SMOLUCHOWSKI’S COAGULATION EQUATION: UNIQUENESS, NON-UNIQUENESS AND A HYDRODYNAMIC LIMIT FOR THE

STOCHASTIC COALESCENT

JAMES R. NORRIS

Abstract. Sufficient conditions are given for existence and uniqueness in Smoluchowski’s coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski’s equation.

1. Introduction

Coagulation of particles, in pairs and over time, within a large system of particles, is a phenomenon widely observed and widely postulated in scientific models. Examples arise in the study of aerosols, of phase separation in liquid mixtures, in polymerization and astronomy. Typically, it is argued that the rate at which pairs of particles coagulate depends, for physical reasons, in a given way, on some positive parameter associated to each particle, such as size or mass. In the models we shall consider, it is further argued that the effects of spatial fluctuations in the size or mass density are negligible—for example, by supposing that the particles perform independent random motions on a time scale faster than the process of coagulation.

The first mathematical model of this sort of process was proposed by Smoluchowski [vS16] in 1916, see also Chandrasekhar [Cha43]. Smoluchowski argued that particles of radius r would perform independent Brownian motions of variance proportional to 1/r, so pairs of particles of radii r_{1} and r_{2} would meet at a rate proportional to

(r_{1} + r_{2})(1/r_{1} + 1/r_{2}).

Expressed in terms of masses, this leads to the coagulation kernel

K(x, y) = (x^{1/3} + y^{1/3})(x^{−1/3} + y^{−1/3})

for particles of masses x and y. Then, making some implicit assumptions about ergodic averages, Smoluchowski wrote down the following infinite system of differential equations for the evolution of densities μ(x) of particles of mass x = 1, 2, 3, …

This research was supported financially by the European Union under contract FMRX CT96 0075A and by the Mathematical Sciences Research Institute. Research at MSRI is supported in part by NSF grant DMS-9701755.

\frac{d}{dt}μ_{t}(x) = \frac{1}{2} ∑_{y=1}^{x−1} K(y, x − y)μ_{t}(y)μ_{t}(x − y) − μ_{t}(x) ∑_{y=1}^{∞} K(x, y)μ_{t}(y).

Here, the first sum on the right corresponds to coagulation of smaller particles to produce one of mass x, whereas the second sum corresponds to removal of particles of mass x as they in turn coagulate to produce larger particles.

More generally, in other models, such systems of equations are considered for many different coagulation kernels K, see for example [Ald]. Also, analogous integrodifferential equations are considered which allow for a continuum of masses x. It is known by now that, for a suitable initial mass distribution μ_{0}, Smoluchowski’s original equations have a unique solution. Much progress has been made in determining when existence and uniqueness holds for more general coagulation kernels, see [McL62], [McL64], [Whi80], [BC90], [Hei92] for discrete mass distributions. Nevertheless many fundamental questions remain open, even for certain coagulation kernels studied extensively in applied sciences.

In this paper we give some new positive results on the existence and uniqueness problem for Smoluchowski-type equations. In particular:

• we prove existence of solutions for continuous coagulation kernels K such that

K(x, y)/xy → 0 as (x, y) → ∞

extending a result of Jeon [Jeo] for the discrete case;

• we prove local existence and uniqueness of solutions when K(x, y) ≤ φ(x)φ(y) for some continuous sublinear function φ : E → (0, ∞), provided that the initial mass distribution μ_{0} satisfies

∫_{(0,∞)} φ(x)^{2}μ_{0} (dx) < ∞;

• this allows us to treat the case where K(x, y) blows up as x → 0 or y → 0, also to prove uniqueness in some cases when the mass distribution has no second, or even first, moment;

• we can do without any local regularity conditions on K;

• we do not have to assume that the initial mass distribution is discrete, nor that it has a density with respect to Lebesgue measure.

We also construct in §3 an example of a coagulation kernel K and an initial mass distribution μ_{0}, such that Smoluchowski’s equation has at least two distinct solutions, both of which are conservative, in the sense that

∫_{(0,∞)} xμ_{t} (dx) = ∫_{(0,∞)} xμ_{0} (dx) < ∞

for all t.

Then in §4 we consider a stochastic system of coagulating particles, where particles of masses x and y coagulate at a rate proportional to K(x, y). We show that, when we can establish uniqueness in Smoluchowski’s equation, the particle system, suitably normalized, converges weakly to the solution of the deterministic equation. Thus we obtain a statistical derivation of Smoluchowski’s equation. This goes some way towards resolving Problem 10 in Aldous’ survey article [Ald].

2. Existence and uniqueness in Smoluchowski’s coagulation equation

Let E = (0, ∞) and let K : E × E → [0, ∞) be a symmetric measurable function, the coagulation kernel. Denote by ℳ = ℳ_{E} the space of signed Radon measures on E, that is to say, those signed measures having finite total variation on each compact subset of E. Denote by ℳ^{+} the set of (non-negative) measures in ℳ. If μ ∈ ℳ^{+} satisfies, for all compact sets B ⊆ E

∫_{B×E} K(x, y)μ (dx)μ(dy) < ∞,

then we define L(μ) ∈ ℳ by

⟨f, L(μ)⟩ = \frac{1}{2} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}K(x, y)μ(dx)μ(dy)

for all bounded measurable functions f of compact support.

We consider the following weak form of Smoluchowski’s coagulation equation

μ_{t} = μ_{0} + ∫_{0}^{t} L(μ_{s}) ds. (2.1)

We admit as a local solution any map

t ↦ μ_{t} : [0, T) ↦ ℳ^{+}

where T ∈ (0, ∞], such that:

(i) we have

∫_{E} x1_{x≤1} μ_{0}(dx) < ∞;

(ii) for all compact sets B ⊆ E, the following map is measurable

t ↦ μ_{t}(B) : [0, T) → [0, ∞);

(iii) we have, for all t < T and all compact sets B ⊆ E

∫_{0}^{t} ∫_{B×E} K(x, y)μ_{s}(dx)μ_{s}(dy)ds < ∞;

(iv) for all bounded measurable functions f of compact support and also for f(x) = x1_{x≤1}, for all t < T

⟨f, μ_{t}⟩ = ⟨f, μ_{0}⟩ + ∫_{0}^{t}⟨f, L(μ_{s})⟩ds. (2.2)

In the case T = ∞, we have a solution.

The condition that, for f(x) = x1_{x≤1}, we have ⟨f, μ_{0}⟩ < ∞ and that (2.2) holds is a boundary condition, expressing that no mass enters at 0. We obtain an equivalent condition on replacing f by any non-vanishing sublinear function E → [0, ∞) of bounded support, which is linear near 0. A function f : E → [0, ∞) is sublinear if

f(λx) ≤ λf(x) for all x ∈ E, λ ≥ 1.

Note that such a function f is always subadditive:

f(x + y) ≤ f(x) + f(y) for all x, y ∈ E.

Hence ⟨f, L(μ)⟩ ≤ 0 for all μ ∈ ℳ^{+}. Note also that, if φ : E → [0, ∞) is any sublinear function and if

φ_{n}(x) = \{ \begin{matrix}nxφ(n^{−1}), & 0 < x ≤ n^{−1}, \\ φ(x), & n^{−1} < x ≤ n, \\ 0, & x > n, \end{matrix}

then φ_{n}(x) ↑ φ(x) for all x, and φ_{n} is sublinear of bounded support, linear near 0. So, for t < T

⟨φ_{n}, μ_{t}⟩ − ⟨φ_{n}, μ_{0}⟩ = ∫_{0}^{t}⟨φ_{n}, L(μ_{s})⟩ ≤ 0.

Hence, using monotone convergence on the left and Fatou’s lemma on the right

⟨φ, μ_{0}⟩ ≥ ⟨φ, μ_{t}⟩ − ∫_{0}^{t}⟨φ, L(μ_{s})⟩ ds. (2.3)

In particular, ⟨φ, μ_{t}⟩ is non-increasing in t. In particular, the total mass density

∫_{E} xμ_{t} (dx)

is non-increasing in t; if it is finite and constant, we say that (μ_{t})_{t<T} is conservative. Throughout this section we make the basic assumption that

K(x, y) ≤ φ(x)φ(y) for all x, y ∈ E (2.4)

where φ : E → (0, ∞) is a continuous sublinear function. We also assume that the initial measure μ_{0} satisfies

⟨φ, μ_{0}⟩ < ∞. (2.5)

We call any local solution (μ_{t})_{t<T} such that

∫_{0}^{t}⟨φ^{2}, μ_{s}⟩ ds < ∞ for all t < T

a strong local solution.

Here is a summary of what is known so far about existence, uniqueness and conservation of mass in Smoluchowski’s equation. The picture is more complete for discrete mass distributions—that is when μ_{0} is supported on ℕ. Then, provided μ_{0} has a finite second moment, Ball and Carr [BC90] proved existence and mass conservation when K(x, y) ≤ x + y, Heilman [Hei92] added uniqueness under the same hypotheses. Jeon [Jeo] has recently proved global existence when K(x, y)/xy → 0 as (x, y) → ∞. McLeod [McL62] long ago proved local existence when K(x, y) ≤ xy. For general mass distributions μ_{0}, less is known. Dubovskii and Stewart [DS96] have shown existence and uniqueness provided μ_{0} has an exponential moment and a continuous density with respect to Lebesgue measure, and provided K is continuous with K(x, y) ≤ 1 + x + y. Recently, Clark and Katsouros [CK] proved existence and uniqueness for a particular choice of kernel which blows up when x or y is small.

Theorem 2.1. Assume conditions (2.4) and (2.5). If (μ_{t})_{t<T} and (ν_{t})_{t<T} are local solutions, starting from μ_{0}, and if (μ_{t})_{t<T} is strong, then μ_{t} = ν_{t} for all t < T. If φ(x) ≥ εx for all x, for some ε > 0, then any strong solution is conservative. Moreover, if ⟨φ^{2}, μ_{0}⟩ < ∞, then

(i) there exists a unique maximal strong solution (μ_{t})_{t<ζ(μ_{0})}, with ζ(μ_{0}) ≥ ⟨φ^{2}, μ_{0}⟩^{−1}, (ii) if φ^{2} is sublinear or if K(x, y) ≤ φ(x) + φ(y) for all x, y ∈ E, then ζ(μ_{0}) = ∞.

The proof will occupy the remainder of this section. The method is to find an approximation to Smoluchowski’s equation by a system depending on K and φ only through their values on a given compact set. The idea of the approximation may be readily understood in terms of the stochastic coalescent, for which a directly analogous approximation is discussed in §4. Moreover, the finite particle interpretation explains certain crucial non-negativity statements, which are given less transparent analytical proofs below.

We remark that this theorem provides examples where uniqueness holds, even when the solution fails to be conservative, in the trivial sense that the total mass density is infinite. We have not yet found an example of a strong solution which has finite initial mass density and which fails to conserve mass. The example of §3 shows, on the other hand, that uniqueness can fail while solutions remain conservative.

