Measure dynamics with Probability Vector Fields and sources
We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass.
The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.
Department of Mathematical Sciences, Rutgers University - Camden
Camden, NJ, USA.
Dipartimento di Matematica “Tullio Levi–Civita”
Università degli Studi di Padova
Keywords: Measure dynamics, Probability Vector Fields, Wasserstein distance, generalized Wasserstein distance
MSC2010: 35S99, 35F20, 35F25
The problem of optimal transportation, also called Monge-Kantorovich problem, has been intensively studied in the mathematical community. Related to such problem, the definition of the Wasserstein distance in the space of probability measure has revealed itself to be a powerful tool, in particular for dealing with dynamics of measures (like the transport PDE, see e.g. ). For a complete introduction to Wasserstein distances, see [12, 13].
This approach has at least two main limits. The first is that the use of transport equation, together with their counterpart in terms of ordinary differential equations [1, 2], does not allow to model neither mass diffusion nor concentration phenomena. The second one is that the Wasserstein distance is defined only if the two measures have the same mass, then PDEs with sources cannot be studied with such tools.
Both limits were recently overcome by a variety of contributions. The first was addressed in , in which a generalization of the concept of vector fields was introduced. Such tool, called Probability Vector Field (PVF in the following), allows to model concentration and diffusion phenomena in the formalism of the transport equation, then being able to translate several useful techniques from dynamical system.
The second limit was addressed by a series of papers introducing generalizations of the Wasserstein distance to measures with different masses. In  we defined a generalized Wasserstein distance , combining the standard Wasserstein and distances. In rough words, for an infinitesimal mass of (or ) can either be removed at cost , or moved from to at cost . This distance is a generalization of the so-called flat distance. Other generalizations of the Wasserstein distance, with the same spirit of allowing sources of mass, are studied in [5, 6, 8, 7]. As a consequence, sources of mass can be introduced in the transport equation, even when they depend on the measure itself, see .
The goal of this article is to define a new class of equations,
which are able to describe complex dynamics in the space of measures,
including mass diffusion, concentration and sources.
The idea is to merge two different dynamics, already individually described in [9, 10], and couple them.
The first contribution is given by dynamics induced by Probability Vector Fields (PVF in the following), recently introduced in . There, the equation
is considered, where is a function from the space of probability measures to the space of probability measures of the tangent space . The idea of such function is to describe the infinitesimal spreading of the mass in a point along the velocities described by the measure on the fiber . Given the projection defined by , we also require , i.e. that the projection of from to coincides with . This is the measure counterpart of the fact that a vector field is a section of the tangent bundle. The main contribution of  is to introduce conditions ensuring existence and/or uniqueness of the solution of the Cauchy problem with dynamics (1). In particular, two key tools are defined: the first is a new non-negative operator , based on the Wasserstein distance and enjoying some of its properties, on the space . The idea is that measures the cost of the minimizing transference plan on fibers, among plans whose projections are optimal on the base space. The formal definition is given in Definition 17. If one assumes that from endowed with the Wasserstein distance to endowed with is Lipschitz, then there exists at least one solution to (1). The second tool is the definition of Dirac germs, that are specific choices of solutions to (1) for measures composed of Dirac deltas only. Fixed a Dirac germ for (1), then for each initial measure there exists at most one solution to (1) that is compatible with such chosen germ. In some specific but relevant cases, the coupling of Lipschitz continuity of with the choice of a compatible Dirac germ ensures both existence and uniqueness of a solution to (1).
The second contribution is given by sources and sinks. In this case, the dynamics reads as
where is a measure on the space , representing a source/sink of mass. The description of such Partial Differential Equation with a fixed source is very classical, since the solution is clearly . Instead, we introduced in  new conditions to ensure that the dynamics (2) is well posed even when the source depends on the whole measure itself. The key tool is the introduction of a new distance on the space of measures with finite mass, called the generalized Wasserstein distance . If is Lipschitz with respect to this distance, then one has existence and uniqueness of the solution to the Cauchy problem with dynamics (2).
