We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each particle is characterized by its position xRd and internal parameter (spin) σxR. While the positions of particles form a fixed (“quenched”) locally-finite set (configuration) γRd, the spins σx and σy interact via a pair potential whenever xy<ρ, where ρ > 0 is a fixed interaction radius. The number nx of particles interacting with a particle in position x is finite but unbounded in x. The growth of nx as |x| → creates a major technical problem for solving our SDE system. To overcome this problem, we use a finite volume approximation combined with a version of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a scale of Banach spaces of weighted sequences. As an application example, we construct stochastic dynamics associated with Gibbs states of our particle system.

In recent decades, there has been an increasing interest in studying countable systems of particles randomly distributed in the Euclidean space Rd. In such systems, each particle is characterized by its position xXRd and an internal parameter (spin) σxSRn, see for example Refs. 34, 32 (Sec. 11), 7, 14, and 15 pertaining to modeling of non-crystalline (amorphous) substances, e.g., ferrofluids and amorphous magnets. Throughout the paper we suppose, mostly for simplicity, that n = 1.

Let us denote by Γ(X) the space of all locally finite subsets (configurations) of X and consider a particle system with positions forming a given fixed (“quenched”) configuration γ ∈ Γ(X). Two spins σx and σy, x, yγ, are allowed to interact via a pair potential if the distance between x and y is no more than a fixed interaction radius ρ > 0, that is, they are neighbors in the geometric graph defined by γ and ρ. The evolution of spins is described then by a system of coupled stochastic differential equations (SDE).

Namely, we consider, for a fixed γ ∈ Γ(X), a system of stochastic differential equations in S=R of the following form:
(1.1)
where Ξt=ξx,txγ and Wx,txγ are, respectively, families of real-valued stochastic processes and independent Wiener processes on a suitable probability space. Here the drift and diffusion coefficients Φx and Ψx are real-valued functions, defined on the Cartesian power Sγσ̄=(σx)xγσxS,xγ. Both Φx and Ψx are constructed using pair interaction between the particles and their self-interaction potentials, see Sec. II, and are independent of σy if yx>ρ.

The aim of including the diffusion term in (1.1) is two-fold. On the one hand, it allows to consider the influence of random forces on our particle system and, on the other hand, to construct and study stochastic dynamics associated with the equilibrium (Gibbs) states of the system. The Gibbs states of spin systems on unbounded degree graphs have been studied in Refs. 14, 15, and 27, see also references given there.

The case where vertex degrees of the graph are globally bounded (in particular, if γ has a regular structure, e.g., γ=Zd) has been well-studied (in both deterministic and stochastic cases), see e.g., Refs. 25, 17, 19, 20, 23, 24, 28, 29, 37, and 39, and references therein. However, the aforementioned applications to non-crystalline substances require dealing with unbounded vertex degree graphs. An important example of such graphs is served by configurations γ distributed according to a Poisson or, more generally, Gibbs measure on Γ(X) with a superstable low regular interaction energy, in which case the typical number of “neighbors” of a particle located at xX is proportional to 1+log|x|, see e.g., Refs. 38 and 26 (p. 1047).

There are two main technical difficulties in the study of system (1.1). The first one is related to the fact that the number of particles interacting with a tagged particle x is finite but unbounded in xγ. Consequently, the system cannot be considered an equation in a fixed Banach space and studied by standard methods of e.g., Refs. 12 and 16.

The way around it has been proposed in Ref. 11, where a deterministic version of system (1.1) (with Ψ ≡ 0) was considered in an expanding scale of embedded Banach spaces of weighted sequences and solved using a version of the Ovsjannikov method.

Originally, the Ovsjannikov method was developed for a linear equation
(1.2)
in a scale of densely embedded Banach spaces Bα, αA, where A is a real interval, such that BαBβ if α < β, and A: BαBβ is bounded with norm satisfying the estimate
(1.3)
for any α,βA. Then, for X0Bα, Eq. (1.2) has a solution XtBβ, t < T, for finite T depending on α and β.

It was noticed in Ref. 11 that, under a stronger norm bound with q < 1 in (1.3), the lifetime of the solution XtBβ is infinite. That fact allows to find a global uniform bound for a sequence of finite volume approximations of the system of differential equations in question and prove its convergence, thus proving the existence and uniqueness of the global solution of the deterministic version of (1.1).

The first advances in the study of stochastic equations in the scale Bα,αA, were made in Refs. 10 and 8, where, respectively, local and global strong solutions of a general stochastic equation had been constructed. In those works, the coefficients are assumed to be Lipschitz mappings BαBβ for any α < β, with Lipschitz constants L(βα)q, q=12 and q<12, respectively. Observe that the threshold value of q here is 12 instead of 1 as in (1.3) because of the presence of the Itô integral, which makes it necessary to work in L2 spaces instead of L1.

The results of Refs. 10 and 8 are applicable to system (1.1) only in the case where the drift coefficients Φx, xγ, are globally Lipschitz. However, to construct the dynamics associated with Gibbs states of interacting particle systems, one has to consider the drift coefficients that are only locally Lipschitz. The existence of such dynamics, under certain dissipativity conditions on the drift, is known in the situation of a regular lattice, see Refs. 4 and 5 (observe that those works deal with the more complicated quantum systems but are applicable to classical systems, too, albeit only for the additive noise).

