Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

A first notion of vector field can be described by maps $\boldsymbol{b}:\mathcal{P}_{b}(\mathsf{X})\to\mathrm{C}(\mathsf{X};\mathsf{X})$, typically taking values in some subset of continuous vector fields in $\mathsf{X}$ (as the locally Lipschitz ones of $\mathrm{Lip}_{loc}(\mathsf{X};\mathsf{X})$), and satisfying suitable growth-continuity conditions. In this respect, the evolution driven by $\boldsymbol{b}$ can be described by a continuous curve $t\mapsto\mu_{t}\in\mathcal{P}_{b}(\mathsf{X})$, $t\in[0,T]$, starting from an initial measure $\mu_{0}\in\mathcal{P}_{b}(\mathsf{X})$ and satisfying the continuity equation

in the distributional sense, i.e.

If $\boldsymbol{b}$ is sufficiently smooth, solutions to (1.1a,b) can be obtained by many techniques. Recent contributions in this direction are given by the papers [Pic19, Pic18, BF21, Cav+20], we also mention [PR14, PR19] for the analysis in presence of sources. In particular, in [BF21] the aim of the authors is to develop a suitable Cauchy-Lipschitz theory in Wasserstein spaces for differential inclusions which generalizes (1.1b) to multivalued maps $\boldsymbol{b}:\mathcal{P}_{b}(\mathsf{X})\rightrightarrows\mathrm{Lip}_{loc}(\mathsf{X};\mathsf{X})$ and requires (1.1b), (1.2) to hold for a suitable measurable selection of $\boldsymbol{b}$. As it occurs in the classical finite-dimensional case, the differential-inclusion approach is suitable to describe the dynamics of control systems, when the velocity vector field involved in the continuity equation depends on a control parameter.

A natural approach, that is suitable for a great generalization, is to approximate (1.1a,b) by a measure-theoretic version of the Explicit Euler scheme. Choosing a step size $\tau>0$ and a partition $\{0,\tau,\cdots,n\tau,\cdots,N\tau\}$ of the interval $[0,T]$, $N={\mathrm{N}(T,\tau)}=\left\lceil T/\tau\right\rceil$, we construct a sequence $M^{n}_{\tau}\in\mathcal{P}_{b}(\mathsf{X})$, $n=0,\cdots,N,$ by the algorithm

where $\boldsymbol{i}_{\mathsf{X}}(x):=x$ is the identity map and $\boldsymbol{r}_{\sharp}\mu$ denotes the push forward of $\mu\in\mathcal{P}(\mathsf{X})$ induced by a Borel map $\boldsymbol{r}:\mathsf{X}\to\mathsf{X}$ and defined by $\boldsymbol{r}_{\sharp}\mu(B):=\mu(\boldsymbol{r}^{-1}(B))$ for every Borel set $B\subset\mathsf{X}$. If $\bar{M}_{\tau}$ is the piecewise constant interpolation of the discrete values $(M^{n}_{\tau})_{n=0}^{N}$, one can then study the convergence of $\bar{M}_{\tau}$ as $\tau\downarrow 0$, hoping to obtain a solution to (1.1a,b) in the limit.

It is then natural to investigate a few relevant questions:

1. $\langle$E.1$\rangle$

   what is the most general framework where the Explicit Euler scheme can be implemented,

2. $\langle$E.2$\rangle$

   what are the structural conditions ensuring its convergence,

3. $\langle$E.3$\rangle$

   how to characterize the limit solutions and their properties.

Concerning the first question $\langle$E.1$\rangle$, one immediately realizes that each iteration of (1.3) actually depends on the probability distribution on the tangent bundle $\mathsf{T\kern-1.5ptX}$ (which we may identify with $\mathsf{X}\times\mathsf{X}$, where the second component plays the role of velocity)

whose first marginal is $M^{n}_{\tau}$. If we denote by $\mathsf{x},\mathsf{v}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the projections $\mathsf{x}(x,v)=x,\ \mathsf{v}(x,v)=v$, and by $\operatorname{\mathsf{exp}}^{\tau}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the exponential map in the flat space $\mathsf{X}$ $\operatorname{\mathsf{exp}}^{\tau}(x,v):=x+\tau v$, we recover $M^{n+1}_{\tau}$ by a single step of “free motion” driven by $\Phi^{n}_{\tau}$ and given by

