Solutions to Hamilton–Jacobi equation on a Wasserstein space

Abstract

We consider a Hamilton–Jacobi equation associated to the Mayer optimal control problem in the Wasserstein space $\mathscr {P}_2(\mathbb {R}^{d})$ and define its solutions in terms of the Hadamard generalized differentials. Continuous solutions are unique whenever we focus our attention on solutions defined on explicitly described time dependent compact valued tubes of probability measures. We also prove some viability and invariance theorems in the Wasserstein space and discuss a new notion of proximal normal.

Appendix

We equip the space $[0,1] \times \mathscr {P}_{2}(\mathbb {R}^{d})$ with the following metric: for $(s_i,\mu _i) \in [0,1] \times \mathscr {P}_{2}(\mathbb {R}^{d}), \, i=1,2$

$$\begin{aligned} d_2((s_1,\mu _1) ,(s_2,\mu _2) ) = \left( (s_1-s_2)^2+ W_2^2(\mu _1,\mu _2) \right) ^{1/2} \end{aligned}$$

and denote by $B((s,\mu ), R)$ the closed ball in this space with center at $(s,\mu ) \in [0,1] \times \mathscr {P}_{2}(\mathbb {R}^{d})$ and radius $R \ge 0$.

Proof of Lemma 6.1

Set $K = \mathscr {E}p(w)$ and let $(s,\nu ), \, ({\bar{t}},{\bar{\mu }}) \in \Delta _r$ and $z \in \mathbb {R}$ be as in the statement of the lemma. Then for any $(t,\mu , z') \in K$,

$$\begin{aligned} d ((t,\mu ,z'),(s,\nu ,z)) \ge d (({\bar{t}},{\bar{\mu }}, w({\bar{t}}, {\bar{\mu }})),(s,\nu ,z)). \end{aligned}$$

(7.1)

We first consider the case $z \ne w({\bar{t}}, {\bar{\mu }})$. Pick any $(\kappa , F, m) \in \overset{\circ }{T}_{K}({\bar{t}}, {\bar{\mu }}, w({\bar{t}}, {\bar{\mu }}))$. Then there exists a sequence $h_i \rightarrow 0+$ such that

$$\begin{aligned}&({\bar{t}} +h_i \kappa - s)^2 + W_{2}^{2}( (Id+h_iF)_{\#}{\bar{\mu }}, \nu )\\&\quad + (w({\bar{t}}, {\bar{\mu }}) +\,h_im - z)^2 - ({\bar{t}} - s)^2 - W_{2}^{2}({\bar{\mu }}, \nu ) - (w({\bar{t}}, {\bar{\mu }}) - z)^2 \ge o(h_i). \end{aligned}$$

Let $\alpha \in \Pi _{0}({\bar{\mu }}, \nu )$ and ${\bar{\alpha }}$ be its barycentric projection with respect to ${\bar{\mu }}$. Dividing by $2h_i>0$ and taking the limit in the above inequality as $i \rightarrow \infty$ we obtain, using the same arguments as in the proof of Proposition 2.14,

$$\begin{aligned} ({\bar{t}} - s)\kappa + \int _{\mathbb {R}^{d}\times \mathbb {R}^{d}} \langle x-y, F(x) \rangle \mathrm {d}\alpha (x,y) + (w({\bar{t}}, {\bar{\mu }}) - z)m \ge 0. \end{aligned}$$

Consider $p \in L^2({\bar{\mu }};\mathbb {R}^{d})$ such that $p(x)={\bar{\alpha }} (x)- x$ ${\bar{\mu }}$-a.e. Then the above inequality implies

$$\begin{aligned} (s -{\bar{t}}, p, z - w({\bar{t}}, {\bar{\mu }}) ) \in [\overset{\circ }{T}_{K}({\bar{t}}, {\bar{\mu }}, w({\bar{t}}, {\bar{\mu }}))]^{-}. \end{aligned}$$