Let B ⊆ E be compact. We will eventually pass to the limit B ↑ E. Denote by ℳ_{B} the space of finite signed measures supported on B. Note that φ is bounded on B. We define L^{B} : ℳ_{B} × ℝ → ℳ_{B} × ℝ by the requirement

⟨(f, a), L^{B}(μ, λ)⟩ =

\frac{1}{2} ∫_{E×E}\{f(x + y)1_{x+y∈B} + aφ(x + y)1_{x+y∉B} − f(x) − f(y)\}K(x, y)μ(dx)μ(dy)

+ λ ∫_{E}{aφ(x) − f(x)}φ(x)μ(dx) for all bounded measurable functions f on E and all a ∈ ℝ. Here we used the notation ⟨(f, a), (μ, λ)⟩ = ⟨f, μ⟩ + aλ.

Consider the equation

(μ_{t}, λ_{t}) = (μ_{0}, λ_{0}) + ∫_{0}^{t} L^{B}(μ_{s}, λ_{s}) ds. (2.6)

We admit as a local solution any continuous map

t ↦ (μ_{t}, λ_{t}) : [0, T] → ℳ_{B} × ℝ

where T ∈ (0, ∞), which satisfies the equation for all t ∈ [0, T]. When [0, T] is replaced by [0, ∞) we get the notion of solution.

Proposition 2.2. Suppose μ_{0} ∈ ℳ_{B} with μ_{0} ≥ 0 and that λ_{0} ∈ [0, ∞). The equation (2.6) has a unique solution (μ_{t}, λ_{t})_{t≥0} starting from (μ_{0}, λ_{0}). Moreover μ_{t} ≥ 0 and λ_{t} ≥ 0 for all t.

Proof. Our basic assumption (2.4) remains valid when φ is replaced by φ + 1, so we may assume without loss that φ ≥ 1. By a scaling argument we may assume, also without loss, that

⟨φ, μ_{0}⟩ + λ_{0} ≤ 1

which implies that

∥μ_{0}∥ + |λ_{0}| ≤ 1.

We shall show, by a standard iterative scheme, that there is a constant T > 0, depending only on φ and B, and a unique local solution (μ_{t}, λ_{t})_{t≤T} starting from (μ_{0}, λ_{0}). Then we shall show, moreover, that μ_{t} ≥ 0 for all t ∈ [0, T].

First of all, let us see that this is enough to prove the proposition. If we put f = 0 and a = 1 in (2.6), we obtain

\frac{d}{dt}λ_{t} = \frac{1}{2} ∫_{E×E} φ(x + y)1_{x+y∉B}K(x, y)μ_{t}(dx)μ_{t}(dy) + λ_{t} ∫_{E} φ(x)^{2}μ_{t}(dx).

So, since μ_{t} ≥ 0, we deduce λ_{t} ≥ 0 for all t. Next, we put f = φ and a = 1 to see that

\frac{d}{dt}(⟨φ, μ_{t}⟩ + λ_{t}) = \frac{1}{2} ∫_{E×E}\{φ(x + y) − φ(x) − φ(y)\}K(x, y)μ_{t}(dx)μ_{t}(dy) ≤ 0.

Hence

∥μ_{T} ∥ + |λ_{T} | ≤ ⟨φ, μ_{T} ⟩ + λ_{T} ≤ ⟨φ, μ_{0}⟩ + λ_{0} ≤ 1.

We can now start again from (μ_{T} , λ_{T}) at time T to extend the solution to [0, 2T], and so on, to prove the proposition.

We use the following norm on ℳ_{B} × ℝ:

∥(μ, λ)∥ = ∥μ∥ + |λ|.

We note the following estimates: there is a constant C < ∞, depending only on φ and B such that, for all μ, μ′ ∈ ℳ_{B} and all λ, λ′ ∈ ℝ,

∥L^{B}(μ, λ)∥ ≤ C∥(μ, λ)∥^{2}, (2.7)

∥L^{B}(μ, λ) − L^{B}(μ′, λ′)∥ ≤ C∥(μ, λ) − (μ′, λ′)∥ (∥(μ, λ)∥ + ∥(μ′, λ′)∥). (2.8)

We turn to the iterative scheme. Set (μ_{t}^{0}, λ_{t}^{0}) = (μ_{0}, λ_{0}) for all t and define inductively a sequence of continuous maps

t ↦ (μ_{t}^{n}, λ_{t}^{n}) : [0, ∞) → ℳ_{B} × ℝ

by

(μ_{t}^{n+1}, λ_{t}^{n+1}) = (μ_{0}, λ_{0}) + ∫_{0}^{t} L^{B}(μ_{s}^{n}, λ_{s}^{n}) ds.

Set

f_{n}(t) = ∥(μ_{t}^{n}, λ_{t}^{n})∥

then f_{0}(t) = f_{n}(0) = ∥(μ_{0}, λ_{0})∥ ≤ 1 and by the estimate (2.7)

f_{n+1}(t) ≤ 1 + C ∫_{0}^{t} f_{n}(s)^{2} ds.

Hence

f_{n}(t) ≤ (1 − Ct)^{−1}, t ≤ C^{−1}

for all n, so, setting T = (2C)^{−1}, we have

∥(μ_{t}^{n}, λ_{t}^{n})∥ ≤ 2, t ≤ T. (2.9)

Next, set g_{0}(t) = f_{0}(t) and for n ≥ 1

g_{n}(t) = ∥(μ_{t}^{n}, λ_{t}^{n}) − (μ_{t}^{n−1}, λ_{t}^{n−1})∥.

By the estimates (2.8) and (2.9), there is a constant C < ∞, depending only on φ and B, such that

g_{n+1}(t) ≤ C ∫_{0}^{t} g_{n}(s) ds, t ≤ T.

Hence, by the usual arguments, (μ_{t}^{n}, λ_{t}^{n}) converges in ℳ_{B} × ℝ, uniformly in t ≤ T, to the desired local solution, which is also unique. Moreover, for some constant C < ∞, depending only on φ and B, we have

∥(μ_{t}, λ_{t})∥ ≤ C, t ≤ T.

It remains to show that μ_{t} ≥ 0 for all t. For this we need the following result.

Proposition 2.3. Let

(t, x) ↦ f_{t}(x) : [0, T] × B → ℝ

be a bounded measurable function, having a bounded partial derivative ∂f/∂t. Then for all t ≤ T

\frac{d}{dt} ⟨f_{t}, μ_{t}⟩ = ⟨∂f/∂t, μ_{t}⟩ + ⟨(f_{t}, 0), L^{B}(μ_{t}, λ_{t})⟩.

Proof. Fix t ≤ T and set ⌊s⌋_{n} = (n/t)^{−1}⌊ns/t⌋ and ⌈s⌉_{n} = (n/t)^{−1}⌈ns/t⌉. Then

⟨f_{t}, μ_{t}⟩ = ⟨f_{0}, μ_{0}⟩ + ∫_{0}^{t}⟨∂f/∂s, μ_{⌊s⌋_{n}}⟩ ds + ∫_{0}^{t}⟨(f_{⌈s⌉_{n}}, 0), L^{B}(μ_{s}, λ_{s})⟩ ds

and the proposition follows on letting n → ∞.

For t ≤ T, set

θ_{t}(x) = exp ∫_{0}^{t} ( ∫_{E} K(x, y)μ_{s}(dy) + λ_{s}φ(x)) ds

and define G_{t} : ℳ_{B} → ℳ_{B} by

⟨f, G_{t}(μ)⟩ = \frac{1}{2} ∫_{E×E}(fθ_{t})(x + y)1_{x+y∈B}K(x, y)θ_{t}(x)^{−1}θ_{t}(y)^{−1}μ(dx)μ(dy).

We note that G_{t}(μ) ≥ 0 whenever μ ≥ 0 and, for some C < ∞, depending only on φ and B, we have

∥G_{t}(μ)∥ ≤ C∥μ∥^{2}, ∥G_{t}(μ) − G_{t}(μ′)∥ ≤ C∥μ − μ′∥ (∥μ∥ + ∥μ′∥).

Set μ̃_{t} = θ_{t}μ_{t}. By Proposition 2.3, for all bounded measurable functions f, we have

\frac{d}{dt}⟨f, μ̃_{t}⟩ = ⟨f∂θ/∂t, μ_{t}⟩ + ⟨(fθ_{t}, 0), L^{B}(μ_{t}, λ_{t})⟩ = ⟨f, G_{t}(μ̃_{t})⟩.

Define inductively a new sequence of measures μ̃_{t}^{n} by setting μ̃_{t}^{0} = μ_{0} and, for n ≥ 0

μ̃_{t}^{n+1} = μ_{0} + ∫_{0}^{t} G_{s}(μ̃_{s}^{n}) ds.

By an argument similar to that used for the original iterative scheme, we can show, first, and possibly for a smaller value of T > 0, but with the same dependence, that ∥μ̃_{t}^{n}∥ is bounded, uniformly in n, for t ≤ T, and then that ∥μ̃_{t}^{n} − μ̃_{t}∥ → 0 as n → ∞. Since μ̃_{t}^{n} ≥ 0 for all n, we deduce μ̃_{t} ≥ 0 and hence μ_{t} ≥ 0 for all t ≤ T. This completes the proof of Proposition 2.2

We remark that the arguments used to prove Proposition 2.2 apply with no essential change to the case where the coagulation kernel is time-dependent provided that (2.4) holds uniformly in time. We remark also that, in the iterative scheme

μ_{t}^{0} = μ_{0}, μ_{t}^{n+1} ≪ μ_{0} + ∫_{0}^{t}(μ_{s}^{n} + μ_{s}^{n} ∗ μ_{s}^{n}) ds

for all n ≥ 0, so by induction we have

μ_{t}^{n} ≪ γ_{0} = ∑_{k=1}^{∞} μ_{0}^{∗k}

where μ_{0}^{∗n} denotes the n-fold convolution of μ_{0}. On letting n → ∞, we see that, if (μ_{t}, λ_{t})_{t≥0} is the unique solution to (2.6), then μ_{t} ≪ γ_{0}. These remarks will be used in the proof of Proposition 4.2.

We now fix μ_{0} ∈ ℳ^{+} with μ_{0} ≥ 0 and ⟨φ, μ_{0}⟩ < ∞. For each compact set B ⊆ E, let

μ_{0}^{B} = 1_{B}μ_{0}, λ_{0}^{B} = ∫_{E\B} φ(x)μ_{0} (dx)

and denote by (μ_{t}^{B}, λ_{t}^{B})_{t≥0} the unique solution to (2.6), starting from (μ_{0}^{B}, λ_{0}^{B}), provided by Proposition 2.2. We shall show in Proposition 2.4 that for B ⊆ B′ we have

μ_{t}^{B} ≤ μ_{t}^{B′}, ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} ≥ ⟨φ, μ_{t}^{B′}⟩ + λ_{t}^{B′}.

