One-sided semiconvex characteristics give W2 uniqueness and stability

For $\epsilon >0$, choose a coupling ${\gamma }_0$ of ${\mu }_0,{\nu }_0$ with $\int \mathrm{\mid }x-y{\mathrm{\mid }}^{2}d{\gamma }_0\le W_2^2({\mu }_0,{\nu }_0)+\epsilon$, using the defining infimum of $W_2$. By [F1], take path laws ${\mathrm{\Pi }}_{\mu },{\mathrm{\Pi }}_{\nu }$ for the two solutions. For either path law ${\mathrm{\Pi }}_{\alpha }$, every rational-time marginal is concentrated on $Q_r$; closedness of $Q_r$, continuity of paths, and countability of the rational times imply almost every path stays in $Q_r$ throughout $[0,T]$. Since $Q_r$ is convex and ${\sup }_{Q_r}\mathrm{\parallel }d\mathrm{\Phi }\mathrm{\parallel }=A_r$, $\mathrm{\Phi }$ is $A_r$-Lipschitz there, and $\mathrm{\parallel }{m}_{{\alpha }_t}-m_{{\alpha }_s}{\mathrm{\parallel }}_{\mathcal{H}}\le A_r{\mathbb{E}}_{{\mathrm{\Pi }}_{\alpha }}\mathrm{\mid }\gamma (t)-\gamma (s)\mathrm{\mid }\le A_r\mathrm{\mid }t-s{\mathrm{\mid }}^{1\mathrm{/}2}{({\mathbb{E}}_{{\mathrm{\Pi }}_{\alpha }}{\int }_s^t\mathrm{\mid }\dot{\gamma }(q){\mathrm{\mid }}^{2}dq)}^{1\mathrm{/}2}\mathrm{.}$ Thus $m_{{\alpha }_t}$ and $a_{\alpha }(t)=\mathrm{\nabla }R(m_{{\alpha }_t})$ are continuous. Disintegrate each law conditional on its time-zero position, then combine the two kernels with ${\gamma }_0$ using [F2]. This gives a joint law of paths $(X,Y)$ with marginals ${\mathrm{\Pi }}_{\mu },{\mathrm{\Pi }}_{\nu }$ and initial pair law ${\gamma }_0$. Set $D(t)=\mathbb{E}\mathrm{\mid }{X}_t-Y_t{\mathrm{\mid }}^2$. The path equations and the scalar chain rule show $D$ is absolutely continuous and ${\displaystyle \frac{1}{2}{D}^{\mathrm{\prime }}(t)=\mathbb{E}⟨X_t-Y_t,v_t^{\mu }(X_t)-v_t^{\nu }(Y_t)⟩}$ for almost every $t$. The product is integrable by Cauchy–Schwarz because both path actions and second moments are finite.

We construct characteristics for any continuous $a:[0,T]\to \mathcal{H}$, with $M={\max }_t\mathrm{\parallel }a(t)\mathrm{\parallel }$, $g(t,x)=d{\mathrm{\Phi }}_x^{\ast }a(t)$, $f=V+{\iota }_D$ and $\lambda ={\lambda }_V$. The sets $Q_s$ cover $\mathrm{\Omega }$: for $q\in Q_1$ the neighborhood condition puts $x\in \mathrm{\Omega }$ in $Q_{1+\mathrm{\mid }x-q\mathrm{\mid }}$, increasing the radius slightly when necessary. Thus every bounded subset of $\mathrm{\Omega }$ lies in a $Q_s$. Bounded supporting potential subgradients at $x,y\in Q_s$ give $\mathrm{\mid }V(y)-V(x)\mathrm{\mid }\le (C_1+C_{2}s)\mathrm{\mid }y-x\mathrm{\mid }+\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }y-x{\mathrm{\mid }}^2\mathrm{/}2$. Subdivide the segment into $n$ equal pieces and let $n\to \mathrm{\infty }$ to see that $V$ is $(C_1+C_{2}s)$-Lipschitz on $Q_s$. Therefore $f$ is proper, lower semicontinuous and $\lambda$-semiconvex, and its subdifferential is nonempty everywhere on $D$. Its supporting inequalities give $⟨\xi -\eta ,x-y⟩\ge \lambda \mathrm{\mid }x-y{\mathrm{\mid }}^2$. For $1+h\lambda >0$, the function $f(y)+\mathrm{\mid }y-z{\mathrm{\mid }}^2\mathrm{/}(2h)$ is coercive and strongly convex: use a supporting inequality at a fixed point of $D$ for the quadratic lower bound. Finite-dimensional compactness gives its unique minimizer $J_{h}z$, and testing minimality along segments gives $(z-J_{h}z)\mathrm{/}h\in {\mathrm{\partial }}_{\lambda }f(J_{h}z)$. Adding two such inequalities gives $\mathrm{\mid }{J}_{h}z-J_{h}w\mathrm{\mid }\le \mathrm{\mid }z-w\mathrm{\mid }\mathrm{/}(1+h\lambda )$.

