A note on weak convergence of the $n$-point motions of Harris flows

TL;DR

This paper extends previous results on the weak convergence of Harris flows' n-point motions to the Arratia flow, specifically when the covariance functions converge to a measure-zero support, broadening the understanding of these stochastic processes.

Contribution

It generalizes the weak convergence results of Harris flows to cases with covariance functions supported on measure-zero sets, expanding the applicability of previous theorems.

Findings

Weak convergence established for Harris flows with covariance functions supported on measure-zero sets.

Extension of convergence results to more general covariance functions.

Provides theoretical foundation for analyzing Harris flows with singular covariance structures.

Abstract

In this note we extend the main results of [2] and [8], which concern the weak convergence of the $n$-point motions of smooth Harris flows to those of the Arratia flow, to the case when the covariance functions of these Harris flows converge pointwise to a covariance function whose support is of zero Lebesgue measure.

Equations

$${⟨x(u,\cdot ),x(v,\cdot )⟩}_t=\underset{0}{\overset{t}{\int }}\Gamma (x(u,s)-x(v,s))ds,t⩾0.$$

$$\Gamma (0)=1,$$

$$x(u,t)=u+\underset{0}{\overset{t}{\int }}\underset{\mathbb{R}}{\int }\phi (x(u,s)-q)W(dq,ds),t⩾0,$$

$$\Gamma (z):=\underset{\mathbb{R}}{\int }\phi (z+q)\phi (q)dq,z\in \mathbb{R}.$$

$${\phi }_{\epsilon }(q):=\frac{1}{\sqrt{\epsilon }}\phi \left(\frac{q}{\epsilon }\right),q\in \mathbb{R},$$

$$(x_{\epsilon }(u_1,\cdot ),\dots ,x_{\epsilon }(u_n,\cdot ))\stackrel{w}{⟶}(x_0(u_1,\cdot ),\dots ,x_0(u_n,\cdot )),\epsilon \to 0+,$$

$$\forall z\in \mathbb{R}:{\Gamma }_{\epsilon }(z)⟶1{\mathrm{I}}_{\{0\}}(z),\epsilon \to 0+,$$

$${\phi }_{\epsilon }^2⟶{\delta }_0,\epsilon \to 0+,$$

$${\phi }_{\epsilon }(q):=\frac{\sqrt{\alpha }}{\sqrt{\epsilon }}\phi \left(\frac{q-a_1}{\epsilon }\right)+\frac{\sqrt{\beta }}{\sqrt{\epsilon }}\phi \left(\frac{q-a_2}{\epsilon }\right),q\in \mathbb{R},$$

$$(x_{\epsilon }(u_1,\cdot ),\dots ,x_{\epsilon }(u_n,\cdot ))\stackrel{w}{⟶}(x_0(u_1,\cdot ),\dots ,x_0(u_n,\cdot )),\epsilon \to 0+,$$

$$\forall z\in \mathbb{R}:{\Gamma }_{\epsilon }(z)⟶\sqrt{\alpha \beta }\cdot 1{\mathrm{I}}_{\{-b\}}(z)+1{\mathrm{I}}_{\{0\}}(z)+\sqrt{\alpha \beta }\cdot 1{\mathrm{I}}_{\{+b\}}(z),\epsilon \to 0+,$$

$${\phi }_{\epsilon }^2⟶\alpha {\delta }_{a_1}+\beta {\delta }_{a_2},\epsilon \to 0+,$$

$${\nu }^2(\Delta )>0,$$

$${\nu }^2:=\nu \otimes \nu$$

$$\Delta :=\{\vec{q}=(q_1,q_2)\in {\mathbb{R}}^2\mid q_1=q_2\}.$$

$${\psi }_{\epsilon }(z):=c_{\epsilon }\underset{\mathbb{R}}{\int }{\phi }_{\epsilon }(z-q)\nu (dq),z\in \mathbb{R},$$