This operation does not depend on the fact that $\Phi^{n}_{\tau}$ is concentrated on the graph of a map (in this case $\boldsymbol{b}^{n}_{\tau}\in\boldsymbol{b}[M^{n}_{\tau}]$): one can more generally assign a multivalued map ${\boldsymbol{\mathrm{F}}}:\mathcal{P}_{b}(\mathsf{X})\rightrightarrows\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$ such that for every $\mu\in\mathcal{P}_{b}(\mathsf{X})$ every measure $\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]\in\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$ has first marginal $\mu=\mathsf{x}_{\sharp}\Phi$. We call ${\boldsymbol{\mathrm{F}}}$ a multivalued probability vector field (MPVF in the following), which is in good analogy with a Riemannian interpretation of $\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$. The disintegration $\Phi_{x}\in\mathcal{P}_{b}(\mathsf{X})$ of $\Phi$ with respect to $\mu$ provides a (unique up to $\mu$-negligible sets) Borel family of probability measures on the space of velocities such that $\Phi=\int_{\mathsf{X}}\Phi_{x}\,\mathrm{d}\mu(x)$. $\Phi$ is induced by a vector field $\boldsymbol{b}$ only if $\Phi_{x}=\delta_{\boldsymbol{b}(x)}$ is a Dirac mass for $\mu$-a.e.$x$. (1.3) now reads as

Besides greater generality, this point of view has other advantages: working with the joint distribution ${\boldsymbol{\mathrm{F}}}[\mu]$ instead of the disintegrated vector field $\boldsymbol{b}[\mu]$ potentially allows for the weakening of the continuity assumption with respect to $\mu$. This relaxation corresponds to the introduction of Young’s measures to study the limit behaviour of weakly converging maps [CRV04]. Adopting this viewpoint, the classical discontinuous example in $\mathbb{R}$ (see [Fil88]), where $\boldsymbol{b}(x)=-\mathrm{sign}(x)$, admits a natural closed realization as MPVF given by

In particular, ${\boldsymbol{\mathrm{F}}}[\delta_{0}]=\big{\{}\delta_{0}\otimes\left((1-\theta)\delta_{-1}+\theta\delta_{1}\right)\mid\theta\in[0,1]\big{\}}$ (see also [Cam+21, Example 6.2]).

The study of measure-driven differential equations/inclusions is not new in the literature \citesDalMasoRampazzo,SilvaVinter. However, these studies, devoted to the description of impulsive control systems [Bre96] and mainly motivated by applications in rational mechanics and engineering, have been used to describe evolutions in $\mathbb{R}^{d}$ rather than in the space of measures.

A second advantage in considering a MPVF is the consistency with the theory of Wasserstein gradient flows generated by geodesically convex functionals introduced in [AGS08] (Wasserstein subdifferentials are particular examples of MPVFs) and with the multivalued version of the notion of probability vector fields introduced in [Pic19, Pic18], whose originating idea was indeed to describe the uncertainty affecting not only the state of the system, but possibly also the distribution of the vector field itself.

A third advantage is to allow for a more intrinsic geometric view, inspired by Otto’s non-smooth Riemannian interpretation of the Wasserstein space: probability vector fields provide an appropriate description of infinitesimal deformations of probability measures, which should be measured by, e.g., the $L^{2}$-Kantorovich-Rubinstein-Wasserstein distance

where $\Gamma(\mu,\nu)$ is the set of couplings with marginals $\mu$ and $\nu$ respectively. It is well known [AGS08, Vil09, San15] that if $\mu,\nu$ belong to the space $\mathcal{P}_{2}(\mathsf{X})$ of Borel probability measures with finite second moment

then the minimum in (1.5) is attained in a compact convex set $\Gamma_{o}(\mu,\nu)$ and $(\mathcal{P}_{2}(\mathsf{X}),W_{2})$ is a complete and separable metric space. Adopting this viewpoint and proceeding by analogy with the theory of dissipative operators in Hilbert spaces, a natural class of MPVFs for evolutionary problems should at least satisfy a $\lambda$-dissipativity condition, $\lambda\in\mathbb{R}$, as