We shall also show in Proposition 2.5 that, for any local solution (ν_{t})_{t<T} of the coagulation equation (2.1), for all t < T

μ_{t}^{B} ≤ ν_{t}, ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} ≥ ⟨φ, ν_{t}⟩.

We now show how these facts lead to the proof of Theorem 2.1.

Set μ_{t} = lim_{B↑E} μ_{t}^{B} and λ_{t} = lim_{B↑E} λ_{t}^{B}. Note that

⟨φ, μ_{t}⟩ = lim_{B↑E}⟨φ, μ_{t}^{B}⟩ ≤ ⟨φ, μ_{0}⟩ < ∞.

So, by dominated convergence, using (2.4), for all bounded measurable functions f,

∫_{E×E} f(x + y)1_{x+y∉B}K(x, y)μ_{t}^{B} (dx)μ_{t}^{B}(dy) → 0

and we can pass to the limit in (2.6) to obtain

\frac{d}{dt}⟨f, μ_{t}⟩ = \frac{1}{2} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}K(x, y)μ_{t} (dx)μ_{t}(dy) − λ_{t}⟨fφ, μ_{t}⟩.

For any local solution (ν_{t})_{t<T}, for all t < T,

μ_{t} ≤ ν_{t}, ⟨φ, μ_{t}⟩ + λ_{t} ≥ ⟨φ, ν_{t}⟩.

Hence, if λ_{t} = 0 for all t < T, then (μ_{t})_{t<T} is a local solution and moreover is the only local solution on [0, T). If (ν_{t})_{t<T} is a strong local solution, then

∫_{0}^{t}⟨φ^{2}, μ_{s}⟩ ds ≤ ∫_{0}^{t}⟨φ^{2}, ν_{s}⟩ ds < ∞

for all t < T; this allows us to pass to the limit in (2.6) to obtain

\frac{d}{dt}λ_{t} = λ_{t}⟨φ^{2}, μ_{t}⟩ (2.10)

and to deduce from this equation that λ_{t} = 0 for all t < T; it follows that (ν_{t})_{t<T} is the only local solution on [0, T). Note that, for any local solution (ν_{t})_{t<T}

∫_{E} x1_{x≤n}ν_{t} (dx) = ∫_{E} x1_{x≤n}ν_{0} (dx)

+ \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{(x + y)1_{x+y≤n} − x1_{x≤n} − y1_{y≤n}\}K(x, y)ν_{s}(dx)ν_{s}(dy) ds.

Hence, if (ν_{t})_{t<T} is strong and φ(x) ≥ εx for all x, for some ε > 0, then by dominated convergence the second term on the right tends to 0 as n → ∞, showing that (ν_{t})_{t<T} is conservative.

Suppose now that ⟨φ^{2}, μ_{0}⟩ < ∞ and set T = ⟨φ^{2}, μ_{0}⟩^{−1}. For any compact set B ⊆ E we have

\frac{d}{dt}⟨φ^{2}, μ_{t}^{B}⟩ ≤ \frac{1}{2} ∫_{E×E}\{φ(x + y)^{2} − φ(x)^{2} − φ(y)^{2}\}K(x, y)μ_{t}^{B}(dx)μ_{t}^{B}(dy) ≤ ⟨φ^{2}, μ_{t}^{B}⟩^{2}

so, for t < T,

⟨φ^{2}, μ_{t}⟩ ≤ lim_{B↑E}⟨φ^{2}, μ_{t}^{B}⟩ ≤ (T − t)^{−1}.

Hence (2.10) holds and forces λ_{t} = 0 for t < T as above, so (μ_{t})_{t<T} is a strong local solution.

If φ^{2} is sublinear, then

⟨φ^{2}, μ_{t}⟩ ≤ ⟨φ^{2}, μ_{0}⟩ < ∞.

If, on the other hand, K(x, y) ≤ φ(x) + φ(y), then

\begin{aligned}\frac{d}{dt}⟨φ^{2}, μ_{t}^{B}⟩ & ≤ ∫_{E×E} φ(x)φ(y)(φ(x) + φ(y))μ_{t}^{B}(dx)μ_{t}^{B}(dy) \\ & ≤ 2⟨φ, μ_{t}^{B}⟩⟨φ^{2}, μ_{t}^{B}⟩ ≤ 2⟨φ, μ_{0}⟩⟨φ^{2}, μ_{t}^{B}⟩, \end{aligned}

so

⟨φ^{2}, μ_{t}⟩ ≤ exp{2⟨φ, μ_{0}⟩t}.

In either case we can deduce that (μ_{t})_{t≥0} is a strong solution.

Proposition 2.4. Suppose B ⊆ B′. Then, for all t ≥ 0,

μ_{t}^{B} ≤ μ_{t}^{B′}, ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} ≥ ⟨φ, μ_{t}^{B′}⟩ + λ_{t}^{B′}.

Proof. Set

\begin{aligned}θ_{t}(x) & = exp ∫_{0}^{t} (∫_{E} K(x, y)μ_{s}^{B}(dy) + λ_{s}^{B}φ(x)) ds, \\ π_{t} & = θ_{t}(μ_{t}^{B′} − μ_{t}^{B}), \\ χ_{t} & = ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} − ⟨φ, μ_{t}^{B′}⟩ − λ_{t}^{B′}. \end{aligned}

Note that π_{0} ≥ 0 and χ_{0} = 0. By Proposition 2.3, for any bounded measurable function f

\frac{d}{dt}⟨f, π_{t}⟩ = ⟨f ∂θ/∂t, μ_{t}^{B′} − μ_{t}^{B}⟩ + ⟨(fθ_{t}, 0), L^{B′}(μ_{t}^{B′}, λ_{t}^{B′}) + L^{B}(μ_{t}^{B}, λ_{t}^{B})⟩

= \frac{1}{2} ∫_{E×E} fθ_{t}(x + y)K(x, y)

× (1_{x+y∈B′}μ_{t}^{B′}(dx)μ_{t}^{B′}(dy) − 1_{x+y∈B}μ_{t}^{B}(dx)μ_{t}^{B}(dy))

+ ∫_{E×E} fθ_{t}(x)(φ(x)φ(y) − K(x, y))μ_{t}^{B′}(dx)(μ_{t}^{B′}(dy) − μ_{t}^{B}(dy))

+ χ_{t} ∫_{E} fθ_{t}(x)φ(x)μ_{t}^{B′}(dx).

Also

\frac{d}{dt} χ_{t} = \frac{1}{2} ∫_{E×E}\{φ(x) + φ(y) − φ(x + y)\}K(x, y)(μ_{t}^{B′}(dx)μ_{t}^{B′}(dy) − μ_{t}^{B}(dx)μ_{t}^{B}(dy)).

So (π_{t}, χ_{t}) satisfies an equation of the form

\frac{d}{dt}(π_{t}, χ_{t}) = H_{t}(π_{t}, χ_{t}) + (α_{t}, 0)

where H_{t} : ℳ_{B′} × ℝ → ℳ_{B′} × ℝ is linear, H_{t}(π, χ) ≥ 0 whenever (π, χ) ≥ 0, where α_{t} ∈ ℳ_{B′} with α_{t} ≥ 0, and where we have estimates, for t ≤ 1,

∥H_{t}(π, χ)∥ ≤ C∥(π, χ)∥ and ∥α_{t}∥ ≤ C

for some constant C < ∞ depending only on φ and B′. Therefore we can apply the same sort of argument that we used for non-negativity to see that π_{t} ≥ 0 and χ_{t} ≥ 0 for all t ≤ 1, and then for all t < ∞, as required. Explicitly, H_{t} is given by

⟨(f, a), H_{t}(π, χ)⟩ = \frac{1}{2} ∫_{E×E} fθ_{t}(x + y)1_{x+y∈B}K(x, y)
× (θ_{t}(x)^{−1}π(dx)μ_{t}^{B′}(dy) + θ_{t}(y)^{−1}μ_{t}^{B}(dx)π(dy))

+ ∫_{E×E} fθ_{t}(x)(φ(x)φ(y) − K(x, y))μ_{t}^{B′}(dx)θ_{t}(y)^{−1}π(dy)

+ χ ∫_{E} fθ_{t}(x)φ(x)μ_{t}^{B′}(dx)

+ \frac{1}{2}a ∫_{E×E}\{φ(x) + φ(y) − φ(x + y)\}K(x, y)

(θ_{t}(x)^{−1}π(dx)μ_{t}^{B′}(dy) + θ_{t}(y)^{−1}μ_{t}^{B}(dx)π(dy))

and α_{t} is given by

⟨f, α_{t}⟩ = \frac{1}{2} ∫_{E×E} fθ_{t}(x + y)1_{x+y∈B′\B}K(x, y)μ_{t}^{B′}(dx)μ_{t}^{B′}(dy).

Proposition 2.5. Suppose that (ν_{t})_{t<T} is a local solution of the coagulation equation (2.1), starting from μ_{0}. Then, for all compact sets B ⊆ E and all t < T,

μ_{t}^{B} ≤ ν_{t}, ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} ≥ ⟨φ, ν_{t}⟩.

Proof. Set

\begin{aligned}θ_{t}(x) & = exp ∫_{0}^{t} (∫_{E} K(x, y)μ_{s}^{B}(dy) + λ_{s}^{B}φ(x)) ds, \\ ν_{t}^{B} & = 1_{B}ν_{t}, \\ π_{t} & = θ_{t}(ν_{t}^{B} − μ_{t}^{B}), \\ χ_{t} & = ⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B} − ⟨φ, ν_{t}⟩. \end{aligned}

We have to show that π_{t} ≥ 0 and χ_{t} ≥ 0. By an obvious modification of Proposition 2.3 we have, for all bounded measurable functions f,

\frac{d}{dt}⟨f, π_{t}⟩ = ⟨f ∂θ/∂t, ν_{t}^{B} − μ_{t}^{B}⟩ + ⟨fθ_{t}1_{B}, L(ν_{t})⟩ − ⟨(fθ_{t}, 0), L^{B}(μ_{t}^{B}, λ_{t}^{B})⟩.

By (2.3), we have

⟨φ, ν_{t}⟩ ≤ ⟨φ, ν_{0}⟩ + \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{φ(x + y) − φ(x) − φ(y)\}K(x, y)ν_{s}(dx)ν_{s}(dy) ds.

On the other hand

\frac{d}{dt}(⟨φ, μ_{t}^{B}⟩ + λ_{t}^{B}) = \frac{1}{2} ∫_{E×E}\{φ(x + y) − φ(x) − φ(y)\}K(x, y)μ_{t}^{B}(dx)μ_{t}^{B}(dy).

So χ_{t} ≥ ρ_{t} where

ρ_{t} = \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{φ(x) + φ(y) − φ(x + y)\}K(x, y)(ν_{s}(dx)ν_{s}(dy) − μ_{s}^{B}(dx)μ_{s}^{B}(dy)) ds.

