the marginal continuity equation

For every natural number $d$, let $E=\mathbb R^{\{0,\ldots,d-1\}}$, with its usual coordinatewise real vector-space structure, product topology, maximum norm, and Lebesgue measure $\lambda$; let $Q$ be any probability measure on $E$ (so $Q(E)=1$); and let $\rho:\mathbb R\times E\times E\to\mathbb R$ and $v:\mathbb R\times E\times E\to E$ be arbitrary functions satisfying the following assumptions. Write $F(t,x,z)=\rho(t,x,z)v(t,x,z)$ for scalar multiplication of the vector $v(t,x,z)$ by the real number $\rho(t,x,z)$. For every $t\in[0,1]$ and every $x,z\in E$, $\rho(t,x,z)>0$; for every such $t$, the function $(x,z)\mapsto\rho(t,x,z)$ is measurable on $E\times E$; for every such $t$ and every $z\in E$, the function $x\mapsto\rho(t,x,z)$ is everywhere nonnegative, is integrable with respect to $\lambda$, and satisfies $\int_E\rho(t,x,z)\,d\lambda(x)=1$; and for every such $t$ and every $x\in E$, the function $z\mapsto\rho(t,x,z)$ is integrable with respect to $Q$. For every $t\in(0,1)$ and every $x\in E$, additionally assume all of the following time regularity conditions: $z\mapsto\rho(t,x,z)$ is $Q$-integrable; there is a neighborhood of $t$ on which, for every real $s$, $z\mapsto\rho(s,x,z)$ is almost everywhere strongly measurable with respect to $Q$; the function $z\mapsto\left.\frac{d}{ds}\rho(s,x,z)\right|_{s=t}$ is almost everywhere strongly measurable with respect to $Q$; and there exist a set $N\subseteq\mathbb R$ containing a neighborhood of $t$ and a $Q$-integrable real function $b:E\to\mathbb R$ such that, for $Q$-almost every $z$, $\left|\left.\frac{d}{dr}\rho(r,x,z)\right|_{r=s}\right|\le b(z)$ for every $s\in N$, and, for $Q$-almost every $z$, $s\mapsto\rho(s,x,z)$ is differentiable at every $s\in N$. For every $t\in(0,1)$ and every $x\in E$, also assume all of the following spatial regularity conditions: $z\mapsto F(t,x,z)$ is $Q$-integrable; there is a neighborhood of $x$ on which, for every $y$, $z\mapsto F(t,y,z)$ is almost everywhere strongly measurable with respect to $Q$; the function $z\mapsto D_yF(t,x,z)$, taking values in the continuous real linear maps from $E$ to $E$, is almost everywhere strongly measurable with respect to $Q$; and there exist a set $M\subseteq E$ containing a neighborhood of $x$ and a $Q$-integrable real function $c:E\to\mathbb R$ such that, for $Q$-almost every $z$, $\|D_yF(t,y,z)\|_{\mathrm{op}}\le c(z)$ for every $y\in M$, and, for $Q$-almost every $z$, $y\mapsto F(t,y,z)$ is Fréchet differentiable at every $y\in M$. The neighborhoods and bounds in these assumptions may depend on $t$ and $x$; in each bound and differentiability assertion, the exceptional null set is independent of the point ranging over the indicated neighborhood, while the separate assertions may initially have different exceptional null sets. Here $D_yF(t,y,z)$ denotes the real Fréchet derivative of $w\mapsto F(t,w,z)$ at $y$, with operator norm induced by the maximum norm on $E$; scalar derivatives and Fréchet derivatives used as functions are assigned the value zero at points where the corresponding function is not differentiable. Let $e_i$ be the vector with coordinate $1$ at $i$ and $0$ elsewhere, and define the divergence of any function $G:E\to E$ at $x$ to be $\operatorname{div}G(x)=\sum_{i=0}^{d-1}(DG(x)[e_i])_i$, using that same total Fréchet derivative convention. The final assumption is that, for every $t\in(0,1)$ and every $x\in E$, for $Q$-almost every $z$ the scalar function $s\mapsto\rho(s,x,z)$ has a derivative at $t$ equal to $-\operatorname{div}(y\mapsto F(t,y,z))(x)$; the exceptional null set in this assumption is allowed to depend on both $t$ and $x$. Define, for every real $t$ and every $x\in E$, $p(t,x)=\int_E\rho(t,x,z)\,dQ(z)$, $J(t,x)=\int_EF(t,x,z)\,dQ(z)$, and $u(t,x)=p(t,x)^{-1}J(t,x)$, where the vector integral is a Bochner integral and the reciprocal of zero is defined to be zero. The theorem asserts both that, for every $t\in[0,1]$, $x\mapsto p(t,x)$ is everywhere nonnegative, is $\lambda$-integrable, has $\int_Ep(t,x)\,d\lambda(x)=1$, and satisfies $p(t,x)>0$ for every $x\in E$, and that, for every $t\in(0,1)$ and every $x\in E$, the function $y\mapsto p(t,y)u(t,y)$ is Fréchet differentiable at $x$ and the function $s\mapsto p(s,x)$ has a derivative at $t$ equal to $-\operatorname{div}(y\mapsto p(t,y)u(t,y))(x)$. Thus the derivative assertion is pointwise for every interior time and every spatial point; there is no derivative assertion at $t=0$ or $t=1$. The integrals defining $p$, $J$, and $u$ are total integrals, returning zero when the corresponding integrand is not integrable; the stated hypotheses ensure integrability of $p$'s integrand throughout $[0,1]$ and of $J$'s integrand throughout $(0,1)$, but do not require the latter at the two endpoints or either integrability condition at other times. Positivity in the conclusion makes the reciprocal in $u$ nonzero and ordinary division throughout $[0,1]$. No absolute continuity condition on $Q$ is assumed. The dimension $d=0$ is included: $E$ is then a singleton vector space, the divergence is an empty sum and equals zero, and every vector field takes the unique zero-vector value; the derivative asserted in the conclusion is consequently zero. All almost-everywhere statements are with respect to the specified probability measure $Q$; no single common full-measure set for all times and all spatial points is required.