For deterministic systems on unbounded degree graphs, the dissipative case was considered in the aforementioned paper.11 In the present work, we revisit the volume approximation approach of that paper. However, the presence of stochastic terms requires the application of very different techniques. To prove the convergence of finite volume approximations, we have developed a version of the Gronwall inequality suitable for a scale of Banach spaces. In this way, we have been able to prove the existence and uniqueness of global strong solutions of (1.1) and their component-wise time continuity, in the case of dissipative single-particle potentials.

The structure of the paper is as follows. In Sec. II we introduce the framework and formulate our main results. Section III is devoted to the proof of the existence and uniqueness result for (1.1). In a short Sec. III D, we discuss Markov semigroup generated by the solution of (1.1). In Sec. IV, we study stochastic dynamics associated with Gibbs states of our system.

Finally, the  Appendix contains auxiliary results on linear operators in the scales of Banach spaces, estimates of the solutions of system (1.1) and, notably, a crucial for our techniques generalization of the classical comparison theorem and a Gronwall-type inequality, suitable for our framework.

Let us fix a configuration γ ∈ Γ(X) and a family Wx,txγ of independent Wiener processes on a suitable filtered and complete probability space P(Ω,F,F,P). Our aim is to find a strong solution of SDE system (1.1), that is, a family Ξt=ξx,txγ of continuous adapted stochastic processes on P such that the equality
(2.1)
holds for all tT[0,T], T > 0, almost surely, that is, on a common for all t set of probability 1. The coefficients Φx and Ψx are defined explicitly in Assumption II.1 below, and 0tΨx(Ξs)dWx,s is the continuous version of the Ito integral, cf. Remark II.4.
First, we need to introduce some notations. We fix ρ > 0 and denote by nx, xγ, the number of elements in the set
Observe that nx ≥ 1 for all xγ, because xγ̄x. We will also use the notation γxγ̄x\xyγ:xyρ,yx.

For a fixed γ ∈ Γ(X), we will consider the Cartesian product Sγ of identical copies Sx, xγ, of S, and denote its elements by z̄zxxγ, etc. When dealing with multiple configurations η ∈ Γ(X), we will sometimes write z̄ηzxxη, to emphasize the dependence on η.

We will work under the following assumption.

Assumption II.1

  • There exists a constant C > 0 such that
    (2.2)
  • The drift coefficients Φx, xγ, have the form
    (2.3)
    where ϕ: SS is a measurable function and φxy: S2S are also measurable functions satisfying uniform Lipschitz condition
    for some constant ā>0 and all x, yγ, σ1, σ2, s1, s2S.
  • There exist constants c > 0 and R ≥ 2 such that
    (2.4)
  • There exists b > 0 such that
    (2.5)
  • The diffusion coefficients Ψx, xγ, have the form
    (2.6)
    where ψxy: S2S are measurable functions satisfying uniform Lipschitz condition
    (2.7)
    for some constant M > 0 and all x, yγ, σ1, σ2, s1, s2S.

The specific form of the coefficients requires the development of a special framework. Indeed, we will be looking for a solution of (2.1) in a scale of expanding Banach spaces of weighted sequences, which we introduce below.

We start with a general definition and consider a family B of Banach spaces Bα indexed by αĀ[α*,α*] with fixed 0 ≤ α*, α* < , and denote by Bα the corresponding norms. When speaking of these spaces and related objects, we will always assume that the range of indices is [α*, α*], unless stated otherwise. The interval Ā remains fixed for the rest of this work. We will also use the corresponding semi-open interval A[α*,α*).

Definition II.2
The family B is called a scale if
where the embedding means that Bα is a dense vector subspace of Bβ.

For any α,βA, we will use the notation

The two main scales we will be working with are given by the spaces lαp of weighted sequences and lαp-valued random processes, respectively, defined as follows.

  1. For all p ≥ 1 and αĀ let
    (2.8)
    and Lp{lαp}αA be, respectively, a Banach space of weighted real sequences and the scale of such spaces.
  2. For all p ≥ 1 and αĀ let Rαp denote the Banach space of lαp-valued random processes ξ̄t,tT, on probability space P, with progressively measurable components and finite norm
    and let Rp{Rαp}αA be the scale of such spaces.

Remark II.3

The choice of exponential weights in the definition of space lαp is dictated by the logarithmic growth condition on numbers nx, cf. (2.2), which in turn is motivated by the fact that it holds for a typical configuration γ distributed according to a Poisson or, more generally, Gibbs measure on Γ(X) with a superstable low regular interaction energy, in which case nx is proportional to 1+log|x|, see e.g., Refs. 38 and 26 (p. 1047). In general, an informal balance condition between nx and w(|x|) is given by w(|x|) ≈ exp(exp(nx)), see Sec. 2.2 of Ref. 11 for details.

Remark II.4.