Now (π_{t}, ρ_{t}) ∈ ℳ_{B} × ℝ obeys a differential equation of the form

\frac{d}{dt}(π_{t}, ρ_{t}) = H_{t}(π_{t}, ρ_{t}) + (α_{t}, β_{t})

where H_{t} : ℳ_{B} × ℝ → ℳ_{B} × ℝ is linear, H_{t}(π, ρ) ≥ 0 whenever (π, ρ) ≥ 0, where α_{t} ≥ 0, β_{t} ≥ 0 and we have estimates of the form

∥H_{t}(π, ρ)∥ ≤ C∥(π, ρ)∥, ∥α_{t}∥ ≤ C, |β_{t}| ≤ C.

It follows that π_{t} ≥ 0 and ρ_{t} ≥ 0, so also χ_{t} ≥ 0 as required. Explicitly, H_{t} is given by

⟨(f, a), H_{t}(π, ρ)⟩ = \frac{1}{2} ∫_{E×E} fθ_{t}(x + y)1_{x+y∈B}K(x, y)
× (θ_{t}(x)^{−1}π(dx)ν_{t}^{B}(dy) + θ_{t}(y)^{−1}μ_{t}^{B}(dx)π(dy))

+ ∫_{E×E} fθ_{t}(x)(φ(x)φ(y) − K(x, y))θ_{t}(y)^{−1}ν_{t}^{B}(dx)π(dy)

+ ρ ∫_{E} fθ_{t}(x)φ(x)ν_{t}^{B}(dx)

+ \frac{1}{2}a ∫_{E×E}\{φ(x) + φ(y) − φ(x + y)\}K(x, y)

× (θ_{t}(x)^{−1}π(dx)ν_{t}^{B}(dy) + θ_{t}(y)^{−1}μ_{t}^{B}(dx)π(dy))

and α_{t}, β_{t} are given by

⟨f, α_{t}⟩ = ∫_{E×E} fθ_{t}(x)(φ(x)φ(y) − K(x, y))1_{y∉B}ν_{t}^{B}(dx)ν_{t}(dy)

+ (χ_{t} − ρ_{t}) ∫_{E} fθ_{t}(x)φ(x)ν_{t}^{B}(dx)

+ \frac{1}{2} ∫_{E×E} fθ_{t}(x + y)1_{x+y∈B}K(x, y)1_{(x,y)∉B×B}ν_{t}(dx)ν_{t}(dy),

β_{t} = \frac{1}{2} ∫_{E×E}\{φ(x) + φ(y) − φ(x + y)\}K(x, y)1_{(x,y)∉B×B}ν_{t}(dx)ν_{t}(dy).

This concludes the proof of Theorem 2.1.

3. An example of non-uniqueness

We construct in this section an example of Smoluchowski’s coagulation equation having at least two solutions, both of which are moreover conservative.

Consider the system of differential equations

\frac{d}{dt}m_{n}(t) = −λ_{n}m_{n}(t)m_{n+1}(t), n = 1, 2, … (3.1)

For any solution we have

m_{n}(t) = m_{n}(0) exp{−λ_{n} ∫_{0}^{t} m_{n+1}(s) ds}.

Assume that m_{n}(0) ≥ 0 for all n, then m_{n}(t) ≥ 0 for all n and t. Note that, if m_{N} is given, and we consider the system restricted to n ≤ N − 1, then m_{n} is decreasing in m_{N} when N − n is odd, and increasing in m_{N} when N − n is even.

Fix N and consider the case

m_{n}(0) = \{ \begin{matrix}2^{−n}, & n = 1, …, 2N \\ 0, & n = 2N + 1, … \end{matrix}

and

λ_{n} = 8^{n} for all n.

Proposition 3.1. We have for all n and t

\begin{aligned}m_{2n}(t) & ≥ \frac{1}{2}m_{2n}(0), \\ m_{2n+1}(t) & ≤ m_{2n+1}(0) exp\{−4^{2n}t\}. \end{aligned}

Proof. Certainly m_{2N}(t) = m_{2N}(0) ≥ \frac{1}{2}m_{2N}(0). Suppose that n < N and

m_{2n+2}(t) ≥ \frac{1}{2}m_{2n+2}(0) for all t.

Then

\begin{aligned}m_{2n+1}(t) & ≤ m_{2n+1}(0) exp\{−λ_{2n+1}m_{2n+2}(0)t/2\} \\ & = m_{2n+1}(0) exp\{−4^{2n}t\} \end{aligned}

so

∫_{0}^{∞} m_{2n+1}(t) dt ≤ m_{2n+1}(0)4^{−2n} = \frac{1}{2} 8^{−2n}

so

m_{2n}(t) ≥ m_{2n}(0) exp\{−8^{−2n}λ_{2n}/2\} ≥ \frac{1}{2}m_{2n}(0).

Hence the proposition follows by induction.

We denote the solution just considered by m^{2N}. The same arguments establish corresponding inequalities for m^{2N+1}, where the roles of even and odd are swapped. Now we let N → ∞. For N ≥ n, m_{2n}^{2N}(t) is decreasing in N and m_{2n+1}^{2N}(t) is increasing in N for all n. We set

m_{n}^{+}(t) = lim_{N→∞} m_{n}^{2N}(t).

The integral equation

m_{n}^{2N}(t) = m_{n}^{2N}(0) − λ_{n} ∫_{0}^{t} m_{n}^{2N}(s)m_{n+1}^{2N}(s) ds

passes to the limit to give

m_{n}^{+}(t) = m_{n}^{+}(0) − λ_{n} ∫_{0}^{t} m_{n}^{+}(s)m_{n+1}^{+}(s) ds

so m^{+} is differentiable with

\frac{d}{dt}m_{n}^{+}(t) = −λ_{n}m_{n}^{+}(t)m_{n+1}^{+}(t).

So (m_{n}^{+} : n ≥ 1) solves the original system of equations. The same argument produces another solution (m_{n}^{−} : n ≥ 1) given by

m_{n}^{−}(t) = lim_{N→∞} m_{n}^{2N+1}(t).

We have m_{n}^{+}(0) = m_{n}^{−}(0) = 2^{−n} for all n. But

\begin{aligned}m_{2n}^{+}(t) & ≥ \frac{1}{2}m_{2n}^{+}(0), \\ m_{2n}^{−}(t) & ≤ exp\{−4^{2n−1}t\}m_{2n}^{−}(0) \end{aligned}

for all n and t, so m^{+} ≠ m^{−}.

We now use the solutions m^{+} and m^{−} to construct an example of Smoluchowski’s coagulation equation having at least two conservative solutions. Let x_{1}, x_{2}, … be an increasing sequence in (0, ∞) which is linearly independent over ℤ. For

x = x_{n} + (k_{1}x_{1} + … + k_{n−1}x_{n−1}), k_{1}, …, k_{n−1} ∈ ℤ

we write n(x) = n. Denote by I the set of all such x and define n(x) = 0 if x ∉ I. Define K : E × E → [0, ∞) by

K(x, y) = \{ \begin{matrix}λ_{n} & if \{n(x), n(y)\} = \{n, n + 1\} and n ≥ 1, \\ 0 & otherwise. \end{matrix}

Set

μ_{0} = ∑_{n=1}^{∞} ε_{x_{n}}2^{−n}

and consider Smoluchowski’s coagulation equation

\frac{d}{dt}⟨f, μ_{t}⟩ = \frac{1}{2} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}K(x, y)μ_{t}(dy)μ_{t}(dx)

starting from μ_{0}.

According to the definition made in §2, for a solution, we require

∫_{0}^{t} ∫_{B×E} K(x, y)μ_{s}(dx)μ_{s}(dy) ds < ∞

for all t and all compact sets B ⊆ E, and, for all bounded measurable functions f of compact support

⟨f, μ_{t}⟩ = ⟨f, μ_{0}⟩ + \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}K(x, y)μ_{s}(dx)μ_{s}(dy) ds.

Consider for n = 1, 2, …,

m_{n}(t) = μ_{t}({x : n(x) = n}).

Take f(x) = 1_{n(x)=n,x≤k} and let k → ∞ to obtain

\frac{d}{dt}m_{n}(t) = −λ_{n}m_{n}(t)m_{n+1}(t).

We deduce that any solution (μ_{t})_{t≥0} of the coagulation equation gives rise to a solution (m_{n}(t) : n ≥ 1)_{t≥0} of the system (3.1). On the other hand, for any solution (m_{n}(t) : n ≥ 1)_{t≥0} of this system, we obtain a solution (μ_{t})_{t≥0} of the coagulation equation by

\frac{d}{dt}μ_{t}(\{x\}) = −(λ_{n−1}m_{n−1}(t) + λ_{n}m_{n+1}(t))μ_{t}(\{x\}) + \frac{1}{2}λ_{n−1} ∑_{y,z∈I_{y+z=x}} μ_{t}(\{y\})μ_{t}(\{z\})

whenever n(x) = n. Hence the coagulation equation has two distinct solutions (μ_{t}^{+})_{t≥0} and (μ_{t}^{−})_{t≥0} corresponding to (m_{t}^{+})_{t≥0} and (m_{t}^{−})_{t≥0}. We now show these solutions are conservative. The idea of the proof is to show that the proportion of original particles making at least 2n jumps falls off geometrically in n.

Proposition 3.2. Suppose that μ_{0} has finite total mass density. Then the solution (μ_{t}^{+})_{t≥0} is conservative.

Proof. For x = k_{1}x_{1} + … + k_{n−1}x_{n−1} + x_{n} ∈ I, define

k_{m}(x) = \{ \begin{matrix}k_{m} & if m < n \\ 1 & if m = n \\ 0 & if m > n. \end{matrix}

Note that k_{m} : I → ℤ^{+} is additive. For m ≤ n consider

v_{m,n}(t) = ∫_{I} k_{m}(x)1_{n(x)≤n}μ_{t}(dx).

Then v_{m,n}(t) is non-increasing in t. Note that

∫_{I×I} k_{m}(x)1_{n(x)≤n}K(x, y)μ_{t}(dx)μ_{t}(dy) ≤ 8^{n}v_{m,n}(t) < ∞

so, by dominated convergence, since (μ_{t}^{+})_{t≥0} is a solution,

\frac{d}{dt}v_{m,n}(t) = −λ_{n}m_{n+1}(t) ∫_{I} k_{m}(x)1_{n(x)=n}μ_{t}(dx).

Note also

\frac{d}{dt} ∫_{I} k_{m}(x)1_{n(x)=n}μ_{t}(dx) ≤ −\frac{d}{dt}v_{m,n−1}(t)

so

∫_{I} k_{m}(x)1_{n(x)=n}μ_{t}(dx) ≤ r_{m,n−1}(t)

where r_{m,n}(t) = 2^{−m} − v_{m,n}(t). Now for n even

r_{m,n}(t) ≤ r_{m,n−1}(t) ∫_{0}^{t} λ_{n}m_{n+1}(s) ds ≤ \frac{1}{2}r_{m,n−1}(t)

so r_{m,n}(t) → 0 as n → ∞. Hence v_{m,n}(t) → 2^{−m} as n → ∞ and hence

∫_{I} xμ_{t}(dx) = ∑_{m} x_{m} ∫_{I} k_{m}(x)μ_{t}(dx) = ∑_{m} x_{m}2^{−m} = ∫_{I} xμ_{0}(dx).