As in Proposition 2.20 we get

$$\begin{aligned} \left( \frac{s-{\bar{t}}}{|z - w({\bar{t}}, {\bar{\mu }}) |}, \frac{1}{|z - w({\bar{t}}, {\bar{\mu }}) |} \,p\right) \in \partial ^-_H w({\bar{t}}, {\bar{\mu }}). \end{aligned}$$

Therefore, by the assumptions of lemma,

$$\begin{aligned} \frac{{\bar{t}} -s}{|z - w({\bar{t}}, {\bar{\mu }}) |} + \mathscr {H}\left( {\bar{\mu }}, - \frac{1}{|z - w({\bar{t}}, {\bar{\mu }}) |} p \right) \ge 0. \end{aligned}$$

Since the Hamiltonian is positively homogeneous with respect to the second variable, multiplying by $|z - w({\bar{t}}, {\bar{\mu }}) |$ we obtain the desired inequality.

It remains to consider the much more delicate case $z = w({\bar{t}}, {\bar{\mu }})$. Let $\gamma \in \Pi _{0}({\bar{\mu }}, \nu )$ and define

$$\begin{aligned} \nu _n = \left( \frac{1}{n} \pi _1 + \left( 1-\frac{1}{n} \right) \pi _2\right) _{\#}\gamma . \end{aligned}$$

By [4, Theorem 7.2.2], $W_2(\nu _n,\nu )= \frac{1}{n}W_2({\bar{\mu }},\nu )$. That is $\lim _{n \rightarrow \infty }\nu _n=\nu$ in $W_2$ metric. We also define

$$\begin{aligned}&s_n= \frac{1}{n}\,{\bar{t}} + \left( 1-\frac{1}{n} \right) s, \; \rho _n:=d_2 ((s_n,\nu _n),({\bar{t}}, {\bar{\mu }} )) \\&\quad = \left( 1-\frac{1}{n} \right) d ((s,\nu , z),({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) )), \; B_n=B((s_n,\nu _n), \rho _n) \end{aligned}$$

and observe that $\lim _{n \rightarrow \infty }s_n = s$, $z = \frac{1}{n} w({\bar{t}}, {\bar{\mu }}) + \left( 1-\frac{1}{n} \right) z = w({\bar{t}}, {\bar{\mu }}).$

Fix n.

Step 1. We prove here that

$$\begin{aligned} K \cap B((s_n,\nu _n, z),\rho _n) = \{({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) )\}. \end{aligned}$$

(7.2)

Let $(t,\mu ,m) \in K$ be such that $\mathrm{dist} ((s_n,\nu _n, z),K)= d ((s_n,\nu _n, z),(t,\mu ,m) )$. It exists because $\Delta _r$ is compact and K is closed. Then

$$\begin{aligned} t-s_n= \frac{1}{n} (t-{\bar{t}}) + \left( 1-\frac{1}{n} \right) (t-s), \quad m- z = \frac{1}{n} (m-w({\bar{t}}, {\bar{\mu }})) + \left( 1-\frac{1}{n} \right) (m-z). \end{aligned}$$

By [11, Proposition 1.1.3],

$$\begin{aligned} (t-s_n)^2 \ge \frac{1}{n} (t-{\bar{t}})^2 + \left( 1-\frac{1}{n} \right) (t-s)^2 - \frac{1}{n}\left( 1-\frac{1}{n} \right) ({\bar{t}} - s)^2 \end{aligned}$$

and

$$\begin{aligned} (m-z)^2 \ge \frac{1}{n} (m-w({\bar{t}}, {\bar{\mu }}))^2 + \left( 1-\frac{1}{n} \right) (m-z)^2 - \frac{1}{n}\left( 1-\frac{1}{n} \right) (w({\bar{t}}, {\bar{\mu }}) - z)^2. \end{aligned}$$

