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 and internal parameter (spin) . While the positions of particles form a fixed (“quenched”) locally-finite set (configuration) , the spins σx and σy interact via a pair potential whenever , 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.
I. INTRODUCTION
In recent decades, there has been an increasing interest in studying countable systems of particles randomly distributed in the Euclidean space . In such systems, each particle is characterized by its position and an internal parameter (spin) , 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, y ∈ γ, 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).
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., ) has been well-studied (in both deterministic and stochastic cases), see e.g., Refs. 2–5, 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 x ∈ X is proportional to , 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 x ∈ γ. 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.
It was noticed in Ref. 11 that, under a stronger norm bound with q < 1 in (1.3), the lifetime of the solution Xt ∈ Bβ 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 , 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 Bα → Bβ for any α < β, with Lipschitz constants L(β − α)−q, and , respectively. Observe that the threshold value of q here is 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, x ∈ γ, 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.
II. THE SETUP AND MAIN RESULTS
For a fixed γ ∈ Γ(X), we will consider the Cartesian product Sγ of identical copies Sx, x ∈ γ, of S, and denote its elements by , etc. When dealing with multiple configurations η ∈ Γ(X), we will sometimes write , to emphasize the dependence on η.
We will work under the following assumption.
- There exists a constant C > 0 such that(2.2)
- The drift coefficients Φx, x ∈ γ, have the formwhere ϕ: S → S is a measurable function and φxy: S2 → S are also measurable functions satisfying uniform Lipschitz condition(2.3)for some constant and all x, y ∈ γ, σ1, σ2, s1, s2 ∈ S.
- There exist constants c > 0 and R ≥ 2 such that(2.4)
- There exists b > 0 such that(2.5)
- The diffusion coefficients Ψx, x ∈ γ, have the formwhere ψxy: S2 → S are measurable functions satisfying uniform Lipschitz condition(2.6)for some constant M > 0 and all x, y ∈ γ, σ1, σ2, s1, s2 ∈ S.(2.7)
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 of Banach spaces Bα indexed by with fixed 0 ≤ α*, α* < ∞, and denote by 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 remains fixed for the rest of this work. We will also use the corresponding semi-open interval .
The two main scales we will be working with are given by the spaces of weighted sequences and -valued random processes, respectively, defined as follows.
- For all p ≥ 1 and letand be, respectively, a Banach space of weighted real sequences and the scale of such spaces.(2.8)
- For all p ≥ 1 and let denote the Banach space of -valued random processes , on probability space P, with progressively measurable components and finite normand let be the scale of such spaces.
The choice of exponential weights in the definition of space 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 , 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.
Note that for p ≥ 2, the definition of norms in and implies that for any and any x ∈ γ we have . Moreover, since each component of is progressively measurable, from the classical theory of integration with respect to the standard Wiener process we see that for all x ∈ γ the integral is well defined and so is the integral , because Ψx is a finite sum of measurable uniformly Lipschitz functions. Moreover, the process , , has a (unique) continuous version.
Our main result is the following theorem.
Assumption p ≥ R ensures that given the random variable is integrable for any t ≥ 0.
The proof of Theorem II.5 will be given in Sec. III.
The proof of this result will be given in Sec. IV B.
From now on, the constant p ≥ R will be fixed.
III. EXISTENCE, UNIQUENESS AND PROPERTIES OF THE SOLUTION
In this section, we give the proof of Theorem II.5. It will go along the following lines.
Consider a sequence of processes , , that solve finite volume cutoffs of system (2.1), and prove their uniform bound in 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.
The uniform bound above implies the convergence of sequence Ξn, n → ∞, to a process , 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 -valued process ηt that solves an equation describing the dynamics of a tagged particle x, while processes ξy,t, y ∈ γ, y ≠ x, are fixed, and prove that ηt = ξx,t.
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.
A. Truncated system
For any system (3.1) admits a unique (up to indistinguishability) solution with continuous sample paths.
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 follows then from the fact that , for x ∉ Λn.■
Our next goal is to show that the sequence converges in 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.
The sequence is Cauchy in for any β > α.
B. One dimensional special case
In order to establish the existence of a solution of Eq. (3.10) we need the following auxiliary result.
Equation (3.10) admits a unique solution.
C. Proof of existence and uniqueness
D. Markov semigroup
Operator family Tt, t ≥ 0, is a strongly continuous Markov semigroup in for any .
IV. STOCHASTIC DYNAMICS ASSOCIATED WITH GIBBS MEASURES
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.
A. Construction of Gibbs measures
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 and spin space , defined by pair and single-particle potentials Wxy and V, respectively. We assume the following:
- , x, y ∈ X, are measurable functions satisfying the polynomial growth estimateand the finite range condition Wxy ≡ 0 if for all x, y ∈ X 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).(4.1)
- the single-particle potential V satisfies the boundfor some constants aV, bV > 0, and τ > r.(4.2)
By we denote the set of all Gibbs measures on Sγ associated with W and V, which are supported on .
Conditions of the uniqueness of are known only in the case of configuration γ with bounded sequence . Sufficient conditions of non-uniqueness (phase transition) for Poisson-distributed γ are given in Ref. 14.
B. Construction of the stochastic dynamics
In this section, we will construct a process Ξt with invariant measure defined by interaction potentials W and V as in Example IV.1. By Theorem IV.2, the set is not empty if . Then, according to the general paradigm, Ξt will be a solution of the system (2.1) with the coefficients satisfying the following:
for each x ∈ γ, the noise is additive, that is, Ψx = id.
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).
First observe that condition ensures that and semigroup Tt is well-defined.
An alternative way to construct stochastic dynamics associated with 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.
Observe that pairs 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).
ACKNOWLEDGMENTS
We are very grateful to Zdzislaw Brzezniak, Dmitry Finkelshtein, Yuri Kondratiev, and Jiang-Lun Wu for their interest in this work and stimulating discussions.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
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 AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: TECHNICAL DETAILS
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.
(Ref. 11, Theorem 3.1 and Remark 3.3). Let with q < 1. Then, for any such that α < β and f0 ∈ Bα, there exists a unique continuous function f: [0, ∞) → Bβ with f(0) = f0 such that:
f is continuously differentiable on (0, ∞);
Af(t) ∈ Bβ for all t ∈ (0, ∞);
- f solves the differential equation
Let us remark that estimate (A2) generalizes the classical estimate for the exponent of a bounded operator A in a Banach space and does not take into account possible dissipativity properties of A.
The aim of this section is to give a sufficient condition for the linear operator Q, given by an infinite real matrix , to generate an Ovsjannikov operator in the scale of spaces of sequences defined by (2.8).
Assume that is such that for all x, y ∈ γ we have
Qx,y = 0 if ;
- there exist C > 0 and k ≥ 1 such thatThen for any q < 1.(A4)
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.
The next result is an extension of the classical comparison theorem to our framework.
3. Estimates of the solutions
We start with the following auxiliary result.
The proof can be obtained by a direct calculation using assumptions on Φ and Ψ stated in Sec. II.■
Let us fix and consider two processes and , , with initial values .