We make some remarks on the relation between this example and the results of §2. The construction of the example makes it insensitive to the additive structure of E. We require very little of the sequence (x_{n})_{n≥1}. By taking x_{n} ≈ 8^{n} we can satisfy the condition

K(x, y) ≤ x

but we get

∫_{E} xμ_{0}(dx) = ∞.

On the other hand, by taking x_{n} ≈ α^{n}, for some α < 2, we get

∫_{E} xμ_{0}(dx) < ∞

but the relation K(x, y) ≤ Cxy for all x, y ∈ E does not hold for any C < ∞. Thus, however we choose (x_{n})_{n≥1}, we cannot regard (μ_{t}^{±})_{t≥0} as a strong solution, even in small time. This is of course implied also by the uniqueness of strong solutions established in §2.

It would be nice to find an example of this type where the initial mass distibution is supported on ℕ. It may be that for integers x_{n} → ∞ sufficiently fast, the analogous equation exhibits the same sort of behaviour. However we have not established whether this is true.

4. Hydrodynamic limit for the stochastic coalescent

In this section we shall prove some limit theorems for the stochastic coalescent. There are two main results. In Theorem 4.1, generalizing a result of Jeon [Jeo], we prove a tightness result for the stochastic coalescent, which implies a general existence theorem for solutions of Smoluchowski’s equation. Then, in Theorem 4.4, we prove weak convergence of the stochastic coalescent to any strong solution of Smoluchowski’s equation. The methods used are mostly standard tools from the theory of weak convergence on Skorohod spaces. The problem-specific idea which leads to Theorem 4.4 is the construction of a coupled family of particle systems, converging to the stochastic coalescent, in direct analogy with the method of §2. A version of this idea was also discovered independently by Kurtz [Kur]. The case of a discrete mass distribution may also be treated using a differential equation approach instead of weak convergence: this is simpler and more effective, establishing convergence at an exponential rate in the number of particles. The particle system we consider has been considered, in various special cases, by many others. In particular, it was considered in full generality by Marcus [Mar68] and Lushnikov [Lus78]. Recall that E = (0, ∞) and that the coagulation kernel K is a symmetric measurable function K : E × E → [0, ∞).

Let X_{0} be a finite, integer-valued measure on E. We can write X_{0} as a sum of unit masses

X_{0} = ∑_{i=1}^{m} ε_{x_{i}}

for some x_{1}, …, x_{m} ∈ E. We think of X_{0} as representing a system of m particles, labelled by their masses x_{1}, …, x_{m}. A Markov process (X_{t})_{t≥0} of finite, integer-valued measures on E can be constructed as follows: for each pair i < j, take an independent exponential random time T_{ij} of parameter K(x_{i}, x_{j}) and set T = min_{i<j} T_{ij}; set X_{t} = X_{0} for t < T and set

X_{T} = X_{0} − ε_{x_{i}} − ε_{x_{j}} + ε_{x_{i}+x_{j}} if T = T_{ij};

then begin the construction afresh from X_{T}. In this process, each pair of particles {x_{i}, x_{j}} coalesces at rate K(x_{i}, x_{j}) to form a new particle x_{i} + x_{j}. We call (X_{t})_{t≥0} a stochastic coalescent with coagulation kernel K.

Denote by d some metric on ℳ which is compatible with the topology of weak convergence, that is to say d(μ_{n}, μ) → 0 if and only if ⟨f, μ_{n}⟩ → ⟨f, μ⟩ for all bounded continuous functions f : E → ℝ. We choose d so that d(μ, μ′) ≤ ∥μ − μ′∥ for all μ, μ′ ∈ ℳ. When the class of functions f is restricted to those of bounded support we get a weaker topology, also metrizable, and we denote by d_{0} some compatible metric, with d_{0} ≤ d.

The following result is a first attempt at proving weak convergence for the stochastic coalescent. It is less than satisfactory because it does not enable us to show uniqueness of limits. We include it here, partly as a warm-up for the more intricate arguments used later, and partly because it provides the best result on global existence of solutions to Smoluchowski’s equation that we know. A version of the result where μ_{0} is supported on ℕ and where φ(x) = x has been proved already by Jeon [Jeo].

Theorem 4.1. Let K : E × E → [0, ∞) be a symmetric continuous function and let μ_{0} be a measure on E. Assume that, for some continuous sublinear function φ : E → (0, ∞),

K(x, y) ≤ φ(x)φ(y), for all x, y ∈ E,
φ(x)^{−1}φ(y)^{−1}K(x, y) → 0, as (x, y) → ∞.

Assume also that ⟨φ, μ_{0}⟩ < ∞. Let (X_{t}^{n})_{t≥0} be a sequence of stochastic coalescents, with coagulation kernel K. Set X̃_{t}^{n} = n^{−1}X_{n^{−1}t}^{n} and suppose that

d_{0}(φX̃_{0}^{n}, φμ_{0}) → 0

as n → ∞ and that, for some constant Λ < ∞, for all n

⟨φ, X̃_{0}^{n}⟩ ≤ Λ.

Then the sequence of laws of X̃^{n} on D([0, ∞), (ℳ, d_{0})) is tight. Moreover, for any weak limit point X, almost surely, (X_{t})_{t≥0} is a solution of Smoluchowski’s coagulation equation (2.1). In particular, this equation has at least one solution.

Proof. For an integer-valued measure μ on E, denote by μ^{(1)} the integer-valued measure on E × E given by

μ^{(1)}(A × A′) = μ(A)μ(A′) − μ(A ∩ A′).

(This is simply the counting measure for ordered pairs of masses of distinct particles.) Similarly, when nμ is an integer-valued measure, set

μ^{(n)}(A × A′) = μ(A)μ(A′) − n^{−1}μ(A ∩ A′)

and note that n^{2}μ^{(n)} = (nμ)^{(1)}. For a bounded measurable function f on E, set

L^{(n)}(μ)(f) = ⟨f, L^{(n)}(μ)⟩

= \frac{1}{2} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}K(x, y)μ^{(n)}(dx, dy),

Q^{(n)}(μ)(f) = \frac{1}{2} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}^{2}K(x, y)μ^{(n)}(dx, dy).

Then

M_{t}^{f,n} = ⟨f, X_{t}^{n}⟩ − ⟨f, X_{0}^{n}⟩ − ∫_{0}^{t} L^{(1)}(X_{s}^{n})(f) ds

is a martingale, having previsible increasing process

⟨M^{f,n}⟩_{t} = ∫_{0}^{t} Q^{(1)}(X_{s}^{n})(f) ds.

Set M̃_{t}^{f,n} = n^{− \frac{1}{2}} M_{n^{−1}t}^{f,n}, then we have

⟨f, X̃_{t}^{n}⟩ = ⟨f, X̃_{0}^{n}⟩ + n^{− \frac{1}{2}} M̃_{t}^{f,n} + ∫_{0}^{t} L^{(n)}(X̃_{s}^{n})(f) ds, (4.1)

⟨M̃^{f,n}⟩_{t} = ∫_{0}^{t} Q^{(n)}(X̃_{s}^{n})(f) ds.

Since φ is subadditive, we have ⟨φ, X̃_{t}^{n}⟩ ≤ Λ for all n and t. Hence by (2.4)

\begin{aligned}|L^{(n)}(X̃_{t}^{n})(f)| & ≤ 2∥f∥Λ^{2}, \\ Q^{(n)}(X̃_{t}^{n})(f) & ≤ 4∥f∥^{2}Λ^{2}. \end{aligned}

Assume that |f| ≤ φ ∧ 1. Then

|⟨f, X̃_{t}^{n}⟩| ≤ Λ

for all t, so we have compact containment. Moreover, by Doob’s L^{2}−inequality, for all s < t,

𝔼 sup_{s≤r≤t} |M̃_{r}^{f,n} − M̃_{s}^{f,n}|^{2} ≤ 4𝔼 ∫_{s}^{t} Q^{(n)}(X̃_{r}^{n})(f)dr ≤ 16Λ^{2}(t − s)

so

𝔼 sup_{s≤r≤t} |⟨f, X̃_{r}^{n} − X̃_{s}^{n}⟩|^{2} ≤ C{(t − s)^{2} + n^{−1}(t − s)}

where C < ∞ depends only on Λ. Hence, by a standard tightness criterion, the laws of the sequence ⟨f, X̃^{n}⟩ on D([0, ∞), ℝ) are tight. See for example [EK86], Corollary 7.4. We note the bound

∥(φ ∧ 1)X̃_{t}^{n}∥ ≤ ⟨φ, X̃_{t}^{n}⟩ ≤ Λ

for all t. Hence we can apply Jakubowski’s criterion [Jak86] to see that the laws of the sequence (φ ∧ 1)X̃^{n} on D([0, ∞), ℳ_{[0,∞]}) are tight. By consideration of subsequences and a theorem of Skorohod, see, for example, Pollard [Pol84], Chapter IV, it suffices from this point on to consider the case where (φ ∧ 1)X̃^{n} converges almost surely in D([0, ∞), ℳ_{[0,∞]}), with limit (φ ∧ 1)X say. We denote also by X the process in ℳ_{E} obtained by restriction of measures. Note that

∥X̃_{t}^{n} − X̃_{t−}^{n}∥ ≤ 3/n

so X ∈ C([0, ∞), ℳ_{E}). Moreover φ^{δ} = φ1_{(0,δ]} is subadditive, so

⟨φ^{δ}, X̃_{t}^{n}⟩ ≤ ⟨φ^{δ}, X̃_{0}^{n}⟩ ≤ ⟨φ^{δ}, μ_{0}⟩ + |⟨φ^{δ}, X̃_{0}^{n} − μ_{0}⟩|

and so, given ε > 0, we can find δ > 0 so that

sup_{n} sup_{t}⟨φ^{δ}, X̃_{t}^{n}⟩ < ε.

Given a continuous bounded function f : E → ℝ of bounded support, we can write f = f_{1} + f_{2} where f_{1} is continuous of compact support and f_{2} is supported in (0, δ) with ∥f_{2}∥ ≤ ∥f∥. Then

lim sup_{n→∞} sup_{s≤t}⟨φf, X̃_{s}^{n} − X_{s}⟩ ≤ lim_{n→∞} sup_{s≤t}⟨φf_{1}, X̃_{s}^{n} − X_{s}⟩ + ∥f∥ sup_{n} sup_{s}⟨φ_{,}^{δ}X̃_{s}^{n}⟩ ≤ ε∥f∥.