By [4, (7.3.12)] we also have

$$\begin{aligned} W_2^2(\nu _n,\mu ) \ge \frac{1}{n} W_2^2(\mu ,{\bar{\mu }}) + \left( 1-\frac{1}{n} \right) W_2^2(\mu ,\nu ) - \frac{1}{n}\left( 1-\frac{1}{n} \right) W_2^2(\nu , {\bar{\mu }}). \end{aligned}$$

Summing up the above inequalities we obtain

$$\begin{aligned} \rho _n^2&\ge d ^2((s_n,\nu _n, z),( t, \mu , m )) \ge \frac{1}{n} d ^2((t,\mu , m),({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) )) + \\&\quad \left( 1-\frac{1}{n} \right) d ^2((t,\mu , m),(s,\nu , z) ) - \frac{1}{n}\left( 1-\frac{1}{n} \right) d ^2((s,\nu , z),({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) )). \end{aligned}$$

From (7.1) we deduce that $\rho _n \le \left( 1-\frac{1}{n} \right) d ((s,\nu , z),(t, \mu ,m))$ and therefore

$$\begin{aligned} \left( 1-\frac{1}{n} \right) ^2d ^2(( t, \mu ,m ),(s,\nu , z))\ge & {} \frac{1}{n} d ^2((t,\mu , m),({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) ))\\&+ \left( 1-\frac{1}{n} \right) d ^2((t,\mu , m),(s,\nu , z) ) \\&- \frac{1}{n} \left( 1-\frac{1}{n} \right) d ^2((s,\nu , z),(t, \mu ,m)). \end{aligned}$$

The above inequality implies that $( t, \mu , m ) =({\bar{t}}, {\bar{\mu }},w({\bar{t}}, {\bar{\mu }}) )$ and (7.2) follows.

Step 2. Let $(t,\mu ) \notin B_n$. We claim that $d_2((t,\mu ),(s_n,\nu _n))=\mathrm{dist} ((t,\mu ), B_n) + \rho _n.$

Indeed, denoting by $\partial B_n$ the boundary of $B_n$ we have

$$\begin{aligned} \mathrm{dist}((t,\mu ), B_n) = \inf _{(a,\sigma ) \in \partial B_n} d_2((t,\mu ),(a,\sigma ) ) \end{aligned}$$

which yields

$$\begin{aligned} d_2((t,\mu ),(s_n,\nu _n)) \le \mathrm{dist} ((t,\mu ), \partial B_n) + \rho _n = \mathrm{dist} ((t,\mu ), B_n) + \rho _n. \end{aligned}$$

Consider any $\gamma \in \Pi _0(\mu ,\nu _n)$ and let $\mu _\lambda = ((1-\lambda ) \pi _1 +\lambda \pi _2)_\#\gamma .$ Then the mapping

$$\begin{aligned} \varphi (\lambda ):= ( (1-\lambda )t +\lambda s_n - s_n)^2 + W_2^2 (\mu _\lambda , \nu _n) = (1-\lambda )^2 ((t-s_n)^2 + W_2^2(\mu , \nu _n)) \end{aligned}$$

is continuous with respect to $\lambda \in [0,1]$ and $\varphi (0) > \rho _n^2$, $\varphi (1) =0$. This implies that there exists $\lambda \in (0,1)$ such that $\varphi (\lambda )=\rho _n^2.$ Hence $((1-\lambda )t +\lambda s_n, \mu _\lambda )$ is an element of $\partial B_n$ and we deduce that $(1-\lambda )\left( (t-s_n)^2 + W_2^2 (\mu , \nu _n)\right) ^{1/2} = \rho _n$. This and the triangle inequality yield

$$\begin{aligned} \mathrm{dist} ((t,\mu ), B_n) = \lambda \left( (t-s_n)^2 + W_2^2 (\mu , \nu _n)\right) ^{1/2}. \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{array}{ll} \mathrm{dist} ((t,\mu ), B_n ) + \rho _n &{} = \lambda d_2((t,\mu ),(s_n,\nu _n)) +(1-\lambda )d_2((t,\mu ),(s_n,\nu _n))\\ &{} = d_2((t,\mu ),(s_n,\nu _n)). \end{array} \end{aligned}$$