Note that for p ≥ 2, the definition of norms in Rαp and lαp implies that for any ξ̄Rαp and any xγ we have E[0Tξx,t2dt]<. Moreover, since each component of ξ̄ is progressively measurable, from the classical theory of integration with respect to the standard Wiener process we see that for all xγ the integral 0tξx,sdWs is well defined and so is the integral 0tΨx(ξ̄s)dWx,s, because Ψx is a finite sum of measurable uniformly Lipschitz functions. Moreover, the process 0tΨx(ξ̄s)dWx,s, tT, has a (unique) continuous version.

For all p ≥ 1 and αĀ we let
be the space of lαp-valued p-integrable random variables.

Our main result is the following theorem.

Theorem II.5.
Suppose that Assumption II.1 holds. Then, for all pR and any F0-measurable ζ̄(ζx)xγLαp, αA, stochastic system (2.1) admits a unique (up to indistinguishability) strong solution ΞRα+p. Moreover, the map
is continuous for any β > α.

Remark II.6.

Assumption pR ensures that given ξ̄Rβp the random variable ϕ(ξ̄t) is integrable for any t ≥ 0.

The proof of Theorem II.5 will be given in Sec. III.

Our second main result is about the construction of non-equilibrium stochastic dynamics associated with Gibbs states of our system. We consider a Gibbs measure ν on Sγ defined by the pair interaction Wx,y(σx, σy) = a(xy)σxσy, σx, σyS, x, yγ, where a:XR is a measurable function with compact support and a single particle potential V:RR satisfying the lower bound
which is supported on lαp for some αA and pR,R+ε, see Sec. IV A for details. Suppose now that ϕ in (2.3) has a gradient form, that is, ϕ = −∇V, and ψxy=0,xyψxx=1 for all x, yγ, so that our noise is additive, cf. (2.6). Let Tt be the Markov semigroup defined by the process Ξt in a standard way. This semigroup acts in the space Cb(lα+p) of bounded continuous functions on space lα+p=β>αlβp equipped with the projective limit topology, see Sec. III D below for details.

Theorem II.7.
Gibbs measure ν is a symmetrizing (reversible) distribution for the solution of (2.1), that is,
for any αA and f,gCb(lα+p).

The proof of this result will be given in Sec. IV B.

From now on, the constant pR will be fixed.

In this section, we give the proof of Theorem II.5. It will go along the following lines.

  1. Consider a sequence of processes ΞtnnN, tT, that solve finite volume cutoffs of system (2.1), and prove their uniform bound in Rβp for any β > α. For this, we use our version of the comparison theorem and Gronwall-type inequality in the scale of spaces, which is in turn based on the Ovsjannikov method, see Subsection  3 of the  Appendix.

  2. The uniform bound above implies the convergence of sequence Ξn, n, to a process Ξ=(ξx)xγRβp, β > α. Our next goal is to prove that the process Ξ solves system (2.1). The multiplicative noise term does not allow to achieve this by a direct limit transition. Therefore, we construct an R-valued process ηt that solves an equation describing the dynamics of a tagged particle x, while processes ξy,t, yγ, yx, are fixed, and prove that ηt = ξx,t.

  3. The uniqueness and continuous dependence on the initial data is proved by using our version of a Gronwall-type inequality, as above in part (1). The continuity of components of Ξ will follow from our work on the dynamics of a tagged particle x in Sec. III B.

Finally, in Subsection III D, we introduce Markov semigroup defined by the solution of (2.1).

Let us fix an expanding sequence {Λn}nN of finite subsets of γ such that Λn ↑ γ as n and consider the following system of equations:
(3.1)
where ζ̄={ζx}xγLαp, αA, is F0-measurable random initial condition and equality (3.1) holds for all tT, P-a.s. Observe that for each nN system (3.1) is a truncated version of our original stochastic system (2.1).

Theorem III.1.

For any nN system (3.1) admits a unique (up to indistinguishability) solution ΞnRαp with continuous sample paths.

Proof.

The existence and uniqueness of continuous strong solutions of the non-trivial finite dimensional part of system (3.1) is well-known, see Ref. 30, Chap. 3. The inclusion ΞnRαp follows then from the fact that ξx,tn=ζx,tT, for x ∉ Λn.■

Our next goal is to show that the sequence {Ξn}nN converges in Rβp for any β > α. We start with the following uniform estimate, which is rather similar to the one from Ref. 27, adapted to the framework of the scale of Banach spaces using our version of the Gronwall inequality.

Theorem III.2.
Let Ξn=(ξxn)xγ, nN, be the sequence of process defined by Theorem III.1. Then for all β > α we have
(3.2)

Proof
It follows from the first part of Lemma A.10 in the  Appendix (with ξ1ξn) that for all x ∈ Λn and tT we have
(3.3)
We remark that inequality above trivially holds for x ∉ Λn, because in this case ξx,tn=ζx and all terms in the right-hand side of the inequality are non-negative.
We now define a measurable map ηn:Tlα1 via the following formula
It is immediate that its components satisfy inequality similar to (3.3), that is,
Set θx=E|ζx|p+C2x and observe that (θx)xγlα1. Then the map ηn fulfills the conditions of Lemma A.8 in the  Appendix, which implies that for all nN and β > α we have
Observe that the left-hand side forms an increasing sequence, which implies that it converges and
Then, for any finite set ηγ, we have
On the other hand, it is clear that
for any xγ. Thus
The latter inequality holds for all finite ηγ, which implies that
and the proof is complete.■

Theorem III.3.

