Stochastic differential equations in a scale of Hilbert spaces

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in a Euclidean space.


Introduction
Evolution differential and stochastic differential equations in Banach spaces play hugely important role in many parts of mathematics and its applications. This class of equations unifies infinite systems of ordinary differential equations and partial differential equations (realized in l p -type spaces of sequences and Sobolevtype spaces, respectively), and their stochastic counterparts, see e.g. [10], [8] and references therein and modern developments in e.g. [4].
So let us consider a stochastic differential equation (SDE) of the form in a Banach space X, where f and B are given vector and operator fields on X respectively and W a suitable Wiener process in X. The standard approach to such equations usually requires that f = A + φ, where (C1) A is a generator of a C 0 -semigroup in X, and (C2) φ and B satisfy certain Lipschitz or dissipativity conditions in X. Then the existence, uniqueness and regularity of solutions of the corresponding Cauchy problem can be proved.
This classical theory does not cover some important examples motivated by e.g. problems of statistical mechanics and hydrodynamics. In particular, there are situations where A fails to satisfy condition (C1) but is instead bounded in a scale of Banach spaces X α , α ∈ A, where A ⊂R 1 is an interval and X α ⊂ X β if α ≤ β. That is, A is a bounded operator acting from X α to X β for any α < β, and for all x ∈ X α and some constant c > 0 (independent of α and β but possibly dependent on the interval A).
In this framework, equation (1.1) with no diffusion term (B ≡ 0) has been studied by Ovsyannikov's method, see e.g. [10] and modern developments and references in [11], [3]. Moreover, instead of (C2), the non-linear drift term φ is allowed to satisfy a generalized Lipschitz condition in the scale (X α ) α∈A with singularity of the type as in (1.2) (see [17,19,3]). The price to pay here is that the existence of a solution with initial value in X α can only be proved in the bigger space X β , β > α. The lifetime of this solution depends on α and β (and the interval A itself).
The aim of the present work is to extend Ovsyannikov's method to the case of stochastic differential equations. We require the drift f to be a map from X α to X β for any α < β and satisfy a generalized Lipschitz condition with singularity (β − α) −1/2 (and make similar assumption about the diffusion coefficient B), see Condition 2.1 given in the next section, and prove the existence and uniqueness of finite time solutions of the corresponding Cauchy problem. Observe that the singularity allowed here is weaker than in the deterministic case (cf. (1.2)), which is related to the specifics of the Ito integral estimates. As in the deterministic case, the solution will live in the scale X α , α ∈ A. For simplicity, we assume that all X α are Hilbert spaces, although all our results hold in a more general situation of suitable Banach spaces. The proof is based on the contractivity of the corresponding integral transformation of a weighted space of trajectories in ∪ α∈A X α (constructed similar to the ones used in [17,19,3]).
Our main example is motivated by the study of countable systems of particles randomly distributed in a Euclidean space R n (of the type considered in [6], [7]). Each particle is characterized by its position x and an internal parameter (spin) σ x ∈ R. For a given fixed ("quenched") configuration γ of particle positions, which is a locally finite subset of R n , we consider a system of stochastic differential equations describing (non-equilibrium) dynamics of spins σ x , x ∈ γ. Two spins σ x and σ y are allowed to interact via a pair potential if the distance between x and y is no more than a fixed interaction radius r, that is, they are neighbors in the geometric graph defined by γ and r. Vertex degrees of this graph are typically unbounded, which implies that the coefficients of the corresponding equations cannot be controlled in a single Hilbert or Banach space (in contrast to spin systems on a regular lattice, which have been well-studied, see e.g. [9] and modern developments in [12], and references therein). However, under mild conditions on the density of γ (holding for e.g. Poisson and Gibbs point processes in R n ), it is possible to apply the approach discussed above and construct a solution in the scale of Hilbert spaces S γ α of weighted sequences (q x ) x∈γ such that x∈γ |q x | 2 e −α|x| < ∞, α > 0. Observe that the family X α = S γ α , α > 0, forms the dual to nuclear space Φ ′ = ∪ α X α . SDEs on such spaces were considered in [13], [14]. The existence of solutions to the corresponding martingale problem was proved under assumption of continuity of coefficients on Φ ′ and their linear growth (which, for the diffusion coefficient, is supposed to hold in each α-norm). Moreover, the existence of strong solutions requires a dissipativity-type estimate in each α-norm, too, which does not hold in our framework.
In the last subsection, we prove the uniqueness of the infinite-particle dynamics using more classical methods, which generalise those applied to deterministic systems in [16], [5].

