Existence of martingale solutions for stochastic flocking models with local alignment
Abstract
We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker-Smale [6, 7] and Motsch-Tadmor [16] models. By regularizing the coefficients, we first construct approximate solutions obtained as the mean-field limit of the corresponding particle systems. We then establish the compactness in law of this family of solutions by relying on a stochastic averaging lemma. This extends the results obtained in [12, 11] in the deterministic case.
Keywords: Stochastic partial differential equations, mean-field limit, collective motion.
1 Introduction, main results
1.1 Collective motion with local alignment
The emergence of a consensus or ordered motion amongst a population of interacting agents has been drawing a fair amount of attention among the scientific community in recent years. This phenomenon, consistently observed in nature, from schooling fish to swarming bacteria, is usually referred to as flocking. One of the earliest and most commonly studied mathematical models describing this kind of behavior is the celebrated Cucker-Smale model, introduced in [6, 7]. In this model, agents interact in a mean-field manner: for , denoting by the position and velocity of the -th individual, the evolution of the system is given by
where the weight function is even, typically of the form
Equivalently, one may consider the conservation equation
| (1.1) |
where the Cucker-Smale alignment term is given by the convolution
| (1.2) |
Equation (1.1) is naturally associated to the particle system, since it is satisfied by the empirical measure
in the sense of distributions. In [16], Motsch and Tadmor brought to light several drawbacks regarding the physical relevance of the Cucker-Smale model when confronted to strongly non-homegeneous distributions of agents, due to the normalizing constant in the alignment force, which involves the whole group of individuals. To remedy these issues, they proposed a new model where the influence between two agents is normalized by the total influence:
Considering some weight function with compact support, we may naturally consider a hybrid model, letting the Cucker-Smale forcing dictate the long-range interaction and the Motsch-Tadmor term dictate the short-range interaction:
| (1.3) |
where the is given by
| (1.4) |
Note that the Motsch-Tadmor forcing term can also be written
expressing the alignment of the speed with the local average velocity , defined as
As suggested in [12], we may also consider (1.3) in the singular limit where the weight function governing the short-range interaction converges to the Dirac function , leading to
| (1.5) |
In (1.5), the Strong Local Alignment term is given by
| (1.6) |
where the local velocity is given by
The existence of solutions to the kinetic equations (1.3) and (1.5) has been established in [12]. Moreover, in [11], the authors rigorously explore the limit : considering some and weight functions of the form , solutions of (1.3) converge (up to some subsequence) to a solution of (1.5).
In order to take into account random phenomena emerging from the environment, or unpredictable interactions between the agents, it is rather natural to perturb the deterministic equations (1.3) and (1.5) with some noise, driven by a Wiener process , leading to the stochastic conservation equation
with or . Here, for simplicity purposes, we choose to only consider a "one-dimensional" noise driven by a real valued Brownian motion , leading to equations
| (1.7) |
and
| (1.8) |
The methods developed in the present paper shall not rely on this particular form, so that the results may be easily generalized to SPDEs with multiple Brownian motions. Note that these stochastic conservation equations are written in Stratonovitch form, since it is the most physically relevant form. As in the deterministic case, equation (1.7) is naturally associated to the stochastic particle system
| (1.9) |
where and .
The mean-field convergence of (1.9) to the corresponding limiting SPDE has been studied in the litterature in the case of the Cucker-Smale interaction, that is with . The diffusion coefficient for some constant is considered in [1] ; the coefficient , where , is looked upon in [4] and [10] ; some more general (non linear) diffusion coefficients are considered in [18].
As for the Motsch-Tadmor model, that is for , the flocking phenomenon for the particle system (1.9) (alignment of speeds, distance between the individuals bounded over time) is studied in [15] in the case of a multiplicative noise . However, even in the deterministic case, due to the singular ratio involved in the non-linear term , the mean-field limit of the Motsch-Tadmor particle system is a very delicate question (as suggested in [16] and [17]) to which the authors could not find a proper answer in the litterature. It should also be noted that the strong local alignment term given by (1.6) is ill-defined when is a general measure, so that the particle system associated with equation (1.8) cannot in fact be written.
