The logarithmic chain rule and deterministic selection

First restrict $P$ to paths whose complete graphs lie in a fixed compact subset $K$ of $[0,T]\times D$; no positive-density restriction is used at this step. Denote this subprobability by $P_K$. Its time marginals only decrease, so are still at most $C\rho_t\,dx$ and are bounded on the compact spatial region by continuity of $\Psi$. Choose a slightly larger compact region inside the physical domain and locally extend the wave. At the time endpoints a constant extension suffices. Spacetime mollification produces smooth waves $\Psi_n$ converging uniformly on $K$, with strong convergence of spatial first derivatives in $L^2(dt\,dx)$ and of time derivatives in $L^1_tL^2_x$ locally. These follow from the stipulated continuity, local $H^2$ and time-derivative hypotheses.

For fixed $\epsilon>0$, the map $z\mapsto\log(\|z\|^2+\epsilon)$ has bounded first derivative. Uniform convergence of $\Psi_n$ on the compact region, together with its uniformly bounded values, therefore gives uniform convergence of $h_n=\log(\|\Psi_n\|^2+\epsilon)$ to $h=\log(\rho+\epsilon)$. The chain formulas and the strong derivative convergences give

For example, subtract the two bounded coefficient factors multiplying $\partial_t\Psi_n$ and $\partial_t\Psi$; their uniform difference tends to zero and the latter derivative is integrable. The spatial argument is identical in $L^2$.

The smooth chain rule holds for $h_n$ along every admitted absolutely continuous curve. The expected time-derivative error tends to zero by the bounded local marginals and Cauchy on the finite spatial volume. If $\theta_t\,dx=(e_t)_\#P_K$, the spatial error is bounded by

Uniform convergence passes the endpoint values. A subsequence with summable expected derivative error, followed by a countable set of rational endpoints, gives the integral chain identity almost surely. The integral representative and continuity of $h(t,\gamma(t))$ extend it to every endpoint. This proves absolute continuity and (3.2) on the restricted paths.

Every continuous path staying in $D$ has a compact graph in $[0,T]\times D$. Exhaust these graphs by nested compact regions and take the countable intersection of their full-measure chain-rule sets. By marginal domination,

This establishes (3.3). There is no substitution of a weak density into an unproved classical path chain rule.

The entrance has $\rho_0>0$ almost surely by domination. For the countable sequence $\epsilon=1/n$, a path starting there and ever reaching a node has variation at least $\log(\rho_0(\gamma(0))+\epsilon)-\log\epsilon$, which diverges. Fatou and the uniform expectation bound exclude a positive mass of such paths. Since the density is continuous along a node-avoiding path on a compact time interval, it has a positive minimum there. Letting $\epsilon\downarrow0$ now bounds the variation of $\log\rho_t(\gamma(t))$ by $C\mathcal N$ in expectation. A drop from above $a_0$ to $a$ costs at least $\log(a_0/a)$, proving (3.4).

Let $P_1,P_2$ have the same entrance $\mu_0$ and respective finite marginal caps; enlarge $C$ to cover both. Disintegrate over entrance and form the conditional product

Its coordinate marginals are $P_1,P_2$ and $\gamma(0)=\eta(0)$ almost surely. By Lemma 3.2 and boundary avoidance, almost every pair of complete graphs lies in some compact $K_n\subset U$. For example choose nested compacts with $|x|\leq n$, distance to $D^c$ at least $1/n$ when relevant, and $\rho\geq1/n$. Restrict $\Theta$ to pairs whose whole graphs lie in $K_n$; call the restriction $\Theta_n$. It retains marginal bounds $C\rho_t\,dx$. Paths are not stopped and their mass is not placed on a boundary.

Let $B=\chi b$ be the positive-tube Sobolev extension supplied by Assumption 3.1. The Sobolev maximal-function inequality [5, Lemmas A.2–A.3] gives, outside an appropriate Lebesgue-null set for almost every $t$,

Here $\mathcal M$ is the Hardy–Littlewood maximal operator. Marginal absolute continuity makes its exceptional set negligible along the paired paths for almost every time. Their AC equations and $B=b$ on $K_n$ imply, for every $\delta>0$,

Indeed differentiate $\log(1+|\gamma-\eta|/\delta)$ almost everywhere, use (3.7), and integrate the nonnegative upper bound; equality at entrance removes the initial term.

The expectation of the right side under $\Theta_n$ is finite and independent of $\delta$. The marginals are supported in a spatial set $V_n$ of finite volume and are bounded there by $C\sup_{K_n}\rho$. Hölder's inequality and the $L^p$ boundedness of $\mathcal M$ bound the expectation by a finite multiple of $\int_0^T\|\nabla B(t)\|_pdt$. If $\Theta_n$ gave positive mass to $\sup_t|\gamma-\eta|\geq h$ for any $h>0$, the expected left side of (3.8) would diverge as $\delta\downarrow0$. Thus $\Theta_n$ is supported on the diagonal. Exhaustion gives the same conclusion for $\Theta$.

Apply this first with $P_1=P_2=P$. For almost every $x$, the product $P_x\otimes P_x$ is diagonal. A probability on a Polish space whose independent pair is almost surely equal is Dirac: otherwise some member of a countable separating family has probability strictly between zero and one, giving positive product mass off the diagonal. The location of that Dirac mass is a measurable function of $x$, because the Dirac embedding of a Polish space into its probability measures is Borel and has a Borel inverse on its image. Defining an arbitrary path on the entrance null set gives a Borel map. Using two different candidate laws in (3.6) makes their selected curves equal almost everywhere, proving uniqueness.

Take $P=\Pi$ to obtain (3.5). Finally $f_0(e_0)\Pi$ is exactly $\int\delta_{X(\cdot,x)}\,\mu_0(dx)$. Its absolute continuity with respect to $\Pi$ transfers all preceding almost-sure path properties, even if $f_0$ is unbounded.

Hermiticity gives $\partial_i\rho=2\operatorname{Re}\langle\Psi,D_i\Psi\rangle$. On a compact positive tube, $\Psi$ is bounded and $\rho$ has a positive floor. Differentiating the current weakly produces only terms of the types $\Psi D^2\Psi$, $(D\Psi)^2$, $\Psi D\Psi$ and $\Psi^2$, with bounded local coefficients. For $d>2$, local $H^2$ gives $D\Psi\in L^{2d/(d-2)}$; the products therefore belong to $L^p$ with the stated $p$. For $d=2$, $H^1$ gradients have every finite local $L^q$ exponent; choose $q=4$. The one-dimensional case is stronger. In the quotient rule for $b=j/\rho$, the remaining product $j\nabla\rho/\rho^2$ has the same squared-first-derivative form. A cutoff supported inside the positive region gives the required global $W^{1,p}$ extension with integrable time norm.

A Hermitian drift adds $\langle\Psi,a_i\Psi\rangle$, whose first derivative has these same product types. A Pauli curl differentiates a spin density once in $j$ and twice in $\nabla j$, again producing the listed products. This proves the local assertion without differentiating a singular or discontinuous scalar potential.

Finally $|j|\leq2\Lambda\sqrt\rho\,|D\Psi|$ and $|\nabla\rho|\leq2\sqrt\rho\,|D\Psi|$ pointwise. Use $\|\Psi\|_2=1$ and $\int|D\Psi|^2\leq K/\alpha$ to obtain the first three bounds. The identity $\partial_t\rho=2\operatorname{Re}\langle\Psi,\partial_t\Psi\rangle$ and Cauchy give the last. Variable unbounded coefficients instead require their corresponding weighted kinetic estimates.