Step 3. For every integer $k \ge 1$ consider the minimization problem

$$\begin{aligned} \mathrm{minimize}_{(t,\mu ) \in \Delta _r} \left( w(t,\mu ) + k \,\mathrm{dist} ((t,\mu ), B_n) \right) . \end{aligned}$$

Denote by $(t_k,\mu _k)$ a minimizer that exists because $\Delta _r$ is compact and w is lower semicontinuous.

Case 1. We assume here that for some k, $(t_k,\mu _k) \in B_n$ and claim that $(t_k,\mu _k)=({\bar{t}}, {\bar{\mu }})$. Indeed, in this case $w(t_k,\mu _k) \le w({\bar{t}}, {\bar{\mu }})$. Therefore, $(t_k,\mu _k, w({\bar{t}}, {\bar{\mu }})) \in K$. On the other hand,

$$\begin{aligned} d ((t_k,\mu _k, w({\bar{t}}, {\bar{\mu }})), (s_n,\nu _n, z)) \le \rho _n \end{aligned}$$

and our claim follows from (7.2). Hence

$$\begin{aligned} w({\bar{t}}, {\bar{\mu }}) \le w(t,\mu ) + k\, \mathrm{dist} ((t,\mu ),B_n) \quad \forall \, (t,\mu ) \in \Delta _r. \end{aligned}$$

Let $\alpha _n \in \Pi _{0}({\bar{\mu }}, \nu _n)$, ${\bar{\alpha }}_n$ be its barycenter with respect to ${\bar{\mu }}$ and let $p \in L^2({\bar{\mu }} ;\mathbb {R}^{d})$ satisfy $p(x)={\bar{\alpha }}_n (x)- x$ ${\bar{\mu }}$-a.e. Consider any $(\kappa , F) \in \overset{\circ }{T}_{\Delta _r}({\bar{t}}, {\bar{\mu }})$ and $h_i \rightarrow 0+, \, (\tau _i,\xi _i) \in \Delta _r$ such that

$$\begin{aligned} D_{\uparrow }w({\bar{t}}, {\bar{\mu }})(\kappa ,F)= \lim _{i \rightarrow \infty } \frac{w(\tau _i,\xi _i) - w({\bar{t}},{\bar{\mu }})}{h_i} \end{aligned}$$

with $|\tau _i-{\bar{t}} - h_i \kappa | + W_2(\xi _i, (Id+h_i F)_{\#}{\bar{\mu }}) = o(h_i )$.

We discuss three possible occurrences:

• (a) There exists a subsequence $\{i_j\}_j$ such that $d_2((\tau _{i_j},\xi _{i_j}), (s_n,\nu _n)) >\rho _n$ for every j. Then $(\tau _{i_j},\xi _{i_j}) \notin B_n$ and by Step 2

  $$\begin{aligned} w({\bar{t}}, {\bar{\mu }}) + k \rho _n \le w( \tau _{i_j}, \xi _{i_j}) + k d_2(( \tau _{i_j},\xi _{i_j}), (s_n,\nu _n)) \quad \forall \, j. \end{aligned}$$

  (7.3)

  On the other hand, $W_2^2(\xi _{i_j},\nu _n )= W_2^2((Id+h_{i_j} F)_{\#}{\bar{\mu }},\nu _n) +o(h_{i_j})$ and therefore from (7.3) we obtain

  $$\begin{aligned} w({\bar{t}}, {\bar{\mu }}) + k \rho _n \le w( \tau _{i_j}, \xi _{i_j}) + k \sqrt{ d_2^2(({\bar{t}} + h_{i_j}\kappa ,(Id+h_{i_j}F)_{\#}{\bar{\mu }}),(s_n,\nu _n) ) + o(h_{i_j})}.\nonumber \\ \end{aligned}$$