In the present work, we shall consider a diffusion coefficient of the form
| (1.10) |
which corresponds to some random environmental forcing , as well as a random perturbation of the weight function involved in the Cucker-Smale alignment term (1.2) (as considered in [4] and [18] in the case of the Cucker-Smale model only). Note that the choice and leads in particular to the simpler coefficient . The arguments developed in the present work also easily apply to the case looked upon in [15].
In this paper, we extend the work developed in [12] and [11] in the deterministic case to establish the existence of martingale solutions (see Definition 1.1 below) for the stochastic conservation equations (1.7) and (1.8). To this intent, we start by regularizing the coefficients: in section 2, we prove the existence of a unique solution of equation (1.7) with regularized coefficients, which is naturally constructed as the mean-field limit of the corresponding stochastic particle system. Then, in section 3, we prove the tightness of these approximate solutions with respect to the regularizing parameter, and rigorously pass to the limit in the martingale problem associated with (1.7).
1.2 Assumptions and main results
The weight functions involved in (1.2) and (1.10) are assumed to satisfy
| (1.11) |
The weight function involved in (1.4) is assumed to be smooth and compactly supported around zero: and for some ,
| (1.12) |
The forcing involved in (1.10) is assumed to be smooth and sublinear:
Simple calculations show that the proper Itô form associated with SPDE (1.7) is
where we have used the notation
and the additional drift forcing term is given by
This motivates the following definition.
Definition 1.1.
Let and be a filtered probability space equipped with an -brownian motion . Let with .
A process with
| (1.13) |
satisfying the estimate
| (1.14) |
is said to be a solution of (1.7) on (with initial value ) when, for any test function , the process is adapted with a continuous version and satisfies
where denotes the second order operator
| (1.15) |
If there exists some probability tuple and a solution of (1.7) on , we say that equation has a martingale solution (in the sense of [8], Chapter 8).
Estimates (1.13) and (1.14) are quite natural since we expect solutions of (1.7) to be densities. On can easily check that these estimates guarantee that the process is well defined, the stochastic integral being a square integrable martingale. Solutions of equation (1.8) are defined similarly. We now state our main results.
Theorem 1 (Stochastic Motsch-Tadmor flocking).
Let with such that, for some and ,
| (1.16) |
Then there exists a martingale solution of equation (1.7) with initial data .
We also prove the existence of a martingale solution of (1.8), which can be constructed as a weak limit of solutions of (1.7) as the function involved in the Motsch-Tadmor alignment term (1.4) properly approaches the Dirac function , similarly to the result established in [11].
Theorem 2 (Stochastic flocking with strong local alignment).
Let satisfy (1.12). Let us consider the sequence of functions given by
Assuming (1.16), the martingale solution of equation (1.7) with constructed in Theorem 1 satisfies, along some subsequence ,
where defines a martingale solution of (1.8). The weighted Sobolev space , with is introduced in (3.34) below.
Acknowledgment
A. Debussche and A. Rosello are partially supported by the French government thanks to the "Investissements d’Avenir" program ANR-11-LABX-0020-01.
2 Regularized equation
In this section, we prove existence and uniqueness of a solution of equation (1.7) with regularized coefficients. This solution will be naturally obtained as the mean-field limit of the corresponding particle system.
For let us introduce smooth, compactly supported truncation functions
satisfying
| (2.1) |
We may then introduce the following regularized coefficients:
| (2.2) |
Simple calculations show that, for fixed , these regularized coefficients are globally Lipschitz continuous in the following sense: for all , ,
| (2.3) |
where the constants involved in in (2.3) depend on , and denotes the Wasserstein distance. The assumptions made also guarantee the uniform sub-linearity of some coefficients: for all , ,
| (2.4) |
where the constant involved in in (2.4) does not depend on .
2.1 Mean-field limit of the associated particle system
We may now consider the associated mean-field particle system on , :
| (2.5) |
where denotes the empirical measure
From the sub-linearity of the coefficients, we easily deduce the following result.
Proposition 2.1.
For any and , , the SDE system (2.5) with initial condition has a unique global solution which satisfies the following estimates: for any ,
The constants involved in depend on and only.
Proof.
