Uniqueness for an inviscid stochastic dyadic model on a tree

In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.


Introduction
The deterministic dyadic model on a tree was introduced as a wavelet description of Euler equations in [15] and studied in [4] as a model for energy cascade in turbulence.It can be seen as a generalization with more structure of the so called dyadic model of turbulence, studied in [7].As we show in section 2 this deterministic model (1) does not have uniqueness in l 2 .The aim of this paper is to prove that we can restore uniqueness with the introduction of a suitable random noise: with (W j ) j∈J a sequence of independent Brownian motions.Let's also assume deterministic initial conditions for (2): X (0) = x = (x j ) j∈J ∈ l 2 .The main result of this paper is the weak uniqueness of solution for (2), proven in theorem 7.2.This paper can be seen as a generalization to the dyadic tree model of the results proven for the classic dyadic model in [5], but the proof of uniqueness given here relies of a new, different approach (see also [8]) based on a general abstract property instead of a trick (see Section 6).The q−matrix we rely on is closely related to an infinitesimal generator, so the technique is valid for a larger class of models.
The set J is a countable set and its elements are called nodes.We assume for the nodes a tree-like structure, where given j ∈ J,  is the (unique) father of the node j, and O j ⊂ J is the finite set of offsprings of j.In J we identify a special node, called root and denoted by 0. It has no parent inside J, but with slight notation abuse we will nevertheless use the symbol 0 when needed.
We see the nodes as eddies of different sizes, that split and transfer their kinetic energies to smaller eddies along the tree.To formalize this idea we consider the eddies as belonging to discrete levels, called generations, defined as follows.For all j ∈ J we define the generation number |j| ∈ N such that |0| = 0 and |k| = |j| + 1 for all k ∈ O j .
To every eddy j ∈ J we associate an intensity X j (t) at time t, such that X 2 j (t) is the kinetic energy of the eddy j at time t.The relations among intensities are those given in (1) for the deterministic model and (2) for the perturbed stochastic model.The coefficients c j are positive real numbers that represent the speed of the energy flow on the tree.
The idea of a stochastic perturbation of a deterministic model is well established in the literature, see [6] for the classical dyadic model, but also [11], [9] for different models.This stochastic dyadic model falls in the family of shell stochastic models.Deterministic shell models have been studied extensively in [12] while stochastic versions have been investigated for example in [10] and [16].
When dealing with uniqueness of solutions in stochastic shell models, the inviscid case we study is more difficult than the viscous one, since the more regular the space is, the simpler the proof and the operator associated to the viscous system regularizes, see for example [3] about GOY models, where the results are proven only in the viscous case.
In (2) the parameter σ = 0 is inserted just to stress the open problem of the zero noise limit, for σ → 0. This has provided an interesting selection result for simple examples of linear transport equations (see [2]), but it is nontrivial in our nonlinear setting, due to the singularity that arises with the Girsanov transform, for example in (6).
It is worth noting that the form of the noise is unexpected: one could think that the stochastic part would mirror the deterministic one, which is not the case here, since there is a j-indexed Brownian component where we'd expect a  one, and there is a k-indexed one instead of a j one.
One could argue that this is not the only possible choice for the random perturbation.On one hand we chose a multiplicative noise, instead of an additive one, but this is due to technical reasons (see [14]).On the other hand, there are other possible choices, for example the Brownian motion could depend on the father and not on the node itself, so that brothers would share the same Brownian motion.But the choice we made is dictated by the fact that we'd like to have a formal conservation of the energy, as we have in the deterministic case (see [4]).If we use Itô formula to calculate we can sum formally on the first n + 1 generations, taking X 0 (t) = 0: since the series is telescoping in both the drift and the diffusion parts independently.That means we have P-a.s. the formal conservation of energy, if we define the energy as 2 Non-uniqueness in the deterministic case In [7] it has been proven that there exists examples of non uniqueness of l 2 solutions for the dyadic model if we consider solutions of the form Y n (t) = a n t − t 0 , called self-similar solutions, with (a n ) n ∈ l 2 .Thanks to the lifting result (Proposition 4.2 in [4]) that is enough to obtain two different solutions of the dyadic tree model, with the same initial conditions.Following the same idea of self-similar solutions, introduced in [7] and [5] for the classic dyadic model and in section 5.1 in [4] for the tree dyadic model, we can construct a direct counterexample to uniqueness of solutions.In order to do this we need an existence result stronger than the one proven in [4].
Theorem 2.1.For every x ∈ l 2 there exists at least one finite energy solution of (1), with initial conditions X(0) = x and such that j∈J The proof of this theorem is classical, via Galerkin approximations, and follows that of theorem 3.3 in [4].Now we recall the time reversing technique.We may consider the system (1) for t ≤ 0: given a solution X(t) of this system for t ≥ 0, we can define X(t) = −X(−t), which is a solution for t ≤ 0, since We can now consider the self similar solutions for the tree dyadic model, as introduced in , defined for t > t 0 , with t 0 < 0 and with We time-reverse them and we define which, as we pointed out earlier, is a solution of (1) in (−∞, −t 0 ), with −t 0 > 0.
Since lim t→+∞ |X j (t)| = 0 and lim Thanks to theorem 2.1 there is a solution X, with initial conditions x = X(0), and this solution is a finite energy one, so, in particular, doesn't blow up in −t 0 .
Yet it has the same initial conditions of X, so we can conclude that there is no uniqueness of solutions in the deterministic case.