The sequence {Ξn}nN is Cauchy in Rβp for any β > α.

Proof.
Let us fix n,mN and assume, without loss of generality, that Λn ⊂ Λm. We first consider the situation where x ∈ Λn. It follows from the second part of Lemma A.10 in the  Appendix (with ξ(1)ξn and ξ(2)ξm) that for all x ∈ Λn and tT we have
(3.4)
(3.5)
In the case where x ∈ Λmn we see that for all tT
so that
(3.6)
(cf. Theorem III.2). Combining Eqs. (3.4) and (3.6) and taking into account that ξ̄x,tn,m=0 for x ∉ Λm, we obtain the inequality
for all xγ and tT. We can now proceed as in the Proof of Theorem III.2. Define a measurable map ϱn,m:Tlα1 via the formula
and set
Obviously, (bx)xγlα1 for any fixed α′ ∈ (α, β). It therefore follows then from Lemma A.8 in the  Appendix that
So we have shown that the following inequality holds:
(3.7)
It follows from Theorem III.2 that the right hand side of (3.7) is the remainder of the convergent series (3.2) (with α′ in place of β), which completes the proof.■

We have shown in Sec. III A that, for any β > α, the sequence {Ξn}nN is Cauchy in the Banach space Rβp and thus converges in this space. So we are now in a position to define the process
(3.8)
This process is a candidate for a solution of the system (2.1). A standard way to show this would be to pass to the limit on both sides of (3.1). This approach requires however somewhat stronger convergence than that in Rβp. We are going to overcome this difficulty by considering special one-dimensional equations.
Consider an arbitrary xγ. It is convenient to consider elements of Sγ as pairs (σx, Z(x)), where σxS and Z(x)=zyyγ\xSγ\x. In these notations, we can write Φx(Ξs)=Φx(ξx,s,Ξs(x)) and Ψx(Ξs)=Ψx(ξx,s,Ξs(x)), where
(3.9)
Let us now fix process Ξ defined by (3.8) and consider the following one-dimensional equation:
(3.10)
for all tT, P-a.s. The main goal of this section is to prove that the Eq. (3.10) has a unique solution ηx,t.

Remark III.4.

Note that, for a fixed xγ, the principal difference between Eqs. (3.10) and (2.1) is that the process Ξ is fixed in (3.10) and defined by the limit (3.8), which makes (3.10) a one-dimensional equation w.r.t. ηx.

In order to establish the existence of a solution of Eq. (3.10) we need the following auxiliary result.

Theorem III.5.
Let xγ and ξx be an x-component of the process Ξ defined by (3.8). Then sample paths of ξx are a.s. continuous and
(3.11)

Proof.
It is sufficient to show that, for a fixed xγ, the sequence ξxnnN, is Cauchy in the norm E[suptT||p]1/p because then there exists a subsequence ξxnkkN such that
which, together with the path-continuity of processes ξx,tnk, implies the statement of the theorem.
Fix N̄N such that xΛN̄ and n,mN̄ and assume, without loss of generality, that n < m so that x ∈ Λn ⊂ Λm Consider the process ξ̄x,tn,m defined in (3.5) and proceed as in the  Appendix, Lemma A.10, with ξ(1)ξn and ξ(2)ξm. Taking suptT of both sides of the equality (A17) we obtain the bound
(3.12)
where
(3.13)
and
Now using first the Burkholder–Davis–Gundy inequality (see Ref. 31) and then the Jensen inequality we see that the following estimate on the stochastic term from (3.12) holds.
(3.14)
The integrand in the right-hand side of the above inequality can be estimated in a similar way as (A16), so that we obtain
It follows now that inequality (3.14) can be written in the following way:
where
Therefore returning to inequalities (3.12) and (3.13) we see that
(3.15)
Since γ̄x is finite we can now use Theorem III.3 to conclude that, with a suitable choice of n,mN, the right hand side of the inequality (3.15) above can be made arbitrary small hence the proof is complete.■

Theorem III.6.

Equation (3.10) admits a unique solution.

