Superposition for finite-action Euclidean continuity equations

Set ${\epsilon }_n=1\mathrm{/}n$. For each n, let $D_n$ be the Borel full-${\mu }_0^{{\epsilon }_n}$-measure set of initial points whose regularized flow paths exist on all of $[0,T]$. The map $F_n:D_n\to C([0,T];{\mathbb{R}}^d)$, $F_n(x)(t)=X_t^{{\epsilon }_n}(x)$, is Borel by [F1]. Extend it by the zero path off $D_n$ and define ${\eta }_n=(F_n{)}_{\mathrm{\#}}{\mu }_0^{{\epsilon }_n}$; this is the law of a Borel random element and is a Borel probability. For every t, composition with evaluation gives $(e_t{)}_{\mathrm{\#}}{\eta }_n=(X_t^{{\epsilon }_n}{)}_{\mathrm{\#}}{\mu }_0^{{\epsilon }_n}={\mu }_t^{{\epsilon }_n}$ by [F1]. The flow equation, the same marginal identity, and Tonelli give $\int \mathcal{E}(\gamma )d{\eta }_n(\gamma )={\int }_0^T\text{\hspace{0.17em}⁣}\int \mathrm{\mid }{v}_t^{{\epsilon }_n}(x){\mathrm{\mid }}^{2}d{\mu }_t^{{\epsilon }_n}(x)dt\le A,$ where $A={\int }_0^T\text{\hspace{0.17em}⁣}\int \mathrm{\mid }{v}_t(x){\mathrm{\mid }}^{2}d{\mu }_t(x)dt<\mathrm{\infty }$. For every fixed t, ${\mu }_t^{{\epsilon }_n}\Rightarrow {\mu }_t$: for bounded continuous f, first restrict y to a compact set carrying all but arbitrarily small ${\mu }_t$-mass using [F8], then use uniform continuity of f on a one-neighborhood of that compact and concentration of the Gaussian kernels at zero.

The path space C with the uniform metric is Polish: it is complete by the continuous-curve completeness fact, and rational polygonal paths are countable and dense. By [F2], the sequence of initial laws is tight. Given $\delta >0$, choose compact K with ${\mu }_0^{{\epsilon }_n}(K^c)<\delta \mathrm{/}2$ for all n, and choose R so $A\mathrm{/}R<\delta \mathrm{/}2$. The set $K_R=\{\gamma :\gamma (0)\in K,\mathcal{E}(\gamma )\le R\}$ is closed by [F3] and compact by Arzela–Ascoli: its paths obey $\mathrm{\mid }\gamma (t)-\gamma (s)\mathrm{\mid }\le \sqrt{R\mathrm{\mid }t-s\mathrm{\mid }}$ and ${\sup }_t\mathrm{\mid }\gamma (t)\mathrm{\mid }\le {\sup }_{x\in K}\mathrm{\mid }x\mathrm{\mid }+\sqrt{RT}$. Markov's inequality, which follows from ${\mathbf{1}}_{\{\mathcal{E}>R\}}\le \mathcal{E}\mathrm{/}R$, gives ${\eta }_n(K_R)\ge 1-\delta$. Thus $({\eta }_n)$ is tight; [F2] gives a weakly convergent subsequence, still denoted ${\eta }_n$, with Borel probability limit $\eta$.

For each t, evaluation $e_t:C\to {\mathbb{R}}^d$ is continuous, so weak convergence and the marginal convergence ${\mu }_t^{{\epsilon }_n}\Rightarrow {\mu }_t$ imply $(e_t{)}_{\mathrm{\#}}\eta ={\mu }_t$. By [F3], $\mathcal{E}$ is nonnegative and lower semicontinuous. For each integer k define $g_k(\gamma )=\min \{k,{\inf }_{\zeta \in C}[\mathcal{E}(\zeta )+k\mathrm{\parallel }\gamma -\zeta {\mathrm{\parallel }}_{\mathrm{\infty }}]\}$. The triangle inequality makes $g_k$ k-Lipschitz, $0\le g_k\le \mathcal{E}$, and $g_k$ increases with k. If $a<\mathcal{E}(\gamma )$, lower semicontinuity gives a neighborhood of $\gamma$ where $\mathcal{E}>a$; outside that neighborhood the term $k\mathrm{\parallel }\gamma -\zeta {\mathrm{\parallel }}_{\mathrm{\infty }}$ exceeds a for all sufficiently large k. Thus $g_k(\gamma )\uparrow \mathcal{E}(\gamma )$. By [F9], weak convergence gives $\int g_{k}d\eta ={\lim }_n\int g_{k}d{\eta }_n\le {\mathrm{lim\hspace{0.17em}inf}}_n\int \mathcal{E}d{\eta }_n$ for every k. Monotone convergence [F9] yields $\int \mathcal{E}d\eta \le {\mathrm{lim\hspace{0.17em}inf}}_n\int \mathcal{E}d{\eta }_n\le A$. In particular $\eta$-almost every path is absolutely continuous with square-integrable derivative.