Setting
Let us consider a family of separable Hilbert spaces X α indexed by α ∈ [α * , α * ] with fixed 0 ≤ α * , α * < ∞, and denote by · α the corresponding norms. We assume that where the embedding means that X α is a vector subspace of X β . When speaking of these spaces and related objects, we will always assume that the range of indices is [α * , α * ], unless stated otherwise. Let W (t) be a cylinder Wiener process in a separable Hilbert space H defined on a suitable filtered probability space. Introduce notation We will denote by · H β its standard norm. Our aim is to construct a strong solution of equation (1.1), that is, a solution of the stochastic integral equation with coefficients acting in the scale of spaces (2.1). More precisely, we assume that f : X α → X β and B : X α → H β for any α < β, and the following Lipschitztype condition is satisfied. and for any α < β and all u, v ∈ X α .
We denote by GL (1) and GL (2) the sets of mappings f and B under conditions (2.3) and (2.4), respectively.

Remark 2.2
The Lipschitz constant L may depend on α * and α * , as usually happens in applications.

Remark 2.3
In contrast to the classical Ovsyannikov method for deterministic equations, where the right-hand side of (2.3) is proportional to (β − α) −1 , we have to require stronger bounds with the singularity (β − α) −1/2 . This is due to the presence of the Ito stochastic integral in (2.2).

Remark 2.4 Setting v = 0 in (2.3) and (2.4), we obtain linear growth conditions
and for some constant K, any α < β and all u ∈ X α .
This framework can be transformed to our setting by an appropriate change of the parametrization, e.g. α → α * − α.

Main results
Let us fix b > 0 and define the function Obviously, p b (α, t) is decreasing in t and increasing in α, and satisfies inequality [17,19,3].

Remark 3.1 Similar spaces of deterministic functions
consider the spaces E β,T and H β,T of square-integarble progessively measurable random processes u : From now on, we fix f ∈ GL (1) and B ∈ GL (2) . For any . The following theorem states the main existence result of this paper.
Proof. It is sufficient to show that the map Thus the statement of the theorem follows from Theorem 4.1 and Corollary 4.2, which will be proved in the next section.
Of course the choice of the weight function p b is somehow ambiguous. The following statement is a corollary of Theorem 3.3 formulated in a slightly more invariant form (although with some loss of information).

Corollary 3.4 Equation (3.2) has a solution
Theorem 3.3 establishes the uniqueness of the solution in M b . A natural question that arises here is whether there might be a solution that does not belong to any M b . An answer is given by the following (somewhat stronger) uniqueness result.
, is a solution of equation (3.2). Then u ∈ M b and coincides in this space with the solution from Theorem 3.3.
Proof. First observe that E α * ,T ⊂ M b , which implies the statement for β = α * .
Let now β ∈ (α * , α * ) and us consider the Banach space M b,β defined by replacing α * with β in the definition of M b (so that M b = M b,α * ). Then we clearly have OE β,T ⊂ M b,β , with the operator O given by the restriction to time Observe that the proof of Theorem 4.1 (and thus of Theorem 3.3) can be accomplished in the space M b,β instead of M b , which implies that Ou is the unique solution of (3.2) in M b,β . Let now v ∈ M b be the solution constructed in Theorem 3.3. By the uniqueness part of that theorem, we have Ou = Ov, which means that By Lemma 3.6 below we have u ∈ M b , and the statement of the theorem follows from the uniqueness in M b .
In our case, this holds for α < β because of (3.3) and for α ≥ β because of the inclusion u ∈ E β,T and the bound p b (α, t) < 1.
Our main example is given by an infinite system of SDEs describing stochastic dynamics of certain infinite particle spin system and will be discussed in Section 5. Here, we provide an example of a very different type, which can also be dealt with by much simpler methods and thus clarifies up to some extend the statement of Theorem 3.3.
Remark 3.7 For simplicity, we required X α to be Hilbert spaces. This is in fact not essential and the case of a scale of suitable Banach spaces can be treated in a similar way.
Example 3.8 Consider the following SPDE on the 1-dimensional torus T:

4)
where u(t) ∈ C 1 (T), u x (t, x) := ∂ ∂x u(t, x), x ∈ T, c ∈ R and W is a real-valued Wiener process. Denote by v(k), k ∈ Z, the Fourier coefficients of v ∈ L 2 (T) and define the scale of Hilbert spaces It is clear that X α ⊂ X β , α > β (cf. Remark 2.6). Let H := R and define B :

4) can now be written in the form (2.2). Moreover, it can be shown by a direct computation that B satisfies condition (2.4). Thus, by Theorem 3.3 adopted to this
setting, for any β < α and an initial condition u(0) ∈ X α there exists a solution u(t) ∈ X β , t < τ (α − β), where τ is a constant (independent of α and β but possibly dependent on their allowed range).