  (7.4)

  Since $\alpha _n\in \Pi _{0}({\bar{\mu }} ,\nu _n )$ we know that $(\pi _1 +h_{i_j}F\circ \pi _1 , \pi _2 )_{\#}\alpha _n$ is a transport plan between $(Id+h_{i_j}F)_{\#}{\bar{\mu }}$ and $\nu _n$. Thus

  $$\begin{aligned} \begin{array}{ll} W_{2}^{2}( (Id+h_{i_j}F)_{\#}{\bar{\mu }}, \nu _n) &{} \le \displaystyle { \int _{\mathbb {R}^{d}\times \mathbb {R}^{d}}|x+h_{i_j}F(x)-y|^{2} \mathrm {d}\alpha _n(x,y)}\\ &{} \displaystyle { = W_2^2({\bar{\mu }},\nu _n) + 2h_{i_j} \int _{\mathbb {R}^{d}\times \mathbb {R}^{d}} \langle x-y, F(x) \rangle \mathrm {d}\alpha _n(x,y) +o(h_{i_j})} \end{array} \end{aligned}$$

  and

  $$\begin{aligned} ({\bar{t}} + h_{i_j}\kappa&-s_n)^2 + \; W_2^2((Id+h_{i_j} F)_{\#}{\bar{\mu }},\nu _n)\\&\quad \displaystyle \le ({\bar{t}} -s_n)^2 \\&\quad + 2h_{i_j}({\bar{t}} -s_n)\kappa + W_2^2({\bar{\mu }},\nu _n) + 2h_{i_j} \int _{\mathbb {R}^{d}\times \mathbb {R}^{d}} \langle x-y, F(x) \rangle \mathrm {d}\alpha _n(x,y) +o(h_{i_j})\\&\quad \displaystyle { = \rho _n^2\left( 1+ \frac{2h_{i_j}}{ \rho _n^2} \left( ({\bar{t}} -s_n)\kappa + \int _{\mathbb {R}^{d}\times \mathbb {R}^{d}} \langle x-y, F(x) \rangle \mathrm {d}\alpha _n(x,y) \right) + \frac{1}{ \rho _n^2}o(h_{i_j}) \right) . } \end{aligned}$$

  This and (7.4) imply

  $$\begin{aligned} -\frac{k}{\rho _n} \left( ({\bar{t}}-s_n)\kappa + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d} }\langle x-y, F(x) \rangle \mathrm{d }\alpha _n(x,y) \right) \le D_{\uparrow }w({\bar{t}}, {\bar{\mu }})(\kappa ,F). \end{aligned}$$

  (7.5)

• (b) There exists a subsequence $\{i_j\}$ such that for every j, $(\tau _{i_j},\xi _{i_j}) \in B_n$ and $d_2((\tau _{i_j},\xi _{i_j}), (s_n,\nu _n)) =\rho _n + o(h_{i_j}).$ Then the inequality $w({\bar{t}}, {\bar{\mu }}) \le w( \tau _{i_j},\xi _{i_j})$ implies that

  $$\begin{aligned} w({\bar{t}}, {\bar{\mu }}) + k \rho _n \le w( \tau _{i_j}, \xi _{i_j}) + k d_2(( \tau _{i_j},\xi _{i_j}), (s_n,\nu _n)) + k\,|o(h_{i_j})| \quad \forall \, j. \end{aligned}$$

  We deduce (7.5) in the same way as in (a).