Proof.
By standard arguments, see e.g., Ref. 1, Proposition 2.9, we conclude that Eq. (3.10) admits a unique local maximal solution ηx such that
for all tT, P-a.s. Here Ξsτn(x) is as in (3.9) and by construction, for all nN, stopping time τn is the first exit time of ηx from the interval (−n, n), defined as
Hence to complete the proof it is sufficient to establish that almost surely limnτn = T. We will prove this fact along the lines of Ref. 1, Theorem 3.1, using the bound (3.11). We begin by using the Itô Lemma to establish the equality
for all tT. Before proceeding we define for convenience the following shorthand notations:
An application of Lemma A.9 in the  Appendix shows that for all tT we have
(3.16)
where constants b and c are defined in Assumption II.1. In the last inequality we used the simple estimate Cp−1 ≤ (1 + C)p−1 ≤ (1 + C)p ≤ 2p−1(1 + Cp) for any C > 0, which holds because p > 1. We can now use the Hölder inequality and classical estimate k=1makNmN1k=1makN (see e.g., Ref. 25) in conjunction with inequality (3.16) above to see that for all tT we have
In a similar way, we obtain the inequality
Setting
we get the bounds
and
Observe that Ax < by Theorem III.5. Finally letting
we see that for all t ∈ [0, ) we have
(3.17)
Observe that constants K and D are independent of the stopping time τn.
The rest of the proof is standard and can be completed along the lines of Ref. 1, Theorem 3.1. We give its sketch for the convenience of the reader. Using Gronwall’s inequality together with the inequality (3.17) above we see that for all t ∈ [0, T] we have
It follows from the definition of stopping time τn that
so that, for all t ∈ [0, T],
Now convergence in probability and the fact that {τn}nN is an increasing sequence imply that almost surely limnτn = T, hence the proof is complete.■

In this section, we are going to prove Theorem II.5. We will show that, for any β > α, the process
(3.18)
solves system (2.1). For this, we will use auxiliary processes ηx constructed in Theorem III.6.

Proof of the existence.
According to Theorem III.6, for each xγ equation
where Ξs(x) is as in (3.9), admits a unique solution ηx,t. Thus it is sufficient to prove that this solution is indistinguishable from the process ξx. The convergence (3.18) implies that, for any fixed xγ,
(3.19)
Therefore, taking into account that both processes ξx and ηx are continuous, to conclude this proof it remains to show that, for any tT,
(3.20)
Let us fix xγ and tT and assume without loss of generality that x ∈ Λnγ. Define the following processes:
The rest of the proof is rather similar to the Proof of Theorem III.6. The Itô Lemma shows that for all tT we have P-a.s.
(3.21)
Using Lemma A.10 in the  Appendix, we can see that for all tT
and
As in the Proof of Theorem III.6, we see that for all tT
(3.22)
and
(3.23)
where
Now, because γx is finite and p ≥ 2 it is clear from Eq. (3.19) that
so we see that Axn0 as n. Therefore using inequality (3.22) and (3.23) above we can conclude from Eq. (3.21) that for all xγ and all tT we have
where
and consequently Cxn,Āxn0 on T as n. Finally using Gronwall inequality we see that for all tT we have
which shows that for all xγ and uniformly on T
Equation (3.20) now follows immediately hence the proof is complete.■

Proof of the uniqueness and continuous dependence.
Suppose that Ξt1=ξx,t1xγ and Ξt2=ξx,t2xγRα+p, are two solutions of system (2.1), with initial values Ξ01,Ξ02Lαp, respectively. Now, for all tT and all xγ letting ξ̄x,t=ξx,t(1)ξx,t(2) we see from Lemma A.10 that
Fix an arbitrary β > α and α1α,β. An application of Lemma A.8 to a bounded measurable map κ:Tlα11 defined by the formula
shows that
where bx=E|ξ̄x,0|p. Therefore we establish that
which implies both statements.■

In this section we denote by Ξt(ζ̄) the solution of Eq. (2.1) with initial condition ζ̄. This process generates an operator family Tt:Cb(lβp)Cb(lαp), α < β, t ≥ 0, by standard formula
(3.24)
Consider the space lα+p=β>αlβp equipped with the projective limit topology, which makes it a Polish space see e.g., Ref. 21.

Theorem III.7.

Operator family Tt, t ≥ 0, is a strongly continuous Markov semigroup in Cb(lα+p) for any αA.

Proof.
Continuity of the map Lαpζ̄Ξ(ζ̄)Rβp, α < β, for an arbitrary T > 0 (cf. Theorem II.5), implies that operators Tt:Cb(lβp)Cb(lαp), t ≥ 0, are bounded, which in turn implies their boundedness as operators in Cb(lα+p), for any αA. The uniqueness of the solution (cf. Theorem II.5) implies in the standard way the evolution property
Observe that the truncated process Ξtn(ζ̄) generates the strongly continuous semigroup Ttn:Cb(lαp)Cb(lαp), for any αA. It follows from the convergence
in Rβp for any β > α that
which in turn implies that Tt:Cb(lα+p)Cb(lα+p) is strongly continuous.■

Remark III.8.
The dominated convergence theorem implies that
(3.25)
for any probability measure ν on lα+p.

As an application of our results, we will present a construction of stochastic dynamics associated with Gibbs measures on Sγ. Sufficient conditions of the existence of these measures were derived in Ref. 14. For the convenience of the reader, we start with a reminder of the general definition of Gibbs measures, adopted to our framework.