$$\underset{\mathbb{R}}{\int }{\psi }_{\epsilon }^2(z)dz=1.$$

$$c_{\epsilon }={\left[\underset{{\mathbb{R}}^2}{\iint }{\Phi }_{\epsilon }(q_1-q_2){\nu }^2(d{q}_{1}d{q}_2)\right]}^{-1/2},$$

$${\Phi }_{\epsilon }(z):=\underset{\mathbb{R}}{\int }{\phi }_{\epsilon }(z+q){\phi }_{\epsilon }(q)dq,z\in \mathbb{R},$$

$${\psi }_{\epsilon }\in C^{\infty }(\mathbb{R}).$$

$${\Gamma }_{\epsilon }(z):=\underset{\mathbb{R}}{\int }{\psi }_{\epsilon }(z+q){\psi }_{\epsilon }(q)dq,z\in \mathbb{R}.$$

$$(x_{\epsilon }(u_1,\cdot ),\dots ,x_{\epsilon }(u_n,\cdot ))\stackrel{w}{⟶}(x_0(u_1,\cdot ),\dots ,x_0(u_n,\cdot )),\epsilon \to 0+,$$

$${\nu }_0(A):=\frac{\sum _{k:a_k\in A}(\nu (\{a_k\}){)}^2}{\sum _k(\nu (\{a_k\}){)}^2},A\in \mathcal{B}(\mathbb{R}),$$

$$\underset{\epsilon \to 0+}{\lim }\underset{\mathbb{R}}{\int }f(z){\psi }_{\epsilon }^2(z)dz=\underset{\mathbb{R}}{\int }f(z){\nu }_0(dz).$$

$$\underset{\epsilon \to 0+}{\lim }\underset{\mathbb{R}}{\int }f(z){\psi }_{\epsilon }^2(z)dz=$$

$$=\underset{\epsilon \to 0+}{\lim }\left[c_{\epsilon }^2\underset{{\mathbb{R}}^2}{\iint }\left(\underset{\mathbb{R}}{\int }f(z){\phi }_{\epsilon }(z-q_1){\phi }_{\epsilon }(z-q_2)dz\right){\nu }^2(d{q}_{1}d{q}_2)\right].$$

$$\underset{\epsilon \to 0+}{\lim }{c}_{\epsilon }^2={\left[\underset{{\mathbb{R}}^2}{\iint }\left(\underset{\epsilon \to 0+}{\lim }{\Phi }_{\epsilon }(q_1-q_2)\right){\nu }^2(d{q}_{1}d{q}_2)\right]}^{-1}=\frac{1}{{\nu }^2(\Delta )}.$$

$$\left|\underset{\mathbb{R}}{\int }f(z){\phi }_{\epsilon }(z-q_1){\phi }_{\epsilon }(z-q_2)dz\right|⩽{\Vert f\Vert }_{\infty }\cdot {\Phi }_{\epsilon }(q_1-q_2)⩽{\Vert f\Vert }_{\infty }<+\infty ,$$

$${\Vert f\Vert }_{\infty }:=\underset{z\in \mathbb{R}}{\max }\left|f(z)\right|,$$

$$\underset{\epsilon \to 0+}{\lim }\underset{\mathbb{R}}{\int }f(z){\phi }_{\epsilon }(z-q_1){\phi }_{\epsilon }(z-q_2)dz=f(q_1)1\mathrm{I}\{q_1=q_2\}.$$

$$\frac{1}{{\nu }^2(\Delta )}\underset{{\mathbb{R}}^2}{\iint }f(q_1)1\mathrm{I}\{q_1=q_2\}{\nu }^2(d{q}_{1}d{q}_2)=\underset{\mathbb{R}}{\int }f(z){\nu }_0(dz).\text{qed}$$

$${\Delta }_z:=\{\vec{q}=(q_1,q_2)\in {\mathbb{R}}^2\mid q_1-q_2=z\},z\in \mathbb{R}.$$

Full text

The main aim of this note is to generalize the results of [2] and [8] concerning the weak convergence of the $n$-point motions of Harris flows.

We begin by recalling the definition of a Harris flow (e. g., see [3, Definition 1.2]).