• (c) It remains to consider the case when for some $\beta >0$ and all large i, $d_2((\tau _{i},\xi _{i}), (s_n,\nu _n)) ) \le \rho _n - \beta h_i.$ We claim that $D_{\uparrow }w({\bar{t}}, {\bar{\mu }})(\kappa ,F)= +\infty$. Indeed take $m= w({\bar{t}}, {\bar{\mu }}) + \sqrt{\beta \rho _n h_i}$ and observe that

  $$\begin{aligned} d((\tau _{i},\xi _{i},m),(s_n,\nu _n,z)) \le \sqrt{\rho _n^2 - 2\beta \rho _nh_i + \beta ^2 h_i^2 + \beta \rho _n h_i}= \sqrt{\rho _n^2 - \beta \rho _nh_i + \beta ^2 h_i^2 }. \end{aligned}$$

  Then $d((\tau _{i},\xi _{i},m),(s_n,\nu _n,z)) \le \rho _n$ whenever $h_i$ is sufficiently small. This and (7.2) imply that

  $$\begin{aligned} w({\bar{t}}, {\bar{\mu }}) + \sqrt{\beta \rho _n h_i} < w(\tau _{i},\xi _{i}). \end{aligned}$$

  Since the above holds true for any large i our claim follows.

Consequently, (a), (b) and (c) together imply (7.5) for any $(\kappa , F) \in \overset{\circ }{T}_{\Delta _r}({\bar{t}}, {\bar{\mu }})$. Therefore

$$\begin{aligned} -\frac{k}{\rho _n} ({\bar{t}}-s_n, - p ) \in \partial ^-_H w({\bar{t}}, {\bar{\mu }}) \end{aligned}$$

and by our assumption,

$$\begin{aligned} \frac{k}{\rho _n} ({\bar{t}}-s_n) + \frac{k}{\rho _n} \mathscr {H}({\bar{\mu }}, - p ) \ge 0. \end{aligned}$$

Hence for some $u \in U$

$$\begin{aligned} {\bar{t}} -s_n + \int _{\mathbb {R}^{d}} \langle - p(x), f({\bar{\mu }},u)(x)\rangle \mathrm {d} {\bar{\mu }}(x) \ge 0, \end{aligned}$$

which implies

$$\begin{aligned} {\bar{t}} -s_n + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d}} \langle x-y, f({\bar{\mu }},u)(x)\rangle \mathrm {d}\alpha _n(x,y) \ge 0. \end{aligned}$$

(7.6)

In this case we set $\mu _{n} = {\bar{\mu }}, t_{n} ={\bar{t}}, u_n=u$.

Case 2. For all k we have $(t_k,\mu _k) \notin B_n$. Then consider a subsequence $(t_{k_i}, \mu _{k_i})$ converging to some $(t,\mu ) \in \Delta _r$. Since

$$\begin{aligned} w(t_k,\mu _k) +k \,\mathrm{dist} ((t_k,\mu _k) , B_n ) \le w({\bar{t}}, {\bar{\mu }}), \end{aligned}$$

(7.7)

by the lower semicontinuity of w, we have $w(t,\mu ) \le w({\bar{t}}, {\bar{\mu }})$ and $(t,\mu ) \in B_n$. In the same way as in Case 1 we deduce that $(t,\mu )=({\bar{t}}, {\bar{\mu }}).$ Because $(t_{k_i}, \mu _{k_i})$ is an arbitrary converging subsequence it follows that $(t_k,\mu _k)$ converge to $({\bar{t}}, {\bar{\mu }})$ when $k \rightarrow \infty$.

By Step 2 for all $(t,\mu )$ sufficiently close to $(t_k,\mu _k)$ we have

$$\begin{aligned} w(t_k,\mu _k) + kd_2((t_k,\mu _k),(s_n,\nu _n)) \le w(t,\mu ) + k d_2((t,\mu ),(s_n,\nu _n)). \end{aligned}$$

(7.8)

Consider $k_n$ such that $(t_{k_n}-{\bar{t}})^2 + W_2^2 (\mu _{k_n}, {\bar{\mu }}) \le 1/n.$ To simplify the notation, we rename the variables by writing $t_n$ instead of $t_{k_n}$ and $\mu _n$ instead of $\mu _{k_n}$. Thus $\mu _n$ converge to ${\bar{\mu }}$ in $W_2$ metric and $t_n \rightarrow {\bar{t}}$.