In the standard Dobrushin–Lanford–Ruelle (DLR) approach in statistical mechanics,22,33 Gibbs measures (states) are constructed by means of their local conditional distributions (constituting the so-called Gibbsian specification). We are interested in Gibbs measures describing equilibrium states of a (quenched) system of particles with positions γX=Rd and spin space S=R, defined by pair and single-particle potentials Wxy and V, respectively. We assume the following:

  • Wxy:S×SR, x, yX, are measurable functions satisfying the polynomial growth estimate
    (4.1)
    and the finite range condition Wxy ≡ 0 if xyρ for all x, yX and some constants IW, JW, R, r ≥ 0. We assume also that Wxy(u, v) is symmetric with respect to the permutation of (x, u) and (y, v).
  • the single-particle potential V satisfies the bound
    (4.2)
    for some constants aV, bV > 0, and τ > r.

Example IV.1
A typical example is given by the pair interaction in of the form
where a:XR is as in Sec. II. In this case, r = 2 and so we need τ > 2 in (4.2). The method of Ref. 14 does not allow us to control the case of τ = 2, even when the underlying particle configuration γ is a typical realization of a homogeneous Poisson random field on Γ(X).

Let F(γ) be the collection of all finite subsets of γ ∈ Γ(X). For any ηF(γ), σ̄η=(σx)xηSη and z̄γ=(zx)xγSγ define the relative local interaction energy
The corresponding specification kernel Πη(dσ̄γz̄γ) is a probability measure on Sγ of the form
(4.3)
where
(4.4)
is a probability measure on Sη. Here Z(z̄η) is the normalizing factor and δz̄γ\η(dσ̄γ\η) is the Dirac measure on Sγ\η concentrated on z̄γ\η. The family Πη(dσ̄|z̄),ηF(γ),z̄Sγ is called the Gibbsian specification (see e.g., Refs. 22 and 33).
A probability measure ν on Sγ is said to be a Gibbs measure associated with the potentials W and V if it satisfies the DLR equation
(4.5)
for all ηF(γ). For a given γ ∈ Γ(X), by G(Sγ) we denote the set of all such measures.

By Gα,p(Sγ)G(Sγ) we denote the set of all Gibbs measures on Sγ associated with W and V, which are supported on lαp.

Theorem IV.2.

Assume that conditions (4.1) and (4.2) are satisfied and pr,τ. Then the set Gα,p(Sγ) is non-empty for any αA.

Proof.
It follows in a straightforward manner from condition (2.2) that
for any p1,p2N, which is sufficient for the existence of νGα,p(Sγ) for any pr,τ, see Refs. 27 and 14.■

Remark IV.3.

The result of Refs. 27 and 14 is more refined and states in addition certain bounds on exponential moments of vGα,p(Sγ).

Remark IV.4.

Conditions of the uniqueness of vGα,p(Sγ) are known only in the case of configuration γ with bounded sequence nx,xγ. Sufficient conditions of non-uniqueness (phase transition) for Poisson-distributed γ are given in Ref. 14.

In this section, we will construct a process Ξt with invariant measure νGα*,p(Sγ) defined by interaction potentials W and V as in Example IV.1. By Theorem IV.2, the set Gα*,p(Sγ) is not empty if p2,τ. Then, according to the general paradigm, Ξt will be a solution of the system (2.1) with the coefficients satisfying the following:

  1. the drift coefficient has a gradient form, that is, ϕ = −∇V and φx,y(σx,σy)=σxWx,y(σx,σy); moreover, ϕ satisfies Conditions (2.4) and (2.5), a typical example is given by
    in which case R = 2n + 1 and τ = 2n + 2, cf. (2.4);
  2. for each xγ, the noise is additive, that is, Ψx = id.

Thus the system (2.1) obtains the form
According to Theorem II.5, this system admits a unique strong solution ΞRα+ for any initial condition σ̄γlαp with arbitrary αA and pR.

A standard way of rigorously proving the invariance of ν would require dealing with Markov processes and semigroups in nuclear spaces. This difficulty can be avoided by using the limit transition (3.25).

Theorem IV.5
Assume that pmax2,R,τ and let Tt be the semigroup defined by the process Ξt, cf. (3.24). Then any νGα*,p(Sγ) is a reversible (symmetrizing) measure for Tt, that is,
for all f,gCb(lα*+p).

Proof.

First observe that condition pmax2,R,τ ensures that Gα*,p(Sγ) and semigroup Tt is well-defined.

Consider the solution Ξn=(ξxn)xγ of the truncated system (3.1). Its non-trivial part (ξxn)xΛn is a Markov process in SΛn. We denote by TtΛn the corresponding Markov semigroup in Cb(SΛn) and observe that, for fCb(Sγ) and fz̄γ(σ̄Λn)f(σΛn×z̄γ\Λn), we have
By standard theory of finite dimensional SDEs, μΛn(dσ̄Λn|z̄Λn) given by (4.4) is a reversible (symmetrizing) measure for the semigroup TtΛn. Thus we have
for any z̄γ. The latter implies in turn that
where ΠΛn(dσ̄γ|z̄γ)=μΛn(dσ̄Λn|z̄γ)δz̄γ\Λn(dσ̄η̄) is the specification kernel, cf. (4.3). Integrating with respect to ν(dz̄γ) and applying the DLR equation (4.5) we see that
Passing to the limit as n [cf. (3.25)] we obtain the equality
as required.■

Remark IV.6.