Itô formulation
Let's write the infinite dimensional system (2) in Itô formulation: We will use this formulation since it's easier to handle the calculations, while all results can also be stated in the Stratonovich formulation.So let's now introduce the definition of weak solution.A filtered probability space (Ω, F t , P ) is a probability space (Ω, F ∞ , P ) together with a rightcontinuous filtration (F t ) t≥0 such that F ∞ is the σ-algebra generated by t≥0 F t .Definition 1.Given x ∈ l 2 , a weak solution of (2) in l 2 is a filtered probability space (Ω, F t , P ), a J-indexed sequence of independent Brownian motions (W j ) j∈J on (Ω, F t , P ) and an l 2 -valued process (X j ) j∈J on (Ω, F t , P ) with continuous adapted components X j such that for every j ∈ J, with c 0 = 0 and X 0 (t) = 0. We will denote this solution by (Ω, F t , P, W, X), or simply by X.
Definition 2. A weak solution is an energy controlled solution if it is a solution as in Definition 1 and it satisfies for all t ≥ 0.
Theorem 3.1.There exists an energy controlled solution to We will give a proof of this Theorem at the end of Section 7. It is a weak existence result and uses the Girsanov transform.
We'll prove in the following result that a process satisfying (3) satisfies (2) too.
Proposition 3.2.If X is a weak solution, for every j ∈ J the process (X j (t)) t≥0 is a continuous semimartingale, so the following equalities hold: where the Stratonovich integrals are well defined.So X satisfies the Stratonovich formulation of the problem (2).
Proof.We know that but from (2) we have that the only contribution to [X  , W j ] is given by the −σc j X j • dW j term, so Now if we consider the other integral, we have For each X k we get, with the same computations, that the only contribution to [X k , W k ] t comes from the term σc k X j • dW k , so that we get

Girsanov transform
Let's consider (3) and rewrite it as The idea is to isolate X  dt + σdW j and prove through Girsanov's theorem that they are Brownian motions with respect to a new measure P in (Ω, F ∞ ), simultaneously for every j ∈ J.This way (3) becomes a system of linear SDEs under the new measure P .The infinite dimensional version of Girsanov's theorem can be found in [17] and [13].
Remark 1.We can obtain the same result under Stratonovich formulation.
Let X be an energy controlled solution: its energy E(t) is bounded, so we can define the process which is a martingale.Its quadratic variation is Because of the same boundedness of E(t) stated above, by the Novikov criterion exp(M t − 1 2 [M ] t ) is a (strictly) positive martingale.We now define P on (Ω, F t ) as for every t ≥ 0. P and P are equivalent on each F t , because of the strict positivity of the exponential.
We can now prove the following: ) is an energy controlled solution of the nonlinear equation (3), then (Ω, F t , P , B, X) satisfies the linear equation where the processes are a sequence of independent Brownian motions on (Ω, F t , P ), with P defined by (7).
Proof.Now let's define Under P , (B j (t)) j∈J,t∈[0,T ] is a sequence of independent Brownian motions.Since Then ( 5) can be rewritten in integral form as which is a linear stochastic equation.
Remark 2. We can write our linear equation ( 8) also in Stratonovich form: Remark 3. If we look at (8) we can see that it is possible to drop the σ, considering it a part of the coefficients c j .
We can use Itô formula to calculate This equality will be useful in the following.We now present an existence result also for system (8).
Proof.Fix N ≥ 1 and consider the finite dimensional stochastic linear system This system has a unique global strong solution (X N j ) j∈J .We can compute, using (10) and the definition of N j in ( 11), This implies that there exists a sequence N m ↑ ∞ such that (X Nm j ) j∈J converges weakly to some (X j ) j∈J in L 2 (Ω × [0, T ], l 2 ) and also weakly star in Now for every N ∈ N, (X N j ) j∈J is inProg, the subspace of progressively measurable processes in L 2 (Ω × [0, T ], l 2 ).But Prog is strongly closed, hence weakly closed, so (X j ) j∈J ∈ Prog.
We just have to prove that (X j ) j∈J solves (8).All the one dimensional stochastic integrals that appear in each equation in (9) are linear strongly continuous operators Prog → L 2 (Ω), hence weakly continuous.Then we can pass to the weak limit in (12).Moreover from the integral equations (9) we have that there is a modification of the solution which is continuous in all the components.