1. (i)

for any $u\in\mathbb{R}$ the stochastic process $\{x(u,t),\;t\geqslant 0\}$ is a Brownian motion with respect to the common filtration $(\mathcal{F}_{t}:=\sigma\{x(v,s),\;v\in\mathbb{R},\;0\leqslant s\leqslant t\})_{t\geqslant 0}$ such that $x(u,0)=u$; 2. (ii)

for any $u,v\in\mathbb{R}$, if $u\leqslant v$, then $x(u,t)\leqslant x(v,t)$ for all $t\geqslant 0$; 3. (iii)

for any $u,v\in\mathbb{R}$ the joint quadratic variation of the martingales $\{x(u,t),\;t\geqslant 0\}$ and $\{x(v,t),\;t\geqslant 0\}$ is given by

[TABLE]

Note that the function $\Gamma$ is necessarily non-negative definite and, in particular, symmetric. Besides, without loss of generality we always assume that

[TABLE]

so that the one-point motions of Harris flows we consider are standard Brownian motions.

The existence of random fields satisfying the above conditions (i), (ii) and (iii) under mild assumptions on the covariance function was proved in [4].

A Harris flow with covariance function $\Gamma=1\!\!\,\mathrm{I}_{\{0\}}$ is called the Arratia flow (here $1\!\!\,\mathrm{I}_{A}(z)\equiv 1\!\!\,\mathrm{I}\{z\in A\}$ stands for the indicator function of the set $A$). It is one of the first examples of Harris flows and was initially constructed in [1] as the weak limit of a family of coalescing simple random walks. Throughout this paper the Arratia flow will be denoted by $\{x_{0}(u,t),\;u\in\mathbb{R},\;t\geqslant 0\}$.

It is convenient to construct Harris flows with a smooth covariance function as solutions of stochastic differential or integral equations. To be more precise, let us consider the following stochastic integral equation:

[TABLE]

where $u\in\mathbb{R}$ plays the role of a parameter, $W$ is a Wiener sheet on $\mathbb{R}\times[0;+\infty)$ and the function $\varphi\in C_{0}^{\infty}(\mathbb{R},[0;+\infty))$ (i. e. infinitely differentiable and with compact support) is symmetric and has a unit $L_{2}$-norm.

It is known [2] that under these conditions on the function $\varphi$ this equation has a unique strong solution for every $u\in\mathbb{R}$ and the random field $\{x(u,t),\;u\in\mathbb{R},\;t\geqslant 0\}$ is a Harris flow with covariance function $\Gamma$ given by

[TABLE]

Now we can formulate the main results of [2] and [8]. Although these results were proved for the case of the finite time interval $[0;1]$, their proofs remain valid for the more general case of the infinite time interval $[0;+\infty)$ and it is in this form that we formulate them below.

[TABLE]

and let $\{x_{\varepsilon}(u,t),\;u\in\mathbb{R},\;t\geqslant 0\}$ be the Harris flow formed by the solutions of the stochastic integral equation (1) with $\varphi_{\varepsilon}$ instead of $\varphi$. Then for any $n\in\mathbb{N}$ and for any $u_{1},\ldots,u_{n}\in\mathbb{R}$ the weak convergence

[TABLE]

takes place in the space $C([0;+\infty),\mathbb{R}^{n})$.

Note that in this case for the covariance function $\Gamma_{\varepsilon}$ of the Harris flow $x_{\varepsilon}$ we have

[TABLE]

and also

[TABLE]

in the sense of generalized functions (here and below $\delta_{a}$ stands for the delta function at point $a\in\mathbb{R}$).

In [8] it was shown that the assertion of this theorem still holds true even if $\varphi_{\varepsilon}^{2}$ converges to a generalized function distinct from $\delta_{0}$.

[TABLE]

where $0<\alpha,\beta<1$, $\alpha+\beta=1$, and $a_{1}<a_{2}$, and let $\{x_{\varepsilon}(u,t),\;u\in\mathbb{R},\;t\geqslant 0\}$ be the Harris flow formed by the solutions of the stochastic integral equation (1) with $\