Since f and ε were arbitrary, this shows that

sup_{s≤t} d_{0}(φX̃_{s}^{n}, φX_{s}) → 0 a.s. (4.2)

We now wish to pass to the limit in (4.1). Let us suppose for now that f : E → ℝ is continuous and of compact support B. Then, as n → ∞

𝔼(sup_{s≤t} |n^{− \frac{1}{2}} M̃_{s}^{f,n}|^{2}) ≤ \frac{4}{n}𝔼⟨M̃^{f,n}⟩_{t} ≤ \frac{16Λ^{2}∥f∥^{2}}{n} → 0,
|(L − L^{(n)})(X̃_{s}^{n})(f)| = \frac{1}{n} |∫_{E}\{f(2x) − 2f(x)\}K(x, x)X̃_{s}^{n}(dx)|

≤ \frac{3∥f∥}{n} ∫_{B∪2B} φ(x)^{2}X̃_{s}^{n}(dx)

≤ \frac{3∥f∥}{n} ∥φ1_{B∪2B}∥⟨φ, X̃_{0}^{n}⟩ → 0.

Hence it will suffice to show that as n → ∞

sup_{s≤t} |⟨f, L(X̃_{s}^{n}) − L(X_{s})⟩| → 0 a.s. (4.3)

where we recall that

⟨f, L(μ)⟩ = \frac{1}{2} ∫ \{f(x + y) − f(x) − f(y)\}K(x, y)μ(dx)μ(dy).

Given δ > 0 and N < ∞, we can write K = K_{1} + K_{2}, where K_{1} is continuous of compact support and where 0 ≤ K_{2} ≤ K and K_{2} is supported on

F_{1} ∪ F_{2} ∪ F_{2} = {(x, y) : x ≤ δ} ∪ {(x, y) : y ≤ δ} ∪ {(x, y) : |(x, y)| ≥ N}.

Then, with an obvious notation, by (4.2)

sup_{s≤t} |⟨f, L_{1}(X̃_{s}^{n}) − L_{1}(X_{s})⟩| → 0 a.s.

whereas, for K_{2}, we use the estimates

\begin{aligned}∥K1_{F_{1}}μ ⊗ μ∥ = ∥K1_{F_{2}}μ ⊗ μ∥ & ≤ ⟨φ, μ⟩⟨φ^{δ}, μ⟩, \\ ∥K1_{F_{3}}μ ⊗ μ∥ & ≤ β_{N}⟨φ, μ⟩^{2}, \end{aligned}

where β_{N} = sup_{|(x,y)|≥N} φ(x)^{−1}φ(y)^{−1}K(x, y). Now,

⟨φ, X̃_{t}^{n}⟩ ≤ ⟨φ, X̃_{0}^{n}⟩, ⟨φ^{δ}, X̃_{t}^{n}⟩ ≤ ⟨φ^{δ}, X̃_{0}^{n}⟩

and, given ε > 0, we can find δ and N so that

⟨φ^{δ}, X̃_{0}^{n}⟩ ≤ \frac{1}{3}εΛ^{−1}, for all n,
⟨φ^{δ}, μ_{0}⟩ ≤ \frac{1}{3}εΛ^{−1},
β_{N} ≤ \frac{1}{3}εΛ^{−2}.

Then

|⟨f, L_{2}(X̃_{t}^{n})⟩| ≤ ε, |⟨f, L_{2}(X_{t})⟩| ≤ ε

for all n and t. Hence

lim sup_{n→∞} sup_{s≤t} |⟨f, L(X̃_{s}^{n}) − L(X_{s})⟩| ≤ 2ε.

But ε was arbitrary, so (4.3) is proved. Hence we can let n → ∞ in (3.1) to obtain

⟨f, X_{t}⟩ = ⟨f, X_{0}⟩ + ∫_{0}^{t}⟨f, L(X_{s})⟩ ds

for all continuous functions f : E → ℝ of compact support. By using the bounds (2.4) and ⟨φ, X_{t}⟩ ≤ Λ, and a straightforward limit argument, we can extend this equation to all bounded measurable functions f. In particular, almost surely, X is a solution of Smoluchowski’s equation, in the sense of §2.

A corollary of Theorem of 4.1 is that, whenever we know Smoluchowski’s equation has at most one solution, then, under the hypotheses of Theorem 4.1, we can deduce, for all t

sup_{s≤t} d_{0}(X̃_{s}^{n}, μ_{s}) → 0

in probability as n → ∞, for the solution (μ_{t})_{t≥0} provided by Theorem 4.1. However, we can only prove uniqueness of solutions in the presence of a strong solution. So we prefer to formulate our main limit result, Theorem 4.4, in that context, when a new approach allows certain other hypotheses to be relaxed.

For the remainder of this section we will assume that we have chosen a continuous sublinear function φ : E → (0, ∞) and that K satisfies

K(x, y) ≤ φ(x)φ(y) for all x, y ∈ E. (4.4)

Our further analysis of the stochastic coalescent will rest on an approximation by a coupled family of Markov processes (X_{t}^{B}, Λ_{t}^{B})_{t≥0}, indexed by sets B ⊆ E which we now describe. Each process (X_{t}^{B})_{t≥0} will take values in the finite integer-valued measures on E, whereas (Λ_{t}^{B})_{t≥0} will be a non-decreasing process in [0, ∞). Let us suppose given initial values (X_{0}^{B}, Λ_{0}^{B}), for all B, such that X_{0}^{B} is supported in B and such that, whenever B ⊆ B′

X_{0}^{B} ≤ X_{0}^{B′}, ⟨φ, X_{0}^{B}⟩ + Λ_{0}^{B} ≥ ⟨φ, X_{0}^{B′}⟩ + Λ_{0}^{B′}.

Write X_{0} = X_{0}^{E} as a sum of unit masses

X_{0} = ∑_{i=1}^{m} ε_{x_{i}}

where x_{1}, …, x_{m} ∈ E. There is a unique increasing map

B ↦ I(B) ⊆ {1, …, m}

such that

X_{0}^{B} = ∑_{i∈I(B)} ε_{x_{i}}.

Set

ν^{B} = Λ_{0}^{B} − ∑_{j∉I(B)} φ(x_{j}).

Note that ν^{B} decreases as B increases and that ν^{E} = Λ_{0}^{E} ≥ 0. For i < j, take independent exponential random variables T_{ij} of parameter K(x_{i}, x_{j}). Set T_{ji} = T_{ij}. Also, for i ≠ j, take independent exponential random variables S_{ij} of parameter φ(x_{i})φ(x_{j}) − K(x_{i}, x_{j}). We can construct, independently for each i, a family of independent exponential random variables S_{i}^{B}, increasing in B, with S_{i}^{B} having parameter φ(x_{i})ν^{B}. Set

T_{i}^{B} = min_{j∉I(B)}(T_{ij} ∧ S_{ij}) ∧ S_{i}^{B}

and note that T_{i}^{B} is an exponential random variable of parameter

∑_{j∉I(B)} φ(x_{i})φ(x_{j}) + φ(x_{i})ν^{B} = φ(x_{i})Λ_{0}^{B}.

For each B, the random variables

(T_{ij}, T_{i}^{B} : i, j ∈ I(B), i < j)

form an independent family, whereas, for i ∈ I(B) and j ∉ I(B), we have

T_{i}^{B} ≤ T_{ij}

and for B ⊆ B′ and all i we have

T_{i}^{B} ≤ T_{i}^{B′}.

Now set

T = (min_{i<j} T_{ij}) ∧ (min_{i} T_{i}^{∅}).

We set (X_{t}^{B}, Λ_{t}^{B}) = (X_{0}^{B}, Λ_{0}^{B}) for t < T and set

(X_{t}^{B}, Λ_{T}^{B}) = \{ \begin{matrix}(X_{0}^{B} − ε_{x_{i}} − ε_{x_{j}} + ε_{x_{i}+x_{j}}, Λ_{0}^{B}) & if T = T_{ij}, i, j ∈ I(B), x_{i} + x_{j} ∈ B \\ (X_{0}^{B} − ε_{x_{i}} − ε_{x_{j}}, Λ_{0}^{B} + φ(x_{i} + x_{j})) & if T = T_{ij}, i, j ∈ I(B), x_{i} + x_{j} ∉ B \\ (X_{0}^{B} − ε_{x_{i}}, Λ_{0}^{B} + φ(x_{i})) & if T = T_{i}^{B}, i ∈ I(B) \\ (X_{0}^{B}, Λ_{0}^{B}) & otherwise. \end{matrix}

It is straightforward to check that X_{T}^{B} is supported on B and, for B ⊆ B′

X_{T}^{B} ≤ X_{T}^{B′}, ⟨φ, X_{T}^{B}⟩ + Λ_{T}^{B} ≥ ⟨φ, X_{T}^{B′}⟩ + Λ_{T}^{B′}.

Hence we can repeat the above construction independently from time T, to obtain a family of Markov processes (X_{t}^{B}, Λ_{t}^{B})_{t≥0} such that X_{t}^{B} is supported on B and for B ⊆ B′ and all t

X_{t}^{B} ≤ X_{t}^{B′}, ⟨φ, X_{t}^{B}⟩ + Λ_{t}^{B} ≥ ⟨φ, X_{t}^{B′}⟩ + Λ_{t}^{B′}. (4.5)

At the outset, we assumed that both X_{0}^{B} and Λ_{0}^{B} were given, for all B. From now on we shall suppose simply that X_{0} = X_{0}^{E} is given and take

X_{0}^{B} = 1_{B}X_{0}, Λ_{0}^{B} = ⟨φ1_{B^{c}}, X_{0}⟩.

Of course these relations do not remain valid as time evolves.

For each fixed B, the process (X_{t}^{B}, Λ_{t}^{B})_{t≥0} may be regarded as a finite state-space Markov chain having three sorts of transition. Each pair of particles x_{i}, x_{j} in X_{t}^{B} is, at rate K(x_{i}, x_{j}), removed; if x_{i} + x_{j} ∈ B, the merged particle is added to X_{t}^{B}, if not, φ(x_{i} + x_{j}) is added to Λ_{t}^{B}. Also, each particle x_{i} in X_{t}^{B} is, at rate φ(x_{i})Λ_{t}^{B}, removed and φ(x_{i}) added to Λ_{t}^{B}. In particular, for the choice of initial values made above, Λ_{t}^{E} = 0 for all t and X_{t} = X_{t}^{E} is simply the stochastic coalescent with coagulation kernel K with which we began.

We now proceed to identify some martingales associated with (X_{t}^{B}, Λ_{t}^{B})_{t≥0}. Recall that, when nμ is an integer-valued measure on E, we denote by μ^{(n)} the measure on E × E characterized by

μ^{(n)}(A × A′) = μ(A)μ(A′) − n^{−1}μ(A ∩ A′).

Given an integer-valued measure μ on E and given λ ≥ 0, define for any bounded measurable function f on E and for a ∈ ℝ

L^{B,(1)}(μ, λ)(f, a) = ⟨(f, a), L^{B,(1)}(μ, λ)⟩ =

\frac{1}{2} ∫_{E×E}\{f(x + y)1_{x+y∈B} + aφ(x + y)1_{x+y∉B} − f(x) − f(y)\}K(x, y)μ^{(1)}(dx, dy)

+ λ ∫_{E}{aφ(x) − f(x)}φ(x)μ(dx)

and

Q^{B,(1)}(μ, λ)(f, a) =

\frac{1}{2} ∫_{E×E}\{f(x + y)1_{x+y∈B} + aφ(x + y)1_{x+y∉B} − f(x) − f(y)\}^{2}K(x, y)μ^{(1)}(dx, dy)

+ λ ∫_{E}{aφ(x) − f(x)}^{2}φ(x)μ(dx).