An alternative way to construct stochastic dynamics associated with νGα*,p(Sγ) is via the theory of Dirichlet forms. Indeed, ν satisfies an integration-by-parts formula and thus defines a classical Dirichlet form, which is a closed bilinear form in L2(Sγ, ν). The generator of this form is a non-negative self-adjoint operator in L2(Sγ, ν) and thus defines a strongly continuous semigroup in L2(Sγ, ν), which, in turn, defines a Markov process in Sγ with invariant measure ν (so-called Hunt process), see Ref. 6 for details. The SDE approach that we use in our work is, however, more explicit and gives in general better control on properties of the stochastic dynamics.

Remark IV.7.

Observe that pairs (γ,(σx)xγ) form the marked configuration space Γ(X, S). For the mathematical formalism of these spaces and discussion of the existence and uniqueness of Gibbs measures and phase transitions, see Refs. 9, 13, 35, and 36 and references therein. The paper35 considers in particular the case of marks with values in a path space, which gives a complementary way of defining and studying infinite dimensional interacting diffusions indexed by elements of γ ∈ Γ(X).

We are very grateful to Zdzislaw Brzezniak, Dmitry Finkelshtein, Yuri Kondratiev, and Jiang-Lun Wu for their interest in this work and stimulating discussions.

The authors have no conflicts to disclose.

Georgy Chargaziya: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Alexei Daletskii: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1. Linear operators in the spaces of sequences

We start with the formulation of a general result from Ref. 11 on the existence of (infinite-time) solutions for a special class of linear differential equations, which extends the so-called Ovsjannikov method, see e.g., Ref. 18.

Definition A.1
Let B={Bα}αA be a scale of Banach spaces. A liner operator A:αABααABα is called an Ovsjannikov operator of order q > 0 if A(Bα) ⊂ Bβ and there exists a constant L > 0 such that
(A1)
for all α<βA. The space of such operators will be denoted by O(B,q).

Theorem A.2

(Ref. 11, Theorem 3.1 and Remark 3.3). Let AO(B,q) with q < 1. Then, for any α,βA such that α < β and f0Bα, there exists a unique continuous function f: [0, ) → Bβ with f(0) = f0 such that:

  1. f is continuously differentiable on (0, );

  2. Af(t) ∈ Bβ for all t ∈ (0, );

  3. f solves the differential equation

Moreover,
(A2)
where Kt(α,β)n=0Lntn(βα)qnnqnn!<.

Remark A.3

Let us remark that estimate (A2) generalizes the classical estimate etAetA for the exponent of a bounded operator A in a Banach space and does not take into account possible dissipativity properties of A.

Remark A.4
Function Kt(α, β) can be estimated in the following way, see Ref. 11:
(A3)
The r.h.s. of (A3) is an entire function of order δ = (1 − q)−1 and type σ = (Le)δ()−1(βα). Thus, for any ɛ > 0, there exists tɛ > 0 such that

The aim of this section is to give a sufficient condition for the linear operator Q, given by an infinite real matrix {Qx,y}x,yγ, to generate an Ovsjannikov operator in the scale L1 of spaces of sequences defined by (2.8).

Theorem A.5

Assume that {Qx,y}x,yγ is such that for all x, yγ we have

  • Qx,y = 0 if xy>ρ;

  • there exist C > 0 and k ≥ 1 such that
    (A4)
    Then QO(L1,q) for any q < 1.

Proof.
Since Q is linear, it is sufficient to show that
(A5)
for any α<βA and zlα1. By the definition of the norm in lβ1 we have
Now, using estimate (A4) we see that
(A6)
because Qx,y = 0 for yγ̄x and −|x| ≤ −|y| + ρ for yγ̄x. Here
Our next goal is to estimate the constant U. Using condition (A4) we see that for all yγ
Observe that there exist constants M,NN such that
Then, taking into account that |x|q/2 ≤ |y|q/2 + ρq/2 for xBy, we obtain, assuming without loss of generality that |y| > M, that
(A7)
where P=P(γ,M,q)xγ|x|Mnxq<. Hence for all yγ we have
with a1=CN2ρq+P and a2 = CN2. Now we see that
(A8)
Hence, we can deduce that function heβαqh,hR, attains its supremum when ddhheβαqh=0 that is when h=q(βα). Hence it follows from inequality (A8) that
Now, continuing from Eq. (A6) we finally see that (A5) holds with L=eα*ρ(a1(α*α*)q+a2qq), and the proof is complete.■

2. Comparison theorem and Gronwall-type inequality

In this section, we prove generalizations of the classical comparison theorem for differential equations and, as a consequence, a version of the Gronwall inequality, that works in our scale of Banach spaces of sequences.

Let us consider the linear integral equation
(A9)
in lα*1 where QO(L1,q), q < 1, is a linear operator generated by the infinite matrix {Qx,y}x,yγ and z̄=(zx)xγlα1 for some α < α*. It follows from Theorem A.2 that this equation has a unique solution flα+1.

The next result is an extension of the classical comparison theorem to our framework.

Theorem A.6
(Comparison Theorem). Suppose that Qx,y ≥ 0 for all x, yγ and let g:Tlα1 be a bounded map such that
Then for all tT and all xγ we have the inequality
where f=(fx)xγ is the solution of (A9).