The coefficients of (2.5) being locally Lipschitz-continuous, the local existence and uniqueness of solutions is guaranteed. The estimates of Proposition 2.1 should first be established with the stopping time
which should then be sent to , as goes to infinity. For the sake of simplicity, we omit this stopping time in the following. Given the sub-linearity of the coefficients, Itô’s formula gives
| (2.6) |
with
Averaging (2.6) over , we are led to
| (2.7) |
from which we easily deduce, for all
Burkholder-Davis-Gundy’s inequality (from [3]) gives
so that we may come back to (2.7) and get
We may now apply Gronwall’s lemma to derive the first estimate. Coming back to (2.6), a similar reasoning leads to the second estimate.
∎
Let us now consider the space of trajectories
and view the empirical measure as a (random) probability over :
More precisely, for , we introduce the Wasserstein space
equipped with the usual distance
where
We may now state the following mean-field limit result.
Proposition 2.2.
Let . As , provided that in , we have where solves the regularized SPDE
| (2.8) |
in the following sense: denoting the operator
| (2.9) |
we have, for any test function ,
More precisely, is the unique element of given by the push-forward measure of the initial data by the non-linear characteristics:
| (2.10) |
where is the flow associated with the SDE
| (2.11) |
Proof.
For simplicity, let us consider the case . We start by noticing that, for fixed , the empirical measure naturally satisfies the fixed-point-like equation (2.10). For all , introducing an optimal plan (one may refer to [20] for details) such that
it follows that
| (2.12) |
and we may simply re-write
| (2.13) |
where is the solution of (2.5). Itô’s formula easily leads to
where
Using the Lipschitz estimate (2.3), we deduce (for fixed )
| (2.14) |
Taking the expectation in (2.14) and applying Gronwall’s lemma leads to
Coming back to (2.14), we may write
and therefore
| (2.15) |
Burkholder-Davis-Gundy’s inequality gives
Making use of (2.3) again, we get
and we may come back (2.15) to obtain
Summing over and as in (2.13) finally leads to
Gronwall’s lemma hence gives so that, coming back to (2.12), we get
as are sent to infinity. We have shown that converges to some in the complete space . Let us now prove that satisfies the fixed-point identity (2.10). First, since , one could easily deduce from the sub-linearity of the coefficients that
thereby guaranteeing that the solution of (2.11) is unique and global. Moreover, it is a well known fact (see e.g [13]) that the flow is almost-surely continuous, so that the push-forward measure involved in (2.10) is indeed well-defined. This could be seen in this case by establishing a Kolmogorov estimate
Let us introduce the measure . Introducing an optimal plan so that
we have this time
As in (2.14), Itô’s formula gives an expression of the form
Proceeding as in the first part of the proof, we eventually obtain
Letting go to infinity, we conclude that a.s, that is exactly (2.10).
We may once again use the same arguments to prove that the fixed-point-like equation (2.10) has a unique solution: considering and such that a.s and a.s, we are led to
so that implies a.s.
2.2 Flow of characteristics, regular solutions
For some fixed , let us consider the unique solution of (2.8) constructed in Proposition 2.2. Although this measure, as well as the associated characteristics , depend on , we shall hide this dependence in the following expressions to avoid cluttering notation. One may note from expressions (2.2) that the coefficients
involved in the SDE (2.11) have the regularity in the variable. More precisely, assumption (1.11) guarantees that, for fixed , for ,
uniformly in and . In particular, the first, second and third order -derivatives of the coefficients are globally Lipschitz-continuous. As a result, Theorem 4.4 of [13], Chapter II, yields
| (2.16) |
More precisely, for , denoting by the solution of the SDE
the inverse map satisfies the corresponding backward SDE
where and denotes the backward Stratonovich integral (see again [13] Theorem 7.3 for this result and p.194 for the definition of the backward integral). When , we simply denote
In the particular case where the initial measure admits a density with respect to the Lebesgue measure on , for any test function we may write
where denotes the jacobian determinant
| (2.17) |
Considering a countable separating family of such test functions , from in , we deduce that
where
| (2.18) |
Note that we may drop the absolute value in (2.18) since a.s. Let us now give some estimates regarding the forward and backward characteristics.
Proposition 2.3.
For all , let such that . Let such that as in (2.10). Then for all , and ,
| (2.19) | |||
| (2.20) | |||
| (2.21) | |||
| (2.22) | |||
| (2.23) |
The constants involved in depend on only.