Then, for all f and a,

M_{t} = ⟨f, X_{t}^{B}⟩ + aΛ_{t}^{B} − ⟨f, X_{0}^{B}⟩ − aΛ_{0}^{B} − ∫_{0}^{t} L^{B(1)}(X_{s}^{B}, Λ_{s}^{B})(f, a) ds (4.6)

is a martingale, having previsible increasing process

⟨M⟩_{t} = ∫_{0}^{t} Q^{B,(1)}(X_{s}^{B}, Λ_{s}^{B})(f, a) ds. (4.7)

Recall from §2 that, for B ⊆ E compact, we denote by ℳ_{B} the space of finite signed measures supported on B and we define L^{B} : ℳ_{B} × ℝ → ℳ_{B} × ℝ by the requirement

⟨(f, a), L^{B}(μ, λ)⟩ =

\frac{1}{2} ∫_{E×E}\{f(x + y)1_{x+y∈B} − aφ(x + y)1_{x+y∉B} − f(x) − f(y)\}K(x, y)μ(dx)μ(dy)

+ λ ∫_{E}{aφ(x) − f(x)}φ(x)μ(dx)

for all f and a.

Fix a measure μ_{0} on E with ⟨φ, μ_{0}⟩ < ∞. Set

μ_{0}^{B} = 1_{B}μ_{0}, λ_{0}^{B} = ⟨φ1_{B^{c}}, μ_{0}⟩.

Recall from §2 that, for each compact set B, the equation

(μ_{t}^{B}, λ_{t}^{B}) = (μ_{0}^{B}, λ_{0}^{B}) + ∫_{0}^{t} L^{B}(μ_{s}^{B}, λ_{s}^{B}) ds (4.8)

has a unique solution, which is a continuous map

t ↦ (μ_{t}^{B}, λ_{t}^{B}) : [0, ∞) → ℳ_{B}^{+} × ℝ^{+}.

Consider now a sequence of integer-valued measures X_{0}^{n}. For each n, denote by (X_{t}^{n})_{t≥0} and (X_{t}^{B,n}, Λ_{t}^{B,n})_{t≥0} the stochastic coalescent and the coupled family of approximations constructed above, starting from X_{0}^{n}. Set

\begin{aligned}X̃_{t}^{n} & = n^{−1}X_{n^{−1}t}^{n}, \\ (X̃_{t}^{B,n}, Λ̃_{t}^{B,n}) & = n^{−1}(X_{n^{−1}t}^{B,n}, Λ_{n^{−1}t}^{B,n}). \end{aligned}

We shall need a mild continuity condition on K. Denote by S(K) ⊆ E × E the set of discontinuity points of K and by μ_{0}^{∗n} the n-th convolution power of μ_{0}. Our assumption is that

(μ_{0}^{∗n})^{⊗2}(S(K)) = 0, for all n ≥ 1. (4.9)

This condition is verified, in particular, when S(K) has Lebesgue measure zero and μ_{0} is absolutely continuous.

For the purposes of the next proposition, we also need an analogous condition on the compact set B:

μ_{0}^{∗n}(∂B) = 0, for all n ≥ 1. (4.10)

This condition is verified, for any given μ_{0}, for all but countably many closed intervals in E.

Proposition 4.2. Assume conditions (2.4), (2.5), (4.9), (4.10). Suppose that

d(X̃_{0}^{B,n}, μ_{0}^{B}) → 0, |Λ̃_{0}^{B,n} − λ_{0}^{B}| → 0

as n → ∞. Then, for all t ≥ 0

sup_{s≤t} d(X̃_{s}^{B,n}, μ_{s}^{B}) → 0, sup_{s≤t} |Λ̃_{s}^{B,n} − λ_{s}^{B}| → 0

in probability as n → ∞.

Proof. Set Λ = sup_{n}⟨φ, X̃_{0}^{n}⟩ and note that Λ < ∞. By rescaling (4.6) and (4.7), we see that, for all B, all bounded measurable functions f and all a ∈ ℝ

M_{t}^{n} = \sqrt{n} (⟨f, X̃_{t}^{B,n}⟩ + aΛ̃_{t}^{B,n} − ⟨f, X̃_{0}^{B,n}⟩ − aΛ̃_{0}^{B,n} − ∫_{0}^{t} L^{B,(n)}(X̃_{s}^{B,n}, Λ̃_{s}^{B,n})(f, a) ds) (4.11)

is a martingale, having previsible increasing process

⟨M^{n}⟩_{t} = ∫_{0}^{t} Q^{B,(n)}(X̃_{s}^{B,n}, Λ̃_{s}^{B,n})(f, a) ds (4.12)

where

\begin{aligned}L^{B,(n)}(μ, λ) & = n^{−2}L^{B,(1)}(nμ, nλ), \\ Q^{B,(n)}(μ, λ) & = n^{−2}Q^{B,(1)}(nμ, nλ). \end{aligned}

There is a constant C < ∞, depending only on B, Λ and φ such that

\begin{aligned}|L^{B}(X̃_{t}^{B,n}, Λ̃_{t}^{B,n})(f, a)| & ≤ C(∥f∥ + |a|), \\ |(L^{B} − L^{B,(n)})(X̃_{t}^{B,n}, Λ̃_{t}^{B,n})(f, a)| & ≤ Cn^{−1}(∥f∥ + |a|), \\ |Q^{B,(n)}(X̃_{t}^{B,n}, Λ̃_{t}^{B,n})(f, a)| & ≤ C(∥f∥ + |a|)^{2}. \end{aligned}

Hence by the same argument as in Theorem 4.1, the laws of the sequence (X̃^{n}, Λ̃^{n}) are tight in D([0, ∞), ℳ_{B}×ℝ). Indeed, similarly, the laws of the sequence (X̃^{n}, Λ̃^{n}, I^{n}, J^{n}) are tight in D([0, ∞), ℳ_{B} × ℝ × ℳ_{B×B} × ℳ_{B×B}), where

\begin{aligned}I_{t}^{n}(dx, dy) & = K(x, y)1_{x+y∈B}X̃_{t}^{n}(dx)X̃_{t}^{n}(dy), \\ J_{t}^{n}(dx, dy) & = K(x, y)1_{x+y∉B}X̃_{t}^{n}(dx)X̃_{t}^{n}(dy). \end{aligned}

Denote by (X, Λ, I, J) some weak limit point of this sequence, which, by passing to a subsequence and the usual argument of Skorohod, we may regard as a pointwise limit in D([0, ∞), ℳ_{B} × ℝ × ℳ_{B×B} × ℳ_{B×B}). Then there exist bounded measurable functions

I, J : Ω × [0, ∞) × B × B → [0, ∞)

symmetric on B × B, such that

\begin{aligned}I_{t}(dx, dy) & = I(t, x, y)X_{t}(dx)X_{t}(dy), \\ J_{t}(dx, dy) & = J(t, x, y)X_{t}(dx)X_{t}(dy) \end{aligned}

in ℳ_{B×B} and such that

\begin{aligned}I(t, x, y) & = K(x, y)1_{x+y∈B} \\ J(t, x, y) & = K(x, y)1_{x+y∉B}. \end{aligned}

whenever (x, y) ∉ S(K) and x + y ∉ ∂B. Moreover we can pass to the limit in (4.11) to obtain, for all continuous functions f and all a ∈ ℝ, for all t ≥ 0, almost surely

\begin{aligned}⟨(f, a), (X_{t}, Λ_{t})⟩ & = ⟨(f, a), (X_{0}, Λ_{0})⟩ \\ & + \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{f(x + y) − f(x) − f(y)\}I(s, x, y)X_{s}(dx)X_{s}(dy)ds \\ & + \frac{1}{2} ∫_{0}^{t} ∫_{E×E}\{aφ(x + y) − f(x) − f(y)\}J(s, x, y)X_{s}(dx)X_{s}(dy)ds \\ & + ∫_{0}^{t} Λ_{s} ∫_{E}\{aφ(x) − f(x)\}φ(x)X_{s}(dx)ds \end{aligned} (4.13)

By the remarks following the proof of Proposition 2.2, this equation forces X_{t} ⊗ X_{t} to be absolutely continuous with respect to

∑_{n=1}^{∞}(μ_{0}^{∗n})^{⊗2}.

Hence by the assumptions (4.9), (4.10), we can replace I(t, x, y) by K(x, y)1_{x+y∈B} and J(t, x, y) by K(x, y)1_{x+y∉B} in (4.13). But this is now equation (4.8) which has a unique solution (μ_{t}^{B}, λ_{t}^{B})_{t≥0}. We have shown that the unique weak limit point of (X̃^{n}, Λ̃^{n}) in D([0, ∞), ℳ_{B}) × ℝ is the continuous deterministic path (μ_{t}^{B}, λ_{t}^{B})_{t≥0}, which proves the proposition.

We consider now the special case where μ_{0} is a probability measure on ℕ = {1, 2, … }. Here we can replace the method of weak convergence in Proposition 4.2 by a more direct approach using differential equations. The benefit in this approach, besides greater transparency, is that we can establish a rate of convergence, which is in principle computable. This would be needed if one wished, in practice, to assess whether Smoluchowski’s equation provided a tolerable approximation to the stochastic coalescent. Since the stochastic coalescent already makes a meanfield approximation—it is assumed we neglect spatial variations in the particle mass distribution—we are effectively assuming there is some external spatial mixing. The relevant particle number n is then the number of particles in the largest region which is mixed to equilibrium in unit time.

Proposition 4.3. Assume conditions (2.4) and (2.5). Suppose that μ_{0} is supported on ℕ. Let B be a finite subset of ℕ. Then there is a constant C < ∞, depending only on K, φ, μ_{0}, and B such that, for all n ≥ 1, for X̃_{t}^{B} = n^{−1}X_{n^{−1}t}^{B} and Λ̃_{t}^{B} = n^{−1}Λ_{n^{−1}t}^{B}, for all 0 ≤ δ ≤ t, if δ_{0} = ∥X̃_{0}^{B} − μ_{0}^{B}∥ + |Λ̃_{0}^{B} − λ_{0}^{B}| ≤ 1 then

ℙ (sup_{s≤t}{∥X̃_{s}^{B} − μ_{s}^{B}∥ + |Λ̃_{s}^{B} − λ_{s}^{B}|} > (δ_{0} + δ)e^{Ct}) ≤ Ce^{−nδ^{2}/Ct}.