Closed equation for E
Proposition 5.1.For every energy controlled solution X of the nonlinear equation (3), E P [X 2 j (t)] is finite for every j ∈ J and satisfies Proof.Let (Ω, F t , P, W, X) be an energy controlled solution of the nonlinear equation ( 3), with initial condition X ∈ l 2 and let P be the measure given by Theorem 4.1.Denote by E P the expectation with respect to P in (Ω, F t ).
Notice that For energy controlled solutions from the definition we have that P -a.s.
because of the behavior of the energy we showed.But on every F t , P ∼ P , so and ( 14) holds.
From (14) it follows that M j (t) is a martingale for every j ∈ J. Moreover since X j (t) is an energy controlled solution and the condition is invariant under the change of measure P ↔ P on F t and, in particular, Now let's write (10) in integral form: We can take the P expectation, where the N j term vanishes, since it's a P -martingale.Now we can derive and the proposition is established.
It's worth stressing that E P [X 2 j (t)] satisfies a closed equation.Even more interesting is the fact that this is the forward equation of a continuous-time Markov chain, as we will see in the following section.

Associated Markov chain
We want to show and use this characterization of the second moments equation as the forward equation of a Markov chain, taking advantage of some known results in the Markov chains theory.We follow the transition functions approach to continuous times Markov chains; we don't assume any knowledge of this theory, so we will provide the basic definitions and results we need.More results can be found in the literature, see for example [1].
and it satisfies the semigroup property (or Chapman-Kolmogorov equation) Definition 4. A q-matrix Q = (q jl ) j,l∈J is a square matrix such that q jl ≤ −q jj =: q j ≤ +∞ ∀j ∈ J.
A q-matrix is called stable if all q j 's are finite, and conservative if If Q is a q-matrix, a Q-function is a transition function f jl (t) such that f ′ jl (0) = Q.The q-matrix shows a close resemblance to the infinitesimal generator of the transition function, but they differ, since the former doesn't determine a unique transition function, while the latter does.Still this approach can be seen as a generator approach to Markov chains in continuous times.Now let's see these objects in our framework: let's write (13) in matrix form.Let Q be the infinite dimensional matrix which entries are defined as k for k = j,  Proposition 6.1.The infinite matrix Q defined above is the stable and conservative q-matrix.Moreover Q is symmetric.
Proof.It's easy to check that Q is a stable and conservative q-matrix.First of all q j,j < 0 for all j ∈ J and q j,l ≥ 0 for all j = l.Then Moreover it is very easy to check that the matrix is symmetric: Since Q is a q-matrix we can construct the process associated, as a jump and hold process on the space state, which in our case is the tree of the dyadic model.The process will wait in node j for an exponential time of parameter q j , and then will jump to  or k ∈ O j with probabilities q j, /q j and q j,k /q j respectively.This process is a continuous time Markov chain that has J as a state space and also has the same skeleton as the dyadic tree model, meaning that the transition probabilities are non-zero only if one of the nodes is the father of the other one.
Given a q-matrix Q, it is naturally associated with two (systems of) differential equations: y ′ jl (t) = h∈J q jh y hl (t), called forward and backwards Kolmogorov equations, respectively.Lemma 6.2.Given a stable, symmetric and conservative q-matrix Q, then the unique nonnegative solution of the forward equations (15) in L ∞ ([0, ∞), l 1 ), given a null initial condition y(0) = 0, is y(t) = 0.
Proof.Let y be a generic solution, then We can consider for every node ŷj = +∞ 0 e −t y j (t)dt, the Laplace transform in 1.From the last equation of the system above, we have j ŷj ≤ M , for some constant M > 0, so in particular we can consider k ∈ J such that ŷk ≥ ŷj , for all j ∈ J. Now we want to show that y ′ k (t) is bounded: thanks to the symmetry and stability of Q we have We can integrate by parts where the last equality follows from the conservativeness of Q, and we used the stability and symmetry.Now we have ŷk = 0 and so all ŷj = 0, hence y j (t) = 0 for all j ∈ J, for all t ≥ 0.

Uniqueness
Now we can use the results of the previous section to prove the main results of this paper.
Proof.By linearity of (8) it is enough to prove that for null initial conditions there is no nontrivial solution.Since we have ( 13), proposition 6.1 and lemma 6.2, then E P [X 2 j (t)] = 0 for all j and t, hence X = 0 a.s.Let's recall that we already proved an existence result for (8) with proposition 4.2.
Theorem 7.2.There is uniqueness in law for the nonlinear system (3) in the class of energy controlled L ∞ (Ω × [0, T ], l 2 ) solutions.
We can now conclude with the proof of Theorem 3.1.
Proof of Theorem 3.1.Let (Ω, F t , P , B, X) be the solution of (8) in L ∞ (Ω × [0, T ], l 2 ) provided by theorem 7.1.We follow the same argument as in Section 4, only from P to P .We construct P as a measure on (Ω, F T ) satisfying where M t = 1 σ j∈J t 0 X  (s)dB j (s).Under P the processes are a sequence of independent Brownian motions.Hence (Ω, F t , P, W, X) is a solution of (3) and it is in L ∞ , since P and P are equivalent on F T .