These estimates are deduced in a classical manner from the sub-linearity and the global Lipschitz continuity of the coefficients of the equations satisfied by and . Applying Kolmogorov’s lemma to (2.21) and (2.23), we deduce that, for all ,
| (2.24) |
is continuous . Additionally, we may deduce from (2.19) and (2.20) the following estimate: given a compact set and , for any and ,
| (2.25) |
In (2.25), and denotes the -Hölder semi-norm
which satisfies indeed by applying Kolmogorov’s lemma to (2.20). We may now establish the following result.
Proposition 2.4 (Regular solution).
Proof.
As a consequence of (2.24), it is clear from expression (2.18) that the first condition is met. Furthermore, is almost surely compactly supported, uniformly in , with
Since is a solution of (2.8), for any , -almost surely,
We can then interchange the integrals
Since all functions are compactly supported (for fixed ), the integrals with respect to cause no issue. As for the stochastic integral, we may use a stochastic Fubini theorem, as long as
From expressions (2.2), we deduce (for fixed )
From expression (2.18), (and ) it is clear that
for some . We may hence write
| (2.28) |
thanks to the bound (2.22). It follows that
Consequently, -a.s, (2.27) holds when integrated against any . We deduce that (2.27) holds for almost every . Since both sides of (2.27) are continuous with respect to (thanks to (2.28) again), we conclude that the equality holds for every .
∎
3 Weak convergence of approximate solutions
From now on, let us fix some initial data satisfying (at least) for some ,
For any , considering the particle system (2.5), with initial data satisfying
we may introduce the solution of (2.8) constructed in Proposition 2.2. As previously discussed, we naturally identify with its density defined by (2.18).
3.1 Uniform estimates
In this section, we shall establish some estimates on uniformly on the regularization parameter .
Proposition 3.1.
Let and . Then a.s, with the estimate
The constant involved in depends on and only.
Proof.
Let us start by considering supported in some compact . Then is a regular solution of (2.8) in the sense of Proposition 2.4. For any , applying Itô’s formula to hence gives
| (3.1) |
where denotes the drift coefficient . We can now integrate (3.1) with respect to and interchange the integrals:
Since all the integrands in (3.1) are compactly supported in uniformly for (for fixed ), the integrals with respect to cause no issue. As for the stochastic integral, we may use a stochastic Fubnini theorem if we can justify that
| (3.2) |
As in the proof of Proposition 2.4, one can see that (with )
for some . Denoting , it follows that
and, fixing some ,
Using (2.25) and (2.22), we deduce
We may hence integrate (3.1) with respect to , which leads to
| (3.3) |
where
Simple calculations lead to the classical identity
| (3.4) |
so that
From expressions (2.2), we derive
Assumptions (1.11), (2.1) guarantee that these terms are bounded uniformly in , , , , so that
Similarly, we have the following identity:
so that, again,
Since (3.2) guarantees that defines a (square integrable) martingale, we may take the expectation in (3.3) and apply Gronwall’s lemma to derive
It follows that
and Burkholder-Davis-Gundy’s inequality yields (making use of (3.4) again)
which gives the expected result. We now extend the estimate to any satisfying by considering a sequence of densities such that, as goes to infinity,
It is easy to see that these assumptions imply in particular
for any . Denoting by the solution of (2.8) with initial data constructed in Proposition 2.2, we may deduce (as in the proof of Proposition 2.2),
Up to a subsequence, we may hence assume that
| (3.5) |
From the estimates
we derive that is bounded in and therefore, up to a subsequence
| (3.6) |
where satisfies the bound
| (3.7) |
Let us consider , and introduce . From (3.5) we deduce
and the bound guarantees
Finally, the bound guarantees
so that, according to (3.6),
We easily derive that in and the bound (3.7) concludes the proof. ∎
Proposition 3.2.
For all and ,
The constant involved in depends on only.
Proof.
It is clear from the expression of (2.2) and Jensen’s inequality that
where
The desired estimate hence follows from the inequality
| (3.8) |
where, with assumption (1.12) in mind, is some constant proportional to
| (3.9) |
∎
Proposition 3.3.
Let , and be a density satisfying . Then,
The constants involved in depends on , , and only.
Proof.