Give $[0,T]\times C$ the probability measure $\lambda =(dt\mathrm{/}T)\otimes \eta$ and define the Borel random element $Z(t,\gamma )=(t,\gamma (t),\dot{\gamma }(t))$, setting the last coordinate to zero wherever the derivative limit is undefined. Measurability follows by expressing the derivative as the limit of measurable difference quotients; it agrees with the path derivative for $\lambda$-almost every pair. The law $\mathrm{\Pi }$ of Z has $(t,x)$ marginal $M\mathrm{/}T$ by the time-marginal identity in step 2.1. Disintegrate with the velocity as the conditioned coordinate, and put $b(t,x)=\int u{K}_{t,x}(du)$ where defined, zero otherwise. It is Borel by [F4]. Conditional Jensen and step 2.1 give $\int \mathrm{\mid }b{\mathrm{\mid }}^{2}dM\le T\int \mathrm{\mid }u{\mathrm{\mid }}^{2}d\mathrm{\Pi }=\int \text{\hspace{0.17em}⁣}{\int }_0^T\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}^{2}dtd\eta (\gamma )\le A\mathrm{.}$ The factor T cancels because $\mathrm{\Pi }$ uses normalized time.

Fix a smooth compactly supported vector field $\xi (t,x)$. For a finite partition $P=(t_i)$ of mesh h, set $S_P(\gamma )={\sum }_i\xi (t_i,\gamma (t_i))\cdot (\gamma (t_{i+1})-\gamma (t_i))\mathrm{.}$ For every $A{C}^2$ path, bounded derivatives of $\xi$ imply $\mid {\int }_0^T\xi (t,\gamma (t))\cdot \dot{\gamma }(t)dt-S_P(\gamma )\mid \le C_{\xi }(h\sqrt{T}\mathcal{E}(\gamma {)}^{1\mathrm{/}2}+h\mathcal{E}(\gamma ))\mathrm{.}$ Indeed on each interval $\mathrm{\mid }\xi (t,{\gamma }_t)-\xi (t_i,{\gamma }_{t_i})\mathrm{\mid }\le C_{\xi }(h+\mathrm{\mid }{\gamma }_t-{\gamma }_{t_i}\mathrm{\mid })$, while Cauchy--Schwarz gives ${\int }_{t_i}^{t_{i+1}}\mathrm{\mid }\dot{\gamma }\mathrm{\mid }\le \sqrt{h{E}_i}$ and $\mathrm{\mid }{\gamma }_t-{\gamma }_{t_i}\mathrm{\mid }\le \sqrt{h{E}_i}$; summing yields the bound. Its expectation is uniform over ${\eta }_n$ and $\eta$ by the action bound. For fixed P with N intervals, $S_P$ is continuous and $\mathrm{\mid }{S}_P{\mathrm{\mid }}^2\le N\mathrm{\parallel }\xi {\mathrm{\parallel }}_{\mathrm{\infty }}^2{\sum }_i\mathrm{\mid }\gamma (t_{i+1})-\gamma (t_i){\mathrm{\mid }}^2$. Pathwise Cauchy--Schwarz gives $\int \mathrm{\mid }\gamma (t_{i+1})-\gamma (t_i){\mathrm{\mid }}^{2}d{\eta }_n\le (t_{i+1}-t_i)A$, so these sums have uniformly bounded second moments under ${\eta }_n$ and $\eta$. Thus $S_P$ is uniformly integrable, and weak convergence plus bounded truncations gives $\int S_{P}d{\eta }_n\to \int S_{P}d\eta$. Letting h tend to zero proves convergence of the expected path integrals.

Under ${\eta }_n$, the path derivative is $v_t^{{\epsilon }_n}(\gamma (t))$; hence its expected $\xi$-pairing is $\int \xi (t,x)\cdot v_t^{{\epsilon }_n}(x)d{\mu }_t^{{\epsilon }_n}(x)dt$. By the convolution definitions, $\int \xi \cdot v^{{\epsilon }_n}d{M}^{{\epsilon }_n}={\int }_0^T\text{\hspace{0.17em}⁣}\int v_t(y)\cdot ({\xi }_t\ast {\rho }_{{\epsilon }_n})(y)d{\mu }_t(y)dt⟶\int \xi \cdot vdM,$ since ${\xi }_t\ast {\rho }_{{\epsilon }_n}\to {\xi }_t$ uniformly and $v\in L^1(M)$ by finite action and Holder. Combining with step 3.2 and the conditional-mean definition of b gives $\int \xi \cdot bdM=\int \xi \cdot vdM$. Smooth-test density [F5] implies $b=v$ M-almost everywhere.

The time marginals of $\eta$ are ${\mu }_t$, so $\int \mathrm{\mid }v(t,\gamma (t)){\mathrm{\mid }}^{2}dtd\eta =A$. By step 3.1 the conditional mean of $\dot{\gamma }$ given $(t,\gamma (t))$ is b, which equals v by step 4.1. Therefore $\int \text{\hspace{0.17em}⁣}{\int }_0^T\mathrm{\mid }\dot{\gamma }(t)-v_t(\gamma (t)){\mathrm{\mid }}^{2}dtd\eta =\int \mathcal{E}d\eta +A-2\int v\cdot bdM=\int \mathcal{E}d\eta -A\le 0.$ The left side is nonnegative, so it is zero. Thus $\dot{\gamma }=v_t(\gamma (t))$ for $dtd\eta$-almost every pair, and the expected path action equals A. Together with the time marginals this is the claimed superposition law.