Condition (1.6) in the simple case $\lambda=0$ has a clear interpretation in terms of one step of the Explicit Euler method: it is an asymptotic contraction as the time step goes to $0$. By using the properties of the Wasserstein distance, we will first compute the right derivative of the (squared) Wasserstein distance along the deformation $\operatorname{\mathsf{exp}}^{\tau}$

and we will show that (1.6) admits the equivalent characterization

If we interpret the left hand side of (1.8) as a sort of Wasserstein pseudo-scalar product of $\Phi$ and $\Psi$ along the direction of an optimal coupling between $\mu$ and $\nu$, (1.8) is in perfect analogy with the canonical definition of $\lambda$-dissipativity (also called one-sided Lipschitz condition) for a multivalued map ${\boldsymbol{F}}:\mathsf{X}\rightrightarrows\mathsf{X}$:

It turns out that the (opposite of the) Wasserstein subdifferential $\boldsymbol{\partial}\mathcal{F}$ [AGS08, Section 10.3] of a geodesically $(-\lambda)$-convex functional $\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$ is a MPVF and satisfies a condition equivalent to (1.6) and (1.8). We also notice that (1.8) reduces to (1.9) in the particular case when $\Phi=\delta_{(x,v)},\Psi=\delta_{(y,w)}$ are Dirac masses in $\mathsf{T\kern-1.5ptX}$.

Differently from the Implicit Euler method, however, even if a MPVF satisfies (1.8), every step of the Explicit Euler scheme (1.4) affects the distance by a further quadratic correction according to the formula

which depends on the order of magnitude of $\Phi$ and $\Psi$, and thus of ${\boldsymbol{\mathrm{F}}}$, at $\mu$ and $\nu$.

Our first main result (Theorems 7.5,7.7), which provides an answer to question $\langle$E.2$\rangle$, states that if ${\boldsymbol{\mathrm{F}}}$ is a $\lambda$-dissipative MPVF according to (1.8) then every family of discrete solutions $(\bar{M}_{\tau})_{\tau>0}$ of (1.4) in an interval $[0,T]$ satisfying the abstract CFL condition

is uniformly converging to a limit curve $\mu\in\mathrm{Lip}([0,T];\mathcal{P}_{2}(\mathsf{X}))$ starting from $\mu_{0}$, with a uniform error estimate

and a universal constant $C\leq 14$. Apart from the precise value of $C$, the estimate (1.11) is sharp [Rul96] and reproduces in the measure-theoretic framework the celebrated Crandall-Liggett [CL71] estimate for the generation of dissipative semigroups in Banach spaces. We derive it by adapting to the metric-Wasserstein setting the relaxation and doubling variable techniques of [NS06], strongly inspired by the ideas of Kružkov [Kru70] and Crandall-Evans [CE75].

This crucial result does not require any bound on the support of the measures neither local compactness of the underlying space $\mathsf{X}$, so that we will prove it in a general Hilbert space, possibly with infinite dimension. Moreover, if $\mu,\nu$ are two limit solutions starting from $\mu_{0},\nu_{0}$ we show that

as it happens in the case of gradient flows of $(-\lambda)$-convex functions. Once one has these building blocks, it is not too difficult to construct a local and global existence theory, mimicking the standard arguments for ODEs.

As we stated in question $\langle$E.3$\rangle$, a further important point is to get an effective characterization of the solution $\mu$ obtained as limit of the approximation scheme.

As a first property, considered in [Pic19, Pic18] in the case of a single-valued PVF, one could hope that $\mu$ satisfies the continuity equation (1.1a) coupled with the barycentric condition replacing (1.1b)

This is in fact true, as shown in [Pic19, Pic18] in the finite dimensional case, if ${\boldsymbol{\mathrm{F}}}$ is single valued and satisfies a stronger Lipschitz dependence w.r.t. $\mu$ (see (H1) in Appendix A).