The first estimate should first be established with the stopping time
which should then be sent to , as goes to infinity. For the sake of simplicity, we omit this stopping time in the following. Let us denote and the associated characteristics (with ), satisfying (2.11). We have
and Itô’s formula gives, for fixed ,
From the uniform sublinearity (2.4), we derive
| (3.10) |
where
We may then integrate (3.10) with respect to using a stochastic Fubini theorem, which leads to
| (3.11) |
where
| (3.12) | |||
To deal with , one may write
so that
From Proposition 3.2, we hence deduce
| (3.13) |
From SDE (3.11) with the sublinear terms (3.12) and (3.13), using Gronwall’s lemma and Burkholder-Davis-Gundy’s inequality, we classically obtain the first estimate. Moreover, from
since , we derive the second estimate, which concludes the proof. ∎
3.2 Stochastic averaging lemma
Proposition 3.4.
Let us assume that the initial data satisfies, for some ,
| (3.14) |
For all, , letting , the averaged quantity
lies in almost surely, with the estimate
| (3.15) |
The constant involved in in (3.15) depends on , , and only.
Proof.
The proof of this result is based on a classical averaging lemma (see e.g [2] for the deterministic case), which we adapt here to the stochastic case, in a similar fashion to [9], Lemma 4.3. Let us first consider some initial data so that is a regular solution of (2.8) in the sense of Proposition 2.4, which may be written as follows: -a.s, for all ,
| (3.16) |
where we have introduced the coefficients
| (3.17) |
Let us fix . For simplicity, let us drop the summation signs in (3.16) and integrate it with respect to . This is possible (for every ) thanks to the bound (3.2) with established previously. Denoting the -Fourier transform
we are led to
Therefore, introducing some , we get
from which we deduce the expression
| (3.18) |
We now integrate (3.18) with respect to . This is possible since one could show that
for fixed , by a method similar to the one employed to establish (3.2) in the proof Proposition 3.1. Introducing the -Fourier transform
equation (3.18) leads to
Note that since
| (3.19) | |||
| (3.20) |
forgetting the summation signs again, we may as well only consider terms of the form
| (3.21) |
where and and are multi-indexes of order , and . From (3.21) Itô’s isometry gives
| (3.22) |
where
Using a trace lemma (see e.g [19], Theorem 2.7.2) we get, for some ,
It then follows by Plancherel’s identity, since is compactly supported, that
| (3.23) |
For the second term , Jensen’s inequality gives
and the same manipulation leads to
| (3.24) |
Similarly, for the third term , Jensen’s inequality gives
where we have used . The same manipulation hence leads to
| (3.25) |
Finally, for the fourth term , we get
and the same manipulation leads to
| (3.26) |
Now fixing some , let us consider
where
| (3.27) | |||
| (3.28) |
On one hand, making use of Plancherel’s identity, and since is compactly supported,
thanks to Proposition 3.1, and the initial bound (3.14). On the other hand, from the bounds (3.23), (3.24), (3.25) and (3.26) we deduce (using Plancherel’s identity once again)
| (3.29) |
Expressions (3.17) and the sublinearity estimate (2.4) immediately give
| (3.30) | |||
| (3.31) | |||
| (3.32) |
from which we easily deduce (using the same method as in the proof of Proposition 3.3 for the term involving ), for ,
so that, thanks again to Proposition 3.1, Proposition 3.3 and the initial bound (3.14), (3.29) yields
Considering , this supremum is bounded under the requirements
which can be met as soon as . For such , we have shown that
which concludes the proof in the case of a regular initial data . We may extend this last inequality to general initial data similarly to the proof of Proposition 3.1. ∎
3.3 Tightness
Given an increasing weight function, say
let us introduce the weighted Sobolev space
| (3.33) |
and the dual space
| (3.34) |
We may also define the intermediate Sobolev spaces , for non-integer . Note that so that .
Proposition 3.5.
Let us assume that the initial data satisfies, for some and ,
| (3.35) |
Then for all , the family of random variables is tight in .
Proof.
Without loss of generality, let . For some and , let us introduce the set
where denotes the -Hölder semi-norm
Since the embedding is compact, the dual embedding is compact. Additionally, for , an interpolation inequality (in weighted Sobolev spaces) yields, for some ,
Consequently, Arzelà-Ascoli’s theorem guarantees that is a relatively compact subset of the separable, complete space . Markov’s inequality gives
for . The first term is bounded uniformly in thanks to Proposition 3.1. The bound on the second term results directly from Kolmogorov’s continuity theorem and the following lemma.
Lemma 3.1.