Proof.
Let BaB([0,T],lα1),aA, be the Banach space of bounded measurable functions Tlα1. For any gBα define the function
It is clear that I(g)Bα+, which implies that the composition power In:BaBa+ is well-defined. It follows from (the proof of) Ref. 11, Theorem 3.1 that
(A10)
Indeed,
It was proved in Ref. 11, Theorem 3.1, cf. formula (3.5), that the series n=0tnn!Qnz̄ converges uniformly in any lβ1, β > α, and n=0tnn!Qnz̄=f(t). On the other hand, dividing the interval α,β into n intervals of equal length and using estimate (A1) on each of them, as in Ref. 11, Theorem 3.1, we obtain the bound
with DL(βα)q. Taking into account that n!nen we see that an → 0, n, which implies (A10).
We have therefore limnIxn(g)(t)=fx(t) for all xγ and all tT. Hence to conclude the proof it is sufficient to fix xγ and prove by induction that for all tT we have
(A11)
The case n = 1 is satisfied by the initial assumption on g. Let us now assume that (A11) is true for some n ≥ 1 and proceed by considering the following chain of inequalities:
which (since t above is arbitrary) completes the proof.■

Corollary A.7
(Generalized Gronwall inequality). Suppose in addition that zx ≥ 0 for all xγ. Moreover assume that components of the map g are non-negative functions, that is, gx(t) ≥ 0 for all xγ and all tT. Then for all β > α we have the inequality
where KT(α,β)=n=0LnTn(βα)qnnqnn!<.

Proof.
Using Theorem A.6, we see that for all xγ and all tT we have
Since functions g and therefore f are non-negative we see that for all xγ
Hence it follows that
The right-hand side of the inequality above can be estimated using Ref. 11, Theorem 3.1, cf. Theorem A.2. In particular, we get
Hence letting KT(α,β)=n=0LnTn(βα)qnnqnn! we see that the proof is complete.■

Lemma A.8
Consider a bounded measurable map ρ:Tlα1, αA, and assume that its components satisfy the inequality
(A12)
for some constants B > 0 and k ≥ 1 and b(bx)xγlα1, bx ≥ 0. Then we have the estimate
(A13)
for any β > α, with KT(α,β)=n=0LnTn(βα)qnnqnn!<, cf. Theorem A.2.

Proof.
Inequality (A12) can be rewritten in the form
where
for all xγ. We have ρB(T,lα1), and |Qx,y|Bnxk. Therefore using Theorem A.5 we conclude that for any q ∈ (0, 1) matrix Qx,y generates an Ovsjannikov operator of order q on L1. Therefore we can now use Corollary A.7 to conclude that (A13) holds.■

3. Estimates of the solutions

We start with the following auxiliary result.

Lemma A.9
Suppose that σ1,σ2R and Z1, Z2Sγ. Then for all xγ we have the following inequalities:
and
where constants M, c, b and ā are defined in Assumption II.1.

Proof.

The proof can be obtained by a direct calculation using assumptions on Φ and Ψ stated in Sec. II.■

Let us fix αA and consider two processes Ξt(1)=ξx,t(1)xγ and Ξt(2)=ξx,t(2)xγ, Ξ(1),Ξ(2)Rα+p, with initial values Ξ01,Ξ02Lαp.

Lemma A.10
Let p ≥ 2, xγ be fixed and assume that R-valued processes ξx,t(1),ξx,t(2) satisfy Eq. (2.1). Then there exist universal constants B, C1 and C2x such that
(A14)
and
(A15)
for all tT, where ξ̄x,tξx,t(1)ξx,t(2). The constants B, C1 and C2 are independent of the processes Ξ(1), Ξ(2) and xγ. Moreover C̄2{C2x}xγlαp.

Proof.
We remark that in this proof all inequalities hold for all tT and Pa.s., that is on the same same set of measure 1. We now start with the proof of inequality (A14). Using Itô Lemma we see that if x ∈ Λn then for all tT
Now from assumptions (2.4) and (2.5) and Lemma A.9 we can deduce that for all tT
where ãxānx and constants ā, b and c are defined in Assumption II.1. In the last inequality, we used the simple estimate Cp−1 ≤ (1 + C)p−1 ≤ (1 + C)p for any C > 0, which holds because p > 1. Taking into account that maxyγ̄x|ξy,s(1)|pyγ̄x|ξy,s(1)|p and using inequality (1 + α)p ≤ 2p−1(1 + αp) we arrive at the following:
In a similar way, using assumption (2.7) we obtain the estimate
(A16)
Observe that nx ≥ 1. Thus there exist constants C1,C2x>0 such that
which implies that (A14) holds.
The proof of inequality (A15) can be obtained similarly. Using the relation
tT, and applying the Itô Lemma to |ξ̄x,t|p we obtain the inequality
(A17)
for some constant B > 0, which implies the result. Finally, C̄2lαp because (see Assumption II.1) for some constant W we have C2xW(1+log(1+|x|)) and one can use exponential weight to sum up these terms.■

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