In the framework of dissipative MPVFs, we will replace (1.12) with its relaxation à la Filippov (see e.g. [Vin10, Chapter 2] and [AF09, Chapter 10])

where $\operatorname{cl}({\boldsymbol{\mathrm{F}}})$ is the sequential closure of the graph of ${\boldsymbol{\mathrm{F}}}$ in the strong-weak topology of $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ (see [NS21] and Section 2.2 for more details; in fact, a more restrictive “directional” closure could be considered, see (6.34)) and $\overline{\operatorname{co}}(\operatorname{cl}({\boldsymbol{\mathrm{F}}})[\mu])$ denotes the closed convex hull of the given section $\operatorname{cl}({\boldsymbol{\mathrm{F}}})[\mu]$.

However, even in the case of a single valued map, (1.12) is not enough to characterize the limit solution, as it has been shown by an interesting example in [Pic19, Cam+21] (see also the gradient flow of Example 6.34).

Here we follow the metric viewpoint adopted in [AGS08] for gradient flows and we will characterize the limit solutions by a suitable Evolution Variational Inequality satisfied by the squared distance function from given test measures. This approach is also strongly influenced by the Bénilan notion of integral solutions to dissipative evolutions in Banach spaces [Bén72]. The main idea is that any differentiable solution to $\dot{x}(t)\in{\boldsymbol{F}}[x(t)]$ driven by a $\lambda$-dissipative operator in a Hilbert space as in (1.9) satisfies

In the framework of $\mathcal{P}_{2}(\mathsf{X})$, we replace $w\in{\boldsymbol{F}}[y]$ with $\Psi\in{\boldsymbol{\mathrm{F}}}[\nu]$ and the scalar product $\langle w,y-x(t)\rangle$ with

as in (1.7). According to this formal heuristic, we obtain the $\lambda$-EVI characterization of a limit curve $\mu$ as

As for Bénilan integral solutions, we can considerably relax the apriori smoothness assumptions on $\mu$, just imposing that $\mu$ is continuous and ($\lambda$-EVI) holds in the sense of distributions in $(0,T)$. In this way, we obtain a robust characterization, which is stable under uniform convergence and also allows for solutions taking values in the closure of the domain of ${\boldsymbol{\mathrm{F}}}$. This is particularly important when ${\boldsymbol{\mathrm{F}}}$ involves drift terms with superlinear growth (see Example 6.32).

The crucial point of this approach relies on a general error estimate, which extends the validity of (1.11) to a general $\lambda$-EVI solution $\mu$ and therefore guarantees its uniqueness, whenever the Explicit Euler method is solvable, at least locally in time.

Combining local in time existence with suitable global confinement conditions (see e.g. Theorem 6.31) we can eventually obtain a robust theory for the generation of a $\lambda$-flow, i.e. a semigroup $(\mathrm{S}_{t})_{t\geq 0}$ in a suitable subset $D$ of $\mathcal{P}_{2}(\mathsf{X})$ such that $\mathrm{S}_{t}[\mu_{0}]$ is the unique $\lambda$-EVI solution starting from $\mu_{0}$ and for every $\mu_{0},\mu_{1}\in D$

as in the case of Wasserstein gradient flows of geodesically $(-\lambda)$-convex functionals.

In the framework of contraction semigroups generated by $\lambda$-dissipative operators in Hilbert or Banach spaces, a crucial role is played by the Implicit Euler scheme, which has the advantage to be unconditionally stable, and thus avoids any apriori restriction on the local bound of the operator, as we did in (1.10). In Hilbert spaces, it is well known that the solvability of the Implicit Euler scheme is equivalent to the maximality of the graph of the operator.

In the case of a Wasserstein gradient flow of a geodesically convex $\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$, every step of the Implicit Euler method (also called JKO/Minimizing Movement scheme [JKO98, AGS08]) can be solved by a variational approach: $M^{n+1}_{\tau}$ has to be selected among the solutions of

Notice, however, that in this case the MPVF $\boldsymbol{\partial}\mathcal{F}$ is defined implicitely in terms of $\mathcal{F}$ and each step of (1.13) provides a suitable variational selection in $\boldsymbol{\partial}\mathcal{F}$, leading in the limit to the minimal selection principle.