For some , for all ,
The constant involved in depends on , and only.
We now prove this lemma: let . Let us simply denote and note that
As in the proof of Proposition (3.4), we may apply the -Fourier transform to equation (3.16) to get (forgetting the summation signs)
Itô’s formula results in
so that, integrating against , we get
| (3.36) |
where
From (3.36) we derive
| (3.37) |
Plancherel’s identity gives
so that
Considering (3.30), (3.31), (3.32), we see that we essentially need a bound (uniform in ) on
| (3.38) |
This is possible since, for any ,
so that, for ,
whenever, recalling (3.35), for some ,
These requirements can be met for some close enough to (and close to ). As for the martingale term, Burkholder-Davis-Gundy’s inequality gives
| (3.39) |
Considering (3.32), the first term in (3.39) is again controlled by the bound on (3.38). We may then come back to (3.37) and use Grönwall’s lemma to conclude.
∎
Proposition 3.6.
Let us assume that the initial data satisfies (3.35).
Then, for all , the family of random variables is tight in
.
Proof.
Let us introduce the weight function
and the associated weighted spaces and as in (3.33) and (3.34). One could prove the following lemma as previously.
Lemma 3.2.
For some , for all ,
The constant involved in depends on , , and only.
We may now fix some and naturally introduce the set
where denotes the set of functions satisfying, for and some ,
| (3.40) | |||
| (3.41) | |||
| (3.42) | |||
Markov’s inequality, gives, for some ,
which tends to zero uniformly in as goes to infinity, thanks to Proposition 3.1, 3.3, 3.4 and Lemma 3.2. It only remains to prove that is a relatively compact subset of . Let us introduce a sequence in .
First, let us show that is compact locally in space, that is in for any . Since , we deduce from (3.40) that
Similarly to the proof of Proposition 3.5, we may then use Arzelà-Ascoli’s theorem to deduce that converges in up to some subsequence (which we omit for clarity). An interpolation inequality (in weighted Sobolev spaces: ) yields
where . It follows that
Thanks to (3.42), we deduce that
To derive compactness globally in space, that is in , it is enough to establish a uniform integrability estimate of the form
To this intent, since is compactly supported, we may simply write
where , according to (3.35), and then use the bounds (3.40) and (3.41) to conclude. ∎
3.4 Convergence of the martingale problem
Let us introduce a sequence and a countable subset of , which we assume to contain the truncation functions
| (3.43) |
Recall that the function has been introduced in (2.1). Since is countable, it follows from Proposition 3.5, Proposition 3.6 and Tykhonov’s theorem that the family of random variables is tight in the space
for . Using Skorokhod’s representation theorem, up to a subsequence of which we omit for simplicity, we may introduce random variables defined on some other probability space such that, for all ,
and the following convergences hold -almost surely:
More precisely, since Proposition 3.1 and Proposition 3.4 provide the bounds
| (3.44) |
uniformly in , we derive in particular that the families of random variables and are uniformly integrable, and therefore
| (3.45) | |||
| (3.46) |
Consequently, up to a subsequence, we may also assume that
| (3.47) |
Remark 3.1.
For all , from the equality , -a.s, it is clear that , -a.s. The convergence (3.45) then guarantees that, for all , is indeed given by
Let us now introduce the averaged quantities
By requiring some greater moments for the initial data, we may extend the convergence (3.46) to and .
Lemma 3.3.
Assume that the initial data satisfies, for some ,
| (3.48) |
Then the following convergences hold in :
Consequently, up to a subsequence, we may also assume that these convergences hold almost everywhere, almost surely.
Proof.
Let us, for instance, prove the convergence of . For fixed , denoting ,
| (3.49) |
and we note that, for any and ,
for close enough to . As a result,
Since, from (3.45), we classically derive
we deduce similarly that as goes to infinity. We may hence come back to (3.49) and conclude. The convergence of the convoluted functions is easily deduced.
∎
For fixed , defines a (strong) solution of (2.8) on . In particular, it satisfies the associated martingale problem: recalling the operator defined in (2.9), for all , the process
| (3.50) |
defines a continuous, real valued martingale on with respect to the filtration
Its quadratic variation is given by
| (3.51) |
We are now ready to state the following result.
Proposition 3.7.