Write each full shifted selection as $d{\mathrm{\Phi }}_x^{\ast }\mathrm{\nabla }R(m_{{\mu }_t})+{\xi }_{\mu }(x)$ and $d{\mathrm{\Phi }}_y^{\ast }\mathrm{\nabla }R(m_{{\nu }_t})+{\xi }_{\nu }(y)$, with the two potential sections in ${\mathrm{\partial }}_{{\lambda }_V}(V+{\iota }_D)$; they need not coincide as functions of position. Adding their supporting inequalities gives $⟨{\xi }_{\mu }(x)-{\xi }_{\nu }(y),x-y⟩\ge {\lambda }_V\mathrm{\mid }x-y{\mathrm{\mid }}^2$. For the smooth terms, add and subtract $d{\mathrm{\Phi }}_y^{\ast }\mathrm{\nabla }R(m_{{\mu }_t})$: their difference has norm at most $N_r{B}_r\mathrm{\mid }x-y\mathrm{\mid }+A_r{L}_r\mathrm{\mid }{m}_{{\mu }_t}-m_{{\nu }_t}\mathrm{\mid }$. Along the joint law, $\mathrm{\mid }{m}_{{\mu }_t}-m_{{\nu }_t}\mathrm{\mid }\le A_r\mathbb{E}\mathrm{\mid }{X}_t-Y_t\mathrm{\mid }\le A_{r}D(t{)}^{1\mathrm{/}2}$. Hence [F3] gives ${\displaystyle \frac{1}{2}{D}^{\mathrm{\prime }}(t)\le (C_r-{\lambda }_V)D(t)\mathrm{.}}$ Multiplying by $e^{-2(C_r-{\lambda }_V)t}$ and differentiating proves $D(t)\le e^{2(C_r-{\lambda }_V)t}D(0)$. The time-$t$ path pair couples ${\mu }_t,{\nu }_t$, so $W_2^2({\mu }_t,{\nu }_t)\le D(t)$; using $D(0)\le W_2^2({\mu }_0,{\nu }_0)+\epsilon$, letting $\epsilon \downarrow 0$, and taking square roots proves the stated estimate. If the initial laws agree, its right side is zero, proving uniqueness.