For simplicity, from now on we restrict ourselves to the space of Borel measures with bounded support and finite mass. In this space, the generalized Wasserstein distance is always finite, while the standard Wasserstein distance is defined only if the masses of the two measures coincide, i.e. . We endow the space with the topology of weak convergence; this coincides with the topology induced by the generalized Wasserstein distance, see Proposition 11 below.
We are now ready to define Measure Differential Equations with Source:
where is a PVF and is a source . The goal is to prove existence and/or uniqueness of a solution to the associated Cauchy problem, under the joint hypotheses ensuring existence and/or uniqueness for each of the dynamics (1) and (2). More precisely, we first give the definition of a solution to (3):
Definition 1 (Solution to (3))
A solution to (3) is a continuous curve satisfying the following condition: for each
the integral is defined for almost every ;
the map belongs to ;
the map is absolutely continuous, and it satisfies
for almost every .
Such definition is pretty weak and can not allow uniqueness results, thus we are also interested in stronger properties for solutions to (3). In particular, we focus on existence of semigroups of solutions, whose definition in this setting is given below.
We also need to define a natural tool, merging properties of the operator on with the setting of the generalized Wasserstein distance on . Such non-negative operator, that we denote by , measures the minimal standard Wasserstein distance on the fiber between transference plans whose projections give a minimizing decomposition for the generalized Wasserstein distance on the base space. The operator is precisely defined in Section 2.5.
We are now ready to state the two main results of this article. The first deals with existence of a solution to (3), while the second focuses on uniqueness.
Consider the Measure Differential Equation with Source (3) with the following two sets of hypotheses:
The Probability Vector Field satisfies:
support sublinearity: there exists such that for all it holds
Lipschitz continuity: for each there exists such that implies
The source satisfies:
Lipschitz continuity: there exists such that for all it holds
uniform boundedness of the support: there exists such that for all it holds .
Several corollaries about existence and/or uniqueness of the solutions to (3) can be directly derived from corresponding results about PVFs from . In particular, one can observe that the uniqueness property depends on the PVF only, and not on the source . We then have the two following remarkable cases:
Fix an increasing function. In the space , define , where
is the cumulative distribution of , and is the Lebesgue measure. This choice of the PVF allows to have solutions that diffuse with finite velocities, see [9, Section 7.1] for more details. In this case, for any choice of the source satisfying (s), one has existence of a solution to (3). Even though this solution is not unique, in general, there exists a unique semigroup obtained by the limit of the discretization algorithm described in Section 3.1.
Similarly, condition (s2) can be replaced by any condition ensuring uniform boundedness of the supports, such as the existence of a radius such that with implies .
The structure of the article is the following. In Section 2 we fix the notation and recall main properties of the tools used later: the Wasserstein distance, the generalized Wasserstein distance and Measure Differential Equations with Probability Vector Fields. In the main Section 3, we prove the results of this paper. In Section 3.1, we prove Theorem 3 about existence of a solution to (3), while in Section 3.2, we prove Theorem 4 about uniqueness.
2 Dynamics in generalized Wasserstein Spaces
In this section, we fix the notation and define the main tools used in the rest of the article: the Wasserstein distance, the generalized Wasserstein distance and Measure Differential Equations with Probability Vector Fields.
2.1 The Wasserstein distance
We use to denote the space of positive Borel regular measures with bounded support and finite mass on . Given Radon measures (i.e. positive Borel measures with locally finite mass), we write if is absolutely continuous with respect to , while we write if for every Borel set . We denote by the norm of (also called its mass). More generally, if is a signed Borel measure, we define .