Let us introduce, on , the filtration
Recalling the operator defined in (1.15), for all test functions of the form
| (3.52) |
the process
defines a continuous, real-valued martingale with respect to , with quadratic variation
Remark 3.2.
Proof of Proposition 3.7.
The martingale problem set on may be expressed as
for all , where , and is continuous and bounded, Since the laws of and coincide, it follows that, on ,
| (3.53) | |||
| (3.54) |
where and and are naturally defined on , as and . We may decompose these into
| (3.55) | |||
where, recalling expressions (2.2),
with
We wish to send to infinity in (3.53) and (3.54). Thanks to the convergence (3.45), the first three linear terms in (3.55) cause no issue ; let us hence focus on the remaining terms. Let us consider the term involving : defining the natural limiting term
with , we have
| (3.56) |
where
First, for we have, thanks to (3.44),
| (3.57) |
For all , for fixed , we may write
which converges to as goes to infinity thanks to the convergence (3.45), and the bounds
and, for some , recalling that ,
for all . Note that this last bound also guarantees the uniform integrability in of the integrand in (3.57), so that . Additionally, for small enough, the bound
guarantees that Similarly, for , we write, for all ,
so that
Since, for fixed , , we conclude in a similar fashion that and . Coming back to (3.56), we have shown that
which is sufficient to pass to the limit in the corresponding term of (3.53) and the left-hand side of (3.54).
The terms involving for can be treated with similar arguments. Let us now handle the more delicate term, involving : we wish to prove that
| (3.58) |
converges to the expected limiting term. Let us introduce
| (3.59) |
For some and , we have
as soon as and , which can be met for close enough to . Hence, thanks to Proposition 3.2, we derive
| (3.60) |
As a consequence, up to some subsequence which we omit for simplicity,
| (3.61) |
It is clear that this weak convergence is enough for the term (3.58) to pass to the limit in (3.53). Therefore, it only remains to identify the limit as
| (3.64) |
First, considering the set, for , ,
we have
| (3.65) |
so that, with Proposition 3.2,
From (3.61) and Lemma 3.3, we deduce
Indeed, it is easy to see that, when ,
and the same equality is deduced for any by density. Hence, almost surely, a.e on . Since this holds for any , we deduce that, almost surely, holds whenever , so that we only have to check equality (3.64) whenever .
Recalling (3.47) and Lemma 3.3, we have
almost everywhere on the set , almost surely. It follows that equality (3.64) holds.
Finally, it remains to pass (3.58) to the limit in the quadratic equality (3.54). To this intent, we simply notice that
where is defined as in (3.59). We have the uniform bound
with so that, up to some subsequence
This time, thanks to (3.60)
Using (3.65), it follows that
Using again Lemma 3.3, we deduce that, almost surely, almost everywhere on
The limit is then determined to be on the complementary set using pointwise convergence, as done previously.
∎
From the martingale problem of Proposition 3.7, we classically construct a martingale solution of (1.7), in the sense of Definition 1.1, using a martingale representation theorem. First, note that estimates (1.13) and (1.14) are easily derived from the convergence (3.45) using Fatou’s Lemma. Introducing the process
| (3.66) |
we see that, for all test function of the form (3.52),
which is a continuous martingale with respect to the filtration , with quadratic variation . With Remark 3.2 in mind, by density, we may carefully extend this statement to any test function in some separable Hilbert space . One may convince oneself for instance that the weighted Sobolev space
is suitable. Using a polarization formula, we deduce that for ,
defines a continuous -martingale, where the operator is defined through
The martingale representation theorem from [8] (p222, Theorem 9.2) holds for the -valued process (3.66), giving another probability space equipped with a filtration , and a -brownian motion on such that
It follows that defines a solution of (1.7) on . This concludes the proof of Theorem 1.
3.5 Strong local alignment
The proof of Theorem 2 can be established using the same arguments. Indeed, the only role played by the weight function in the estimates of sections 3.1 and 3.2 is through the constant in Proposition 3.2, given by (3.9). Introducing of the form we notice that
uniformly in . As a result, introducing a weak solution of (1.7) with constructed (in law) as previously, the following estimates hold uniformly in : for any ,
and for some ,
From here, we may use the same arguments as previously to establish the tightness of (resp. ) in for (resp. in ) and then pass to the limit in the martingale problem satisfied by as goes to infinity.
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