Proof. We regard (X̃_{t}^{B}, Λ̃_{t}^{B})_{t≥0} as taking values in the finite-dimensional vector space ℝ^{B} × ℝ. Recall from the proof of Proposition 4.2 that

M_{t} = \sqrt{n}\{(X̃_{t}^{B}, Λ̃_{t}^{B}) − (X̃_{0}^{B}, Λ̃_{0}^{B}) − ∫_{0}^{t} L^{B,(n)}(X̃_{s}^{B}, Λ̃_{s}^{B}) ds\} (4.14)

is a martingale, having previsible increasing process

⟨M⟩_{t} = ∫_{0}^{t} Q^{B,(n)}(X̃_{s}^{B}, Λ̃_{s}^{B}) ds.

We recall a form of the exponential martingale inequality for martingales M whose jumps are bounded uniformly by A ∈ [0, ∞) and which have a continuous previsible increasing process ⟨M⟩: for all θ ≥ 0 we have

ℙ(sup_{t} M_{t} ≥ δ and ⟨M⟩_{∞} ≤ ε) ≤ exp\{−θδ + \frac{1}{2}θ^{2}e^{θA}ε\}.

Let f : B → ℝ and a ∈ ℝ be given, with ∥f∥ + |a| ≤ 1. Consider the martingale

M_{t}^{f,a} = ⟨(f, a), M_{t}⟩.

The jumps of M^{f,a} are bounded uniformly by Cn^{− \frac{1}{2}} for some C < ∞, depending on φ and B. The process ⟨M^{f,a}⟩ is continuous and satisfies

⟨M^{f,a}⟩_{t} ≤ Ct

for some C < ∞, depending on φ, B and μ_{0}. So, by the exponential martingale inequality

ℙ( sup_{s≤t} M_{s}^{f,a} ≥ δ) ≤ exp\{−θδ + \frac{1}{2}Cθ^{2}te^{C θ/\sqrt{n}}\}.

Assume that δ ≤ \sqrt{n}t and take θ = δ/(3Ct). Then Cθ/\sqrt{n} ≤ 1/3, so e^{Cθ/\sqrt{n}} ≤ 3/2 and so

ℙ( sup_{s≤t} M_{s}^{f,a} ≥ δ) ≤ e^{−δ^{2}/4Ct}.

Since B is finite, we deduce that for some C < ∞, depending on φ, B and μ_{0}

ℙ( sup_{s≤t} ∥M_{s}∥ ≥ δ) ≤ Ce^{−δ^{2}/Ct}

whenever δ ≤ \sqrt{n}t. Hence

ℙ( sup_{s≤t} ∥n^{− \frac{1}{2}} M_{s}∥ ≥ δ) ≤ Ce^{−nδ^{2}/Ct}

whenever δ ≤ t. Note also the estimate

∥(L^{B} − L^{B,(n)})(X̃_{t}^{B}, Λ̃_{t}^{B})∥ ≤ Cn^{−1}, for all t ≥ 0,

for some C < ∞, depending on φ and B. Set Y_{t} = (X̃_{t}^{B}, Λ̃_{t}^{B}) − (μ_{t}^{B}, λ_{t}^{B}) and subtract the equations (4.14), (2.6) to obtain

Y_{t} = R_{t} + ∫_{0}^{t} L_{s}^{B}(Y_{s}) ds

where

R_{t} = Y_{0} + n^{− \frac{1}{2}} M_{t} + ∫_{0}^{t}(L^{B} − L^{B,(n)})(X̃_{s}^{B}, Λ̃_{s}^{B}) ds

and where

⟨(f, a), L_{t}^{B}(μ, λ)⟩ = \frac{1}{2} ∫_{E×E}\{f(x + y)1_{x+y∈B} + aφ(x + y)1_{x+y∉B} − f(x) − f(y)\}
× K(x, y)(X̃_{t}^{B} + μ_{t}^{B})(dx)μ(dy)
+ (Λ̃_{t}^{B} + λ_{t}^{B}) ∫_{E}(aφ(x) − f(x))φ(x)μ(dx)
+ λ ∫_{E}(aφ(x) − f(x))φ(x)(X̃_{t}^{B} + μ_{t}^{B})(dx).

Note the estimate

∥L_{t}^{B}(μ, λ)∥ ≤ C∥(μ, λ)∥/2

where C < ∞ depends on φ, B and μ_{0}. Set g(t) = sup_{s≤t} ∥Y_{s}∥ and r(t) = sup_{s≤t} ∥R_{s}∥. Then

g(t) ≤ r(t) + \frac{1}{2}C ∫_{0}^{t} g(s) ds

so g(t) ≤ r(t)e^{Ct/2}. Now, for δ ≤ t, we have

ℙ(r(t) ≥ g(0) + δ/2 + Ct/n) ≤ Ce^{−nδ^{2}/Ct}.

We may assume that C ≥ 4. If δ^{2} ≤ Ct/n we have nothing to prove. Otherwise

Ct/n < δ^{2} ≤ \frac{1}{2}δe^{Ct/2}

so

ℙ(g(t) ≥ (δ_{0} + δ)e^{Ct}) ≤ ℙ(r(t) ≥ g(0) + δ/2 + Ct/n) ≤ Ce^{−nδ^{2}/Ct}.

Here is the main result of this section.

Theorem 4.4. Let K : E × E → [0, ∞) be a symmetric measurable function and let μ_{0} be a measure on E. Assume that

(μ_{0}^{∗n})^{⊗2}(S(K)) = 0 for all n ≥ 1

where S(K) denotes the discontinuity set of K. Assume also that, for some continuous sublinear function φ : E → (0, ∞),

K(x, y) ≤ φ(x)φ(y) for all x, y ∈ E

and that ⟨φ, μ_{0}⟩ < ∞ and ⟨φ^{2}, μ_{0}⟩ < ∞. Denote by (μ_{t})_{t<T} the maximal strong solution provided by Theorem 2.1. Let (X_{t}^{n})_{t≥0} be a sequence of stochastic coalescents, with coagulation kernel K. Set X̃_{t}^{n} = n^{−1}X_{n^{−1}t}^{n} and suppose that

d(φX̃_{0}^{n}, φμ_{0}) → 0

as n → ∞. Then, for all t < T

sup_{s≤t} d(φX̃_{s}^{n}, φμ_{s}) → 0

in probability, as n → ∞.

Moreover, if μ_{0} is supported on ℕ, then for all t < T and all δ > 0 there are constants δ_{0} > 0 and C < ∞, depending only on K, μ_{0}, φ, t and δ, such that, for all n,

∥φ(X̃_{0}^{n} − μ_{0})∥ ≤ δ_{0}

implies

ℙ( sup_{s≤t} ∥φ(X̃_{s}^{n} − μ_{s})∥ > δ) ≤ e^{−n/C}.

Proof. Fix δ > 0 and t < T. Since (μ_{t})_{t<T} is strong, we can find a compact set B satisfying (4.10) and such that λ_{t}^{B} < δ/2. Now

d(φX̃_{0}^{n}, φμ_{0}) → 0

so

d(X̃_{0}^{B,n}, μ_{0}^{B}) → 0, |Λ̃_{0}^{B,n} − λ_{0}^{B}| → 0.

Hence, by Proposition 4.3

sup_{s≤t} d(X̃_{s}^{B,n}, μ_{s}^{B}) → 0, sup_{s≤t} |Λ̃_{s}^{B,n} − λ_{s}^{B}| → 0

in probability as n → ∞. Since {μ_{s}^{B} : s ≤ t} is compact, we also have

sup_{s≤t} d(φX̃_{s}^{B,n}, φμ_{s}^{B}) → 0

in probability as n → ∞. By Proposition 2.5 and by (4.5), for s ≤ t

∥φ(μ_{s} − μ_{s}^{B})∥ = ⟨φ, μ_{s} − μ_{s}^{B}⟩ ≤ λ_{s}^{B} ≤ λ_{t}^{B} < δ/2
∥φ(X̃_{s}^{n} − X̃_{s}^{B,n})∥ = ⟨φ, X̃_{s}^{n} − X̃_{s}^{B,n}⟩ ≤ Λ̃_{s}^{B,n} ≤ Λ̃_{t}^{B,n}
≤ λ_{t}^{B} + |Λ̃_{t}^{B,n} − λ_{t}^{B}|
≤ δ/2 + |Λ̃_{t}^{B,n} − λ_{t}^{B}|.

Now

\begin{aligned}d(φX̃_{s}^{n}, φμ_{s}) & ≤ ∥φ(X̃_{s}^{n} − X̃_{s}^{B,n})∥ + d(φX̃_{s}^{B,n}, φμ_{s}^{B}) + ∥φ(μ_{s} − μ_{s}^{B})∥ \\ & ≤ δ + d(φX̃_{s}^{B,n}, φμ_{s}^{B}) + |Λ̃_{t}^{B,n} − λ_{t}^{B}| \end{aligned}

so

ℙ(sup_{s≤t} d(φX̃_{s}^{n}, φμ_{s}) > δ) → 0

as n → ∞, as required.

In the case where μ is supported on ℕ, we can argue similarly, replacing the weak metric d by the total variation norm and replacing Proposition 4.3 by Proposition 4.4, to arrive at the desired conclusion.

Corollary 4.5. Let K, μ_{0}, and φ be as in Theorem 4.4. Assume in addition that μ_{0} is a probability measure and that X̃_{0}^{n} is the empirical distribution of a sample of size n from μ_{0}. Then, for all t < T

sup_{s≤t} d(φX̃_{s}^{n}, φμ_{s}) → 0

in probability, as n → ∞. Moreover, if μ_{0} is supported on ℕ, and if ⟨e^{αφ}, μ_{0}⟩ < ∞ for some α > 0, then, for all t < T and all δ > 0, there is a constant C < ∞, depending only on K, μ_{0}, φ, t and δ, such that for all n

ℙ(sup_{s≤t} ∥φ(X̃_{s}^{n} − μ_{s})∥ > δ) ≤ e^{−n/C}.

Proof. For general μ_{0}, it suffices to note that

d(φX̃_{0}^{n}, φμ_{0}) → 0

almost surely as n → ∞, by the strong law of large numbers, and to apply Theorem 4.4.

Suppose now that μ_{0} is supported on ℕ. We have

∥φ(X̃_{0}^{n}−μ_{0})∥ ≤ 2⟨φ1_{(N,∞)}, μ_{0}⟩+|⟨φ1_{(N,∞)}, X̃_{0}^{n}−μ_{0}⟩|+⟨φ1_{(0,N]}, |X̃_{0}^{n}−μ_{0}|⟩ = I_{1}+I_{2}+I_{3}.

We can choose N so that I_{1} ≤ δ_{0}/2. Then by standard exponential estimates we can find C < ∞, depending on μ_{0}, φ, N and δ_{0}, such that

ℙ(I_{2} + I_{3} > δ_{0}/2) ≤ e^{−n/C}.

On combining this estimate with that found in Theorem 4.4, we deduce

ℙ(sup_{s≤t} ∥φ(X̃_{s}^{n} − μ_{s})∥ > δ) ≤ 2e^{−n/C}

as required.

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