Given a Borel map , the push forward of a measure is defined by:
Note that the mass of is identical to the mass of . Therefore, given two measures with the same mass, one may look for such that and it minimizes the cost
This means that each infinitesimal mass is sent to and that its infinitesimal cost is the -th power of the distance between them. Such minimization problem is known as the Monge problem. A generalization of the Monge problem is defined as follows. Given a probability measure on , one can interpret as a method to transfer a measure on to another measure on as follows: each infinitesimal mass on a location is sent to a location with a probability given by . Formally, is sent to if the following properties hold:
Such is called a transference plan from to and we denote the set of transference plans from to by . A condition equivalent to (7) is that, for all it holds .
One can define a cost for as follows
and look for a minimizer of in . Such problem is called the Monge-Kantorovich problem. It is important to observe that such problem is a generalization of the Monge problem. The main advantage of this approach is that a minimizer of in always exists. We then denote by the set of transference plans that are minimizers of , that is always non-empty.
One can thus define on the following operator between measures of the same mass, called the Wasserstein distance:
It is indeed a distance on the subspace of measures in with a given mass, see . It is easy to prove that for , by observing that and that does not depend on the mass.
From now on, we only consider the Wasserstein distance with parameter , that will then be denoted by . It satisfies the following fundamental dual property.
[Kantorovich-Rubinstein duality] Let . It then holds
Such property plays a crucial role in the theory of PVF, see . It is then unclear if a corresponding theory can be generalized to any .
2.2 The generalized Wasserstein distance
In this section, we provide a definition of the generalized Wasserstein distance, introduced in [10, 11], together with some useful properties. We consider here the generalized Wasserstein distance with parameters , to simplify the notation, and .
Let be two measures. We define the functional
We now provide some properties of . Proofs can be adapted from those given in .
We recall now some useful topological results related to the metric space when endowed with the generalized Wasserstein distance. We first recall the definition of tightness in this context.
A set of measures is tight if for each there exists a compact such that for all .
We now recall the definition of weak convergence of measures, as well as an important result about convergence with respect to the generalized Wasserstein distance, see [10, Theorem 13].
Let be a sequence of measures in , and a measure. We say that converges to with respect to the weak topology, and we write , if for all functions it holds
Let be a sequence of measures in , and . Then
We also recall the result of completeness, see [10, Proposition 15].
The space endowed with the distance is a complete metric space.
The generalized Wasserstein distance also satisfies a useful dual formula, showing that it coincides with the so-called flat distance. See .
Let . It then holds
We recall that the distance satisfies a dual formula too, that is
We also have this useful estimate to bound integrals. See .
Let . It then holds
Let be two time-varying vector fields, uniformly Lipschitz with respect to the space variable, and the flow generated by respectively. Let be the Lipschitz constant of and , i.e. for all , and similarly for . Let . We have the following estimates for the standard Wasserstein distance
We have the following estimates for the generalized Wasserstein distance
2.3 Measure Differential Equations with Probability Vector Fields
In this section, we summarize the main results and tools about PVFs, introduced in . We slightly enlarge the setting of , since we consider general measures with finite mass and not only probability measures.
We first recall the definition of a solution to the Cauchy problem
Fix a final time . A solution to (15) is a map such that and the following holds: for each
the integral is defined for almost every ;
the map belongs to ;
the map is absolutely continuous, and it satisfies
for almost every .
We now recall the definition of the pseudo-distance , that will be useful in the following.
Let with . Denote by and the projection of the PVF on the base space. Define
Clearly, such functional is not a distance, see examples in . Nevertheless, we will see in the following that the local Lipschitz condition (V2) will ensure existence of solutions to (15). Observe that it also holds
See  for more details.
We now address the problem of existence of solutions to (15). The idea developed in  is to define a semigroup of solutions as the limit of approximated ones. We first describe precisely the discretization method, that will be also useful in the following.
Fix and define the time step size , the velocity step size and the space step size . Define the equispaced discretization points of , and the equispaced discretization points of .
Define the space of measures of with support on the set of points , and the space of measures of with support on the set of points ,
Define the discretization operator in the space variable as follows
where with . Define the discretization operator in the velocity variable as follows
where with .