For $h=T\mathrm{/}n$, $h\mathrm{\mid }\lambda \mathrm{\mid }\le 1\mathrm{/}2$, define $x_0=u\in D\cap Q_r$, $x_k=J_h(x_{k-1}-hg(t_k,x_{k-1}))$, $v_k=(x_k-x_{k-1})\mathrm{/}h$, and ${\xi }_k=-v_k-g(t_k,x_{k-1})\in {\mathrm{\partial }}_{\lambda }f(x_k)$. Put $S_k={\sum }_{j\le k}h\mathrm{\mid }{v}_j\mathrm{\mid }$. The neighborhood condition gives $x_k\in Q_{r+S_k}$. Compare ${\xi }_k$ with a supporting potential vector ${\eta }_{k-1}$ at $x_{k-1}$ of norm at most $C_1+C_2(r+S_{k-1})$. Monotonicity yields $(1+h\lambda )\mathrm{\mid }{v}_k{\mathrm{\mid }}^2\le -⟨g(t_k,x_{k-1})+{\eta }_{k-1},v_k⟩$, so $\mathrm{\mid }{v}_k\mathrm{\mid }\le A+B{S}_{k-1}$, where $A=2(M+1)(C_1+C_{2}r)$ and $B=2(M+1)C_2$. Hence $S_k\le (1+hB)S_{k-1}+hA$ and $S_k\le S_{\ast }=(A\mathrm{/}B)(e^{BT}-1)$, with $S_{\ast }=AT$ when $B=0$. All iterates and affine interpolants $X_h$ lie in $D\cap Q_R$, where $R=r+S_{\ast }+1$, and have speed at most $K=A+B{S}_{\ast }$. For each fixed $u$, their images lie in the compact ball $\bar{B}(u,KT)$ and they are uniformly Lipschitz. Arzelà–Ascoli gives a subsequence converging uniformly to a $K$-Lipschitz $D\cap Q_R$-valued path $X$. Both constant interpolants converge uniformly to $X$, $V(x_k)$ converges uniformly to $V(X)$ by step 1.2, and $g_h(t)=g(t_k,x_{k-1})$ converges uniformly to $g(t,X(t))$ by continuity of $a$ and the Lipschitz bound on $d\mathrm{\Phi }$ on $Q_R$.

Write $x_h^{+}(t)=x_k$ on $(t_{k-1},t_k]$. The discrete supporting inequality is $⟨X_h^{\mathrm{\prime }},x_h^{+}-z⟩+V(x_h^{+})+⟨g_h,x_h^{+}-z⟩+(\lambda \mathrm{/}2)\mathrm{\mid }{x}_h^{+}-z{\mathrm{\mid }}^2\le V(z)$ for every $z\in D$. Replace $x_h^{+}$ by $X_h$ only in the derivative pairing; its error is at most $K^{2}h$. Integrating over any $[b,c]\subset [0,T]$ makes that pairing the exact endpoint difference $(\mathrm{\mid }{X}_h(c)-z{\mathrm{\mid }}^2-\mathrm{\mid }{X}_h(b)-z{\mathrm{\mid }}^2)\mathrm{/}2$. All other terms converge uniformly by step 2.2, yielding $(\mathrm{\mid }X(c)-z{\mathrm{\mid }}^2-\mathrm{\mid }X(b)-z{\mathrm{\mid }}^2)\mathrm{/}2+{\int }_b^c[V(X)+⟨g(t,X),X-z⟩+(\lambda \mathrm{/}2)\mathrm{\mid }X-z{\mathrm{\mid }}^2]dt\le (c-b)V(z)$. Choose countably many dense $z\in D$ and differentiate this interval inequality where the scalar absolutely continuous functions and coordinates of $X$ are differentiable. Outside their common null set the inequality says $-X^{\mathrm{\prime }}-g(t,X)\in {\mathrm{\partial }}_{\lambda }f(X)$, first for these tests and then for every $z\in D$ by continuity of $V$ on bounded sets. This proves the inclusion without any continuity or convergence assertion for the nonsmooth minimum field.

