The small noise limit of order-based diffusion processes
In this article, we introduce and study order-based diffusion processes. They are the solutions to multidimensional stochastic differential equations with constant diffusion matrix, proportional to the identity, and drift coefficient depending only on the ordering of the coordinates of the process. These processes describe the evolution of a system of Brownian particles moving on the real line with piecewise constant drifts, and are the natural generalization of the rank-based diffusion processes introduced in stochastic portfolio theory or in the probabilistic interpretation of nonlinear evolution equations. Owing to the discontinuity of the drift coefficient, the corresponding ordinary differential equations are ill-posed. Therefore, the small noise limit of order-based diffusion processes is not covered by the classical Freidlin-Wentzell theory. The description of this limit is the purpose of this article.
We first give a complete analysis of the two-particle case. Despite its apparent simplicity, the small noise limit of such a system already exhibits various behaviours. In particular, depending on the drift coefficient, the particles can either stick into a cluster, the velocity of which is determined by elementary computations, or drift away from each other at constant velocity, in a random ordering. The persistence of randomness in the small noise limit is of the very same nature as in the pioneering works by Veretennikov (Mat. Zametki, 1983) and Bafico and Baldi (Stochastics, 1981) concerning the so-called Peano phenomenon.
In the case of rank-based processes, we use a simple convexity argument to prove that the small noise limit is described by the sticky particle dynamics introduced by Brenier and Grenier (SIAM J. Numer. Anal., 1998), where particles travel at constant velocity between collisions, at which they stick together. In the general case of order-based processes, we give a sufficient condition on the drift for all the particles to aggregate into a single cluster, and compute the velocity of this cluster. Our argument consists in turning the study of the small noise limit into the study of the long time behaviour of a suitably rescaled process, and then exhibiting a Lyapunov functional for this rescaled process.
Key words and phrases:Order-based diffusion process, small noise, Peano phenomenon, sticky particle dynamics, Lyapunov functional
2000 Mathematics Subject Classification:60H10, 60H30
1.1. Diffusions with small noise
The theory of ordinary differential equations (ODEs) with a regular drift coefficient and perturbed by a small stochastic noise was well developped by Freidlin and Wentzell . For a Lipschitz continuous function , they stated a large deviations principle for the laws of the solutions to the stochastic differential equations , from which it can be easily deduced that converges to the unique solution to the ODE when vanishes. When the ODE is not well-posed, the behaviour of in the small noise limit is far less well understood.
In one dimension of space, Veretennikov  and Bafico and Baldi  considered ODEs exhibiting a Peano phenomenon, i.e. such that and the ODE admits two continuous solutions and such that , and for . Other solutions are easily obtained for the ODE: as an example, for all , the function defined by if and if is also a continuous solution to the ODE. The solutions and are called extremal in the sense that they leave the origin instantaneously. For particular examples of such ODEs, it was proved in  and  that the small noise limit of the law of concentrates on the set of extremal solutions and the weights associated with each such solution was explicitely computed. In this case, large deviations principles were also proved by Herrmann  and Gradinaru, Herrmann and Roynette .
In higher dimensions of space, very few results are available. Buckdahn, Ouknine and Quincampoix  proved that the limit points of the law of concentrate on the set of solutions to the ODE in the so-called Filippov generalized sense. However, an explicit description of this set is not easily provided in general. Let us also mention the work by Delarue, Flandoli and Vincenzi  in the specific setting of the Vlasov-Poisson equation on the real line for two electrostatic particles. For a particular choice of the electric field and of the initial conditions, they showed that the particles collapse in a finite time , so that the ODE describing the Lagrangian dynamics of the two particles is singular at this time. After the singularity, the ODE exhibits a Peano-like phenomenon in the sense that it admits several extremal solutions, i.e. leaving the singular point instantaneously. Similarly to the one-dimensional examples addressed in [37, 2], the trajectory obtained as the small noise limit of a stochastic perturbation is random among these extremal solutions.
1.2. Order-based processes
In this article, we are interested in the small noise limit of the solution to the stochastic differential equation
where , is a function from the symmetric group to , is a standard Brownian motion in and, for , is a permutation such that . A permutation shall sometimes be represented by the word , especially for small values of . As an example, the permutation defined by , and is denoted by .