In the case of more general dissipative evolutions, it is not at all clear how to solve the Implicit Euler scheme, in particular when ${\boldsymbol{\mathrm{F}}}[\mu]$ is not concentrated on a map, and to characterize the maximal extension of ${\boldsymbol{\mathrm{F}}}$ (in the Hilbertian case the maximal extension of a dissipative operator ${\boldsymbol{F}}$ is explicitly computable at least when the domain of ${\boldsymbol{F}}$ has not empty interior, see the Theorems of Robert and Bénilan in [Qi83]). Indeed, the analogy with the Hilbertian theory does not extend to some properties which play a crucial role. In particular, a dissipative MPVF ${\boldsymbol{\mathrm{F}}}$ in $\mathcal{P}_{2}(\mathsf{X})$ is not locally bounded in the interior of its domain (see Example 5.2) and maximality may fail also for single-valued continuous PVFs (see Example 5.3). Even more remarkably, in the Hilbertian case a crucial equivalent characterization of dissipativity reads as

which implies that the resolvent operators $(I-\tau{\boldsymbol{F}})^{-1}$ (and every single step of the Implicit Euler scheme) are contractions in $\mathsf{X}$. On the contrary, if we assume the forward characterizations (1.6) and (1.8) of dissipativity in $\mathcal{P}_{2}(\mathsf{X})$ (with $\lambda=0$) we cannot conclude in general that

since the squared distance map $f(t):=W^{2}_{2}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{t}_{\sharp}\Psi)$, $t\in\mathbb{R}$, is not convex in general (see e.g. [AGS08, Example 9.1.5]) and the fact that its right derivative at $t=0$ (corresponding to $\left[\Phi,\Psi\right]_{r}$) is $\leq 0$ according to (1.8) does not imply that $f(0)\leq f(t)$ for $t<0$ (corresponding to (1.14) for $t=-\tau$).

For these reasons, we decided to approach the investigation of dissipative evolutions in $\mathcal{P}_{2}(\mathsf{X})$ by the Explicit Euler method, and we defer the study of the implicit one to a forthcoming paper.

As we already mentioned, our theory works in a general separable Hilbert space $\mathsf{X}$: we collect some preliminary material concerning the Wasserstein distance in Hilbert spaces and the properties of strong-weak topology for $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ in Section 2.

In Section 3, we will study the semi-concavity properties of $W_{2}$ along general deformations induced by the exponential map $\operatorname{\mathsf{exp}}^{\tau}$ and we introduce and study the pairings $\left[\cdot,\cdot\right]_{r}$, $\left[\cdot,\cdot\right]_{l}$. We will apply such tools to derive the precise expressions of the left and right derivatives of $W_{2}$ along absolutely continuous curves in $\mathcal{P}_{2}(\mathsf{X})$ in Section 3.2.

In Section 4, we will introduce and study the notion of $\lambda$-dissipative MPVF, in particular its behaviour along geodesics (Section 4.2) and its extension properties (Section 4.3). A few examples are collected in Section 5.

The last two sections contain the core of our results. Section 6 is devoted to the notion of $\lambda$-EVI solutions and to their properties: local uniqueness, stability and regularity in Section 6.3, global existence in Section 6.4 and barycentric characterizations in Section 6.5. Section 7 contains the main estimates for the Explicit Euler scheme: the Cauchy estimates between two discrete solutions corresponding to different step sizes in Section 7.2 and the uniform error estimates between a discrete and a $\lambda$-EVI solution in Section 7.3.

G.S. and G.E.S. gratefully acknowledge the support of the Institute of Advanced Study of the Technical University of Munich. The authors thank the Department of Mathematics of the University of Pavia where this project was partially carried out. G.S. also thanks IMATI-CNR, Pavia. G.C. and G.S. have been supported by the MIUR-PRIN 2017 project Gradient flows, Optimal Transport and Metric Measure Structures. G.C. also acknowledges the partial support of the funds FARB 2016 Politecnico di Milano Prog. TDG6ATEN04.