Two solutions $X,Y$ to the same forcing have bounded images in a common $Q_s$. Monotonicity gives $(\mathrm{\mid }X-Y{\mathrm{\mid }}^2\mathrm{/}2{)}^{\mathrm{\prime }}\le (M{B}_s-\lambda )\mathrm{\mid }X-Y{\mathrm{\mid }}^2$, so multiplication by the scalar integrating factor proves uniqueness. Therefore all converging Euler subsequences have the same limit and compactness gives uniform convergence of the entire family. For all starts in $D\cap Q_r$, step 2.2 gives the same radius $R$ and time-Lipschitz constant $K$; the same distance estimate gives $\mathrm{\mid }X(t,u)-X(t,w)\mathrm{\mid }\le e^{(M{B}_R-\lambda )t}\mathrm{\mid }u-w\mathrm{\mid }$. Together these prove joint continuity in $(t,u)$ and uniform Lipschitz dependence on the initial point. Nearby starts lie in a common $Q_s$ by the neighborhood property. For different forcings $a,b$ and paths in $Q_s$, monotonicity gives $(\mathrm{\mid }X-Y{\mathrm{\mid }}^2\mathrm{/}2{)}^{\mathrm{\prime }}\le c\mathrm{\mid }X-Y{\mathrm{\mid }}^2+A_s\mathrm{\parallel }a-b\mathrm{\parallel }\mathrm{\mid }X-Y\mathrm{\mid }$, $c=(\max \mathrm{\parallel }a\mathrm{\parallel })B_s-\lambda$. Apply this to $\sqrt{\mathrm{\mid }X-Y{\mathrm{\mid }}^2+{\epsilon }^2}$ and let $\epsilon \downarrow 0$ to obtain $\mathrm{\mid }X(t)-Y(t)\mathrm{\mid }\le e^{ct}\mathrm{\mid }X(0)-Y(0)\mathrm{\mid }+A_s{\int }_0^t{e}^{c(t-q)}\mathrm{\parallel }a(q)-b(q)\mathrm{\parallel }dq$. This is forcing stability. All finite horizons patch uniquely.

Since $V\circ X$ is Lipschitz, at a time where it and $X$ are differentiable and the inclusion holds, test any $\eta \in {\mathrm{\partial }}_{\lambda }f(X(t))$ against $X(t+h)$ for both signs of $h$. Division by $h$ and passage to zero give $(V\circ X{)}^{\mathrm{\prime }}=⟨\eta ,X^{\mathrm{\prime }}⟩$ for every such $\eta$ simultaneously. For $\xi =-X^{\mathrm{\prime }}-g(t,X)$ this gives $⟨\eta -\xi ,X^{\mathrm{\prime }}⟩=0$, so $\mathrm{\mid }g+\eta {\mathrm{\mid }}^2=\mathrm{\mid }-X^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }\eta -\xi {\mathrm{\mid }}^2$. Thus $-X^{\mathrm{\prime }}$ is the full shifted minimum. Its Borel measurability for fixed forcing follows directly from $q_{\epsilon }(t,x)=(x-J_{\epsilon }(x-\epsilon g(t,x)))\mathrm{/}\epsilon$: these are continuous, and monotonicity gives $(1+\epsilon \lambda )\mathrm{\mid }{q}_{\epsilon }{\mathrm{\mid }}^2\le ⟨g(t,x)+\eta ,q_{\epsilon }⟩$ for every $\eta \in {\mathrm{\partial }}_{\lambda }f(x)$. Therefore $q_{\epsilon }$ is bounded and its prox point tends to $x$. Continuity of $V$ passes the supporting inequalities to every cluster point $q$, placing it in the shifted subdifferential; the displayed inequality becomes its minimum-norm projection inequality. All cluster points equal the unique minimum, so the entire sequence converges. The pointwise limit of the continuous $q_{\epsilon }$ is Borel.

For any finite-action PDE solution with this velocity and fixed continuous forcing, [F1] represents it by absolutely continuous paths. Their positions lie in closed $D$ at every rational time and hence every time. Each path solves the inclusion, so uniqueness from step 4.1 identifies it with $X(\cdot ,u)$, where $u$ is its initial point. Thus ${\mu }_t=X(t,\cdot {)}_{\mathrm{\#}}{\mu }_0$, with the same radius bound from step 2.2 whenever ${\mu }_0$ is concentrated on $Q_r$. This argument uses neither a backward inverse nor continuity of the minimum field. With $a(t)=\mathrm{\nabla }R(m(t))$ it proves the characteristic assertion in the Statement; steps 1.1–2.1 prove the self-consistent law stability assertion.