(3.5)
Fix any β < α and an initial condition u(0) ∈ X α . It follows directly from (3.5) that the solution u(t) belongs to X β for t < c −2 (α − β). It is also clear that the solution does not live in the scale of standard Sobolev spaces. Neither of course does B satisfy condition (2.4) in such a scale.

Proof of the contractivity.
In this section, we will show that F is a contraction in M b with b sufficiently small. Proof. Let u, v ∈ M b and fix β ≤ α * and t ∈ (0, bβ). Then F (u)(t), F (v)(t) ∈ X β , and we have the estimate , and the integral term of (4.1) obtains the form Thus it follows from (4.1) that Let us now show that F preserves the space M b . For this, we set u 0 (t) = 0 ∈ X α * . Then and so In the second inequality we used Remark 2.4 with u = 0 and α = 0. Then This together with (4.2) implies the result.

Stochastic spin dynamics of a quenched particle system
Our main example is motivated by the study of stochastic dynamics of interacting particle systems. Let γ ⊂ X = R d be a locally finite set (configuration) representing a collection of point particles. Each particle with position x ∈ X is characterized by an internal parameter (spin) σ x ∈ S = R 1 . We fix a configuration γ and look at the time evolution of spins σ x (t), x ∈ γ, which is described by a system of stochastic differential equations in S of the form whereσ = (σ x ) x∈γ and W = (W x ) x∈γ is a collection of independent Wiener processes in S. We assume that both drift and diffusion coefficients f x and B x depend only on spins σ y with |y − x| < r for some fixed interaction radius r > 0 and have the form where the mappings ϕ xy : S × S → S and Ψ xy : S × S → S satisfy finite range and uniform Lipschitz conditions, see Definition 5.3 and Condition 5.5 below.
Our aim is to realise system (5.1) as an equation in a suitable scale of Hilbert spaces and apply the results of previous sections in order to find its strong solutions.
We introduce the following notations: n x ≡ n x,r (γ) := number of points in γ x,r ( = number of particles interacting with particle in position x).
Observe that, although the number n x is finite, it is in general unbounded function of x. We assume that it satisfies the following regularity condition.

Condition 5.1 There exists a constant a(γ, r) such that
for all x ∈ X.

Existence of the dynamics
Our dynamics will live in the scale of Hilbert spaces Let us define the corresponding spaces GL (1) and GL (2) (cf. Condition 2.1) and set Observe that W (t) := (W x (t)) x∈γ is a cylinder Wiener process in H.
Let V be a family of mappings V xy : S 2 → S, x, y ∈ γ.

Definition 5.3
We call the family V admissible if it satisfies the following two assumptions: • finite range: there exists constant r > 0 such that V xy ≡ 0 if |x − y| ≥ r; • uniform Lipschitz continuity: there exists constant C > 0 such that for all x, y ∈ γ and q ′ 1 , q ′ 2 , q ′′ 1 , q ′′ 2 ∈ S. Define a map V : S γ → S γ and a linear operator V (q) : S γ → S γ ,q ∈ S γ , by the formula respectively.
The proof of this Lemma is quite tedious and will be given in Section 6.
Now we can return to the discussion of system (5.1). Assume that the following condition holds. By Lemma 5.4 we have ϕ ∈ GL (1) and Ψ ∈ GL (2) . Thus we can write (5.1) in the formσ where W (t) = (W x (t)) x∈γ , and apply the results of Section 3 to its integral counterpart. We summarize the existence results in the following theorem, which follows directly from Theorem 3.3.
Theorem 5.6 System (5.1) has a strong solution u :

and the restriction of
Remark 5.7 Theorem 5.6 can also be proved in the scale of Banach spaces cf. Remark 3.7.

The uniqueness
In this section we establish a stronger uniqueness result, extending to our situation the method applied to deterministic systems in [16], [5]. As before, the main ingredients here are the bound on the density of configuration γ (Condition 5.1) and uniform Lipschitz continuity of the maps ϕ xy and Ψ xy (Condition 5.5). However, in contrast to the previous section, we will consider solutions of a more general type.
Let E(S, T ) be the space of square-integrable progressively measurable random processes q : Definition 5. 8 We call a random processq : [0, T ) → S γ a pointwise (strong) solution of system (5.1) if q x (·) ∈ E(S, T ) and satisfies integral equation It is clear that the solution constructed in Theorem 5.6 is a pointwise strong solution.
To proceed with the proof, we need the following Lemma, which will in turn be proved in Section 6. For any n ∈ N and t ∈ [0, T ) define Lemma 5.10 Assume that conditions of Theorem 5.9 hold. Then there exists µ > 0 such that for any t ∈ [0, T ).

Proofs of auxiliary results
In this section, we present proofs of two technical lemmas used in the previous section.
Then for |x| ≤ nr E (I 1,x (t)) ≤ 4n x C 2 The proof is complete.