The first property of such discretization is that it introduces an arbitrarily small error in the Wasserstein distance.
Given and , for a sufficiently large it holds
Proof. The proof with and being probability measures is given in . The generalization to measures with finite mass is straightforward.
One can then define an approximated solution (called the Lattice Approximate Solution) to (15) via an explicit Euler scheme.
Given the Cauchy problem (15), we define the following Lattice Approximate Solution : we set , then recursively
and for intermediate times we define
We are now ready to state the existence of a solution to (15) as a limit of the Lattice Approximate Solutions introduced above.
Let a PVF be given, satisfying (V) where is replaced by . Then, there exists a Lipschitz semigroup of solutions to (15), obtained as uniform-in-time limit of Lattice Approximate Solutions for the Wasserstein Metric.
Proof. The first key observation is that both and are operators preserving the mass for sufficiently large, i.e. and similarly for the PVF. As a consequence, the mass of coincides with , that in turn coincides with for sufficiently large.
We now recall the definition of Dirac germs, that permits to address the problem of uniqueness of the solution to (15). We also give the definition of semigroup compatible with the germ.
Fix a PVF . Define the space of measures composed of Dirac deltas. A Dirac germ compatible with is a map assigning to each a Lipschitz curve , with the following conditions:
is uniformly positive for measures with uniformly bounded support;
is a solution to (1).
Fix a PVF satisfying (V1), a final time and a Dirac germ . A semigroup for (1) is said to be compatible with if one has the following property: for each there exists such that the space satisfies
We are now ready to prove the main result about uniqueness of solutions to (15).
Consider a PVF satisfying (V1) and fix a Dirac germ . There exists at most one Lipschitz semigroup of solutions to (15) compatible with .
Proof. First observe that uniform boundedness of the support and the weak formulation (16) when choosing on imply that the mass is constant along trajectories of (1). Thus, the Dirac germ satisfies conservation of mass too.
2.4 Measure Equations with sources
In this section, we briefly study the measure equation with source
The goal is to prove that condition (s) in Theorem 3 ensures existence and uniqueness of a solution to (20). This is indeed a particular case of a more general result, stated in , in which a transport term is added too. For our future use, we prove the statement with the same discretization method of Lattice Approximate Solution introduced in Definition 18.
Moreover, such solution is the uniform-in-time Wasserstein limit for of Lattice Approximate Solutions defined as follows: Define , then recursively
We also define the time-interpolated solution for as follows:
Proof. We first prove existence of a solution, based on the Lattice Approximate Solution. We prove that is a sequence of equi-Lipschitz and equi-bounded curves in , where is a compact subset of and the space is endowed with the generalized Wasserstein distance . For it holds
hence recursively .
We now prove that there exists such that for all and . Eventually enlarging the radius given in hypothesis (s2), one can assume that . Thus, the approximation operator satisfies , as well as for any . Since sum of measures with the same support gives a measure with the same support, one can easily prove by induction that measures defined by the scheme (21) all have support contained in with .
Choose now , that is a compact space. Then, is complete when endowed with the generalized Wasserstein distance , see Proposition 12. The sequence is equi-Lipschitz in , due to (22), and equi-bounded, since masses are equi-bounded. Then, there exists a converging subsequence, converging to some . Recall that such convergence with respect to coincides with weak convergence of measures.
We now prove that is a solution to (20). We first observe that for sufficiently large implies . We now prove that for all and with , it holds
The definition (21) implies that, for , it holds
We then have
where is the largest integer such that , i.e. . The first two terms converge to zero since , while the last term is identically zero due to (24). For the third term, observe that it holds
where we used condition (s1) about the Lipschitz continuity of , as well as the dual formulation (14) for . The proof now follows from observing that both the terms in the right hand side of (26) converge to zero: the first satisfies
for the constant given by (22). The second converges to zero since metrizes weak convergence.