Let $(\kappa , F) \in \overset{\circ }{T}_{\Delta _r}(t_n,\mu _n)$ and $\alpha _n \in \Pi _0(\mu _n, \nu _n)$. Set $\delta _n:=d_2((t_n,\mu _n),(s_n,\nu _n)) \ge \rho _n .$ From (7.8), we know that for all $(t,\mu )$ sufficiently close to $(t_n,\mu _n)$ we have

$$\begin{aligned} w(t_n,\mu _n) + k_nd_2((t_n,\mu _n),(s_n,\nu _n)) \le w(t,\mu ) + k_nd_2((t,\mu ),(s_n,\nu _n)). \end{aligned}$$

From this last inequality it follows by the arguments similar to those of Case 1 (a) applied with $(t_n,\mu _n)$ instead of $({\bar{t}},{\bar{\mu }})$ that

$$\begin{aligned} -\frac{k_n}{\delta _n} \left( (t_n-s_n)\kappa + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d}}\langle x-y, F(x) \rangle \mathrm {d} \alpha _n(x,y) \right) \le D_{\uparrow }w(t_n,\mu _n)(\kappa ,F). \end{aligned}$$

Let ${\bar{\alpha }}_n$ denote the barycentric projection of $\alpha _n$ with respect to $\mu _n$ and consider $p_n \in L^2(\mu _n;\mathbb {R}^{d})$ satisfying $p_n(x) = {\bar{\alpha }}_n (x)- x$ $\mu _n$ a.e. Then

$$\begin{aligned} -\frac{k_n}{\delta _n} \left( t_n-s_n, -p_n\right) \in \partial ^-_H w(t_n,\mu _n) \end{aligned}$$

and by our assumptions for some $u_n \in U$

$$\begin{aligned} \frac{k_n}{\delta _n} \left( t_n - s_n + \int _{\mathbb {R}^{d}}\langle - p_n(x), f(\mu _n,u_n) \rangle \mathrm {d} \mu _n(x) \right) \ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{k_n}{\delta _n} \left( t_n - s_n + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d}}\langle x -y, f(\mu _n,u_n) \rangle \mathrm {d} \alpha _n(x,y) \right) \ge 0 \end{aligned}$$

and

$$\begin{aligned} t_n - s_n + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d}}\langle x-y, f(\mu _n,u_n) \rangle \mathrm {d} \alpha _n(x,y) \ge 0 . \end{aligned}$$

This and (7.6) imply that the above inequality holds in both cases 1 and 2.

Step 4. Observe next that for some $R>0$ and all n, $\mathrm{supp} (\alpha _n) \subset B(0,R)\times B(0,R)$. By [4, Proposition 7.1.3] a subsequence $\alpha _{n_i}$ converges narrowly to some $\alpha \in \Pi _0({\bar{\mu }}, \nu )$. Since U is compact, $n_i$ may be chosen in such way that $u_{n_i}$ converge to some ${\bar{u}} \in U.$

Taking the limit in the last inequality we obtain

$$\begin{aligned} {\bar{t}} - s + \int _{\mathbb {R}^{d} \times \mathbb {R}^{d}} \langle x-y, f({\bar{\mu }},{\bar{u}}) \rangle \mathrm {d} \alpha (x,y) \ge 0. \end{aligned}$$

Let ${\bar{\alpha }}$ denote the barycentric projection of $\alpha$ with respect to ${\bar{\mu }}$. Consider $p \in L^2({\bar{\mu }};\mathbb {R}^{d})$ satisfying $p(x) = {\bar{\alpha }}(x)-x$ ${\bar{\mu }}$-a.e. Then the last inequality yields ${\bar{t}} - s + \mathscr {H}({\bar{\mu }},-p) \ge 0$ completing the proof. $\square$