On the set of vectors with non pairwise distinct coordinates, the permutation is not uniquely defined. For the sake of precision, a convention to define in this case is given below, although we prove in Proposition 1.1 that the solution to (1) does not depend on the definition of the quantity on .
The solution to (1) shall generically be called order-based diffusion process, as it describes the evolution of a system of particles moving on the real line with piecewise constant drift depending on their ordering. Note that, in such a system, the interactions can be nonlocal in the sense that a collision between two particles can modify the instantaneous drifts of all the particles in the system.
Section 2 is dedicated to the complete description of the case . Unsurprisingly, if the particles have distinct initial positions , then in the small noise limit they first travel with constant velocity vector .
At a collision, or equivalently when the particles start from the same initial position, various behaviours are observed, depending on . To describe these situations, a configuration is said to be converging if , that is to say, the velocity of the leftmost particle is larger than the velocity of the rightmost particle, and diverging otherwise. If both configurations are converging, which writes
and shall be referred to as the converging/converging case, then, in the small noise limit, the particles stick together and form a cluster. The velocity of the cluster can be explicitely computed by elementary arguments. Except in some degenerate situations, it is deterministic and constant. If one of the configuration is converging while the other is diverging, which shall be referred to as the converging/diverging case, then, in the small noise limit, the particles drift away from each other with constant velocity vector , where is the diverging configuration. Finally, if both configurations are diverging, which writes
and shall be referred to as the diverging/diverging case, then the particles drift away from each other with constant velocity vector , where is a random permutation in with an explicit distribution.
The study of the two-particle case is made possible by the fact that most results actually stem from the study of the scalar process . In particular, our result in the diverging/diverging case is similar to the situation of [37, 2], in the sense that the zero noise equation for admits exactly two extremal solutions and exhibits a Peano phenomenon.
In higher dimensions, providing a general description of the small noise limit of seems to be a very challenging issue. As a first step, Sections 3 and 4 address two cases in which the function satisfies particular conditions. In Section 3, we assume that there exists a vector such that, for all , for all , . In other words, the instantaneous drift of the -th particle does not depend on the whole ordering of , but only on the rank of among . In particular, the interactions are local in the sense that a collision between two particles does not affect the instantaneous drifts of the particles not involved in the collision. Such particle systems are generally called systems of rank-based interacting diffusions. They are of interest in the study of equity market models [12, 3, 31, 14, 20, 23, 22, 13, 15, 21] or in the probabilistic interpretation of nonlinear evolution equations [5, 24, 26, 27, 9].
A remarkable property of such systems is that the reordered particle system, defined as the process such that, for all , is the increasing reordering of , is a Brownian motion with constant drift vector , normally reflected at the boundary of the polyhedron . By a simple convexity argument, we prove that the limit of when vanishes is the deterministic process with the same drift , normally reflected at the boundary of .
The small noise limit turns out to coincide with the sticky particle dynamics introduced by Brenier and Grenier , which describes the evolution of a system of particles with unit mass, travelling at constant velocity between collisions, and such that, at each collision, the colliding particles stick together and form a cluster, the velocity of which is determined by the global conservation of momentum. This provides an effective description of the small noise limit of .
An important fact in the rank-based case is that, whenever some particles form a cluster in the small noise limit, then for any partition of the cluster into a group of leftmost particles and a group of rightmost particles, the average velocity of the leftmost group is larger than the average velocity of the rightmost group. In Section 4 we provide an extension of this stability condition to the general case of order-based diffusions. We prove that, when all the particles have the same initial position, this condition ensures that in the small noise limit, all the particles aggregate into a single cluster. However the condition is no longer necessary and we give a counterexample with particles.
To determine the motion of the cluster, we reinterpret the study of the small noise limit of as a problem of long time behaviour for the process , thanks to an adequate change in the space and time scales. In the rank-based case, it is well known that does not have an equilibrium [31, 26] as its projection along the direction is a Brownian motion with constant drift. However, under a stronger version of the stability condition, the orthogonal projection of on the hyperplane admits a unique stationary distribution . We extend both the strong stability condition and the existence and uniqueness result for to the order-based case, and thereby express the velocity of the cluster in terms of .
1.3. Preliminary results and conventions
1.3.1. Definition of
For all , we denote by the set of permutations such that . The set is nonempty, and it contains a unique element if and only if . The permutation is defined as the lowest element of for the lexicographical order on the associated words.
1.3.2. Well-posedness of (1)
Throughout this article, refers to the initial positions of the particles, and a standard Brownian motion in is defined on a given probability space . The filtration generated by is denoted by . The expectation under is denoted by .
For all , for all , the stochastic differential equation (1) admits a unique strong solution on the probability space provided with the filtration . Besides, -almost surely,
1.3.3. Convergence of processes
Let . For all , the space of continuous functions is endowed with the sup norm in time associated with the norm on . Let be a continuous process in defined on the probability space .
If is a continuous process in defined on the probability space , then for all , is said to converge to in if
If is a continuous process in defined on some probability space , the process is said to converge in distribution to if, for all , for all bounded continuous function ,
where denotes the expectation under , and, for the sake of brevity, the respective restrictions of and to are simply denoted by and .
Finally, the deterministic process shall simply be denoted by .
2. The two-particle case
In this section, we assume that . Then, (1) rewrites
In the configuration , that is to say whenever , the instantaneous drift of the -th particle is . Thus, in the small noise limit, the particles tend to get closer to each other if , and to drift away from each other else. As a consequence, the configuration is said to be converging if and diverging if . Similarly, the configuration is said to be converging if and diverging if . The introduction of the quantities and is motivated by the fact that the reduced process satisfies the scalar stochastic differential equation
where , and is a standard Brownian motion in defined on , adapted to the filtration .
The description of the small noise limit of is exhaustively made in Subsection 2.1. Some proofs are postponed to Appendix A. In Subsection 2.2, the small noise limit of is discussed. In the sequel, we use the terminology of  and call extremal solution to the zero noise version of (2) a continuous function such that
and, for all , .
2.1. Small noise limit of the system of particles
To describe the small noise limit of , we first address the case in which both particles have the same initial position, i.e. . The zero noise version of (2) rewrites
In the diverging/diverging case , , the equation above admits two extremal solutions and defined by and . In the converging/diverging case , , the only extremal solution is , and symmetrically, in the case , , the only extremal solution is . In all these cases, the small noise limit of concentrates on the set of extremal solutions to the zero noise equation, similarly to the situations addressed in [37, 2].
Assume that , and recall that , .
If , , the process converges in distribution to where is a Bernoulli variable with parameter .
If , , the process converges in to .
If , , the process converges in to .
In the converging/converging case , , there is no extremal solution to the zero noise version of (2). Informally, in both configurations the instantaneous drifts of each particle tend to bring the particles closer to each other. Therefore, in the small noise limit, the particles are expected to stick together and form a cluster; that is to say, the limit of the distribution of is expected to concentrate on . The motion of the cluster is described in the following proposition.
Assume that , and that , .
If , the process converges in to , where is the unique deterministic constant in such that, for all , .
If , the process converges in to , where is the random process in defined by
Note that, in both cases, the small noise limit of takes its values in .
In other words, in case (1), the cluster has a deterministic and constant velocity given by
In case (2), both particles have the same instantaneous drift in each of the two configurations, and the instantaneous drift of the cluster is a random linear interpolation of these drifts, with a coefficient distributed according to the Arcsine law.
A common feature of Propositions 2.1 and 2.2 is that, in all cases, the small noise limit of is a linear interpolation of and with coefficients . Depending on the case at stake, exhibits a wide range of various behaviours: in case (1), it is random in and constant in time, in cases (2) and (3) it is deterministic in and constant in time, in case (1) it is deterministic in and constant in time, and in case (2) it is random in and nonconstant in time.
In view of (2), appears as the natural small noise limit of the quantity , where denotes the occupation time of in the configuration :
Assume that .
If , , then converges in distribution to the process , where is a Bernoulli variable with parameter .
If , , then converges in to .
If , , then converges in to .
If , and , then converges in to , where .
If , then for all ,
In cases (2), (3), and (4) above, the convergence is stated either in or in as these modes of convergence appear most naturally in the proof. However, all our arguments can easily be extended to show that all the convergences hold in , for all . As a consequence, all the convergences in Proposition 2.1 and 2.2, except in case (1), actually hold in , for all .
On the contrary, the convergence in the diverging/diverging case (1) cannot hold in probability. Indeed, let us assume by contradiction that there exists such that the convergence in case (1) of Lemma 2.3 holds in probability in . Then, for all , converges in probability to . Let us fix . By Proposition 1.1, for all , the random variable is measurable with respect to the -field generated by . Thus, we deduce that the random variable is measurable with respect to . As a consequence, is measurable with respect to , which is contradictory with the Blumenthal zero-one law for the Brownian motion .
Let us now address the case of particles with distinct initial positions. Let . If , a pair of particles travelling at constant velocity vector with initial positions never collides, and the natural small noise limit of is given by , for all .
If , a pair of particles travelling at constant velocity vector with initial positions collides at time . The natural small noise limit of is now described by for , and for , is the small noise limit of , where is a copy of started at . In that case, at least the configuration is converging, therefore there is neither random selection of a trajectory as in case (1), nor random and nonconstant velocity of the cluster as in case (2).
These statements are straightforward consequences of the description of the small noise limit of the process with carried out in Corollary 2.6 below.
2.2. The reduced process
By Veretennikov , strong existence and pathwise uniqueness hold for (3); therefore, for all , is adaptated to the filtration generated by the Brownian motion . As a consequence, the probability of a measurable event with respect to the -field generated by for some shall be abusively denoted by instead of .
To describe the small noise limit of , we define and . Let us begin with the case , which corresponds to .
Assume that . Then,
if and , then converges in distribution to , where is a Bernoulli variable of parameter ;
if and , then converges to in ;
if and , then converges to in ;
if and , then converges to in ; more precisely,
We now give a direct proof of case (4). By the Itô formula, for all ,
If and , then for all one has , therefore
For all , let us define
and for all , let . The process is a martingale, therefore, for all , , and by the Fatou lemma, . As a consequence, is a martingale. For all ,
where we have used the Cauchy-Schwarz inequality at the second line, the Doob inequality at the third line and the Itô isometry at the fourth line. This completes the proof of case (4). ∎
We now describe the small noise limit of in the case . Due to the same reasons as in Remark 2.4, all the convergences below are stated in but can easily be extended to for all . The proof of Corollary 2.6 is postponed to Appendix A.
Assume that . Let us define if , and if . Then converges in to the process defined by:
A symmetric statement holds if .
For a given continuous and bounded function on , the function defined by
is continuous on owing to the Girsanov theorem and the boundedness of . Following [16, Chapter II], it is a viscosity solution to the parabolic Cauchy problem
Attanasio and Flandoli  addressed the limit of when vanishes, for a particular function such that the corresponding hyperbolic Cauchy problem
admits several solutions. In the diverging/diverging case , , we recover their result of [1, Theorem 2.4] as converges pointwise to the function defined by
Note that, in general, is discontinuous on the half line .
In the converging/converging case , , converges pointwise to the function defined by
Note that is continuous on , and constant on the cone .
3. The rank-based case
In this section, we assume that there exists a vector such that, for all , for all , . In other words, the instantaneous drift of the -th particle at time only depends on the rank of among . We recall in Subsection 3.1 that, in this case, the increasing reordering of the particle system is a Brownian motion with constant drift, normally reflected at the boundary of the polyhedron . Its small noise limit is obtained through a simple convexity argument, and identified as the sticky particle dynamics in Subsection 3.2. The description of the small noise limit of the original particle system is then derived in Subsection 3.3.
3.1. The reordered particle system
For all , let
refer to the increasing reordering of
i.e. with . The increasing reordering of the initial positions is denoted by . The process shall be referred to as the reordered particle system. It is continuous and takes its values in the polyhedron . The following lemma is an easy adaptation of [24, Lemma 2.1, p. 91].
For all , there exists a standard Brownian motion
in , defined on , such that
where the continuous process in is associated with in in the sense of Tanaka [35, p. 165]. In other words, is a Brownian motion with constant drift vector and constant diffusion matrix , normally reflected at the boundary of the polyhedron ; where refers to the identity matrix.
By Tanaka [35, Theorem 2.1, p. 170], there exists a unique solution
to the zero noise version of the reflected equation (4) given by
where is associated with in . An explicit description of as the sticky particle dynamics started at with initial velocity vector is provided in Subsection 3.2 below.
For all ,
By the Itô formula,
Let refer to the total variation of on . Then, by the definition of (see [35, p. 165]), -almost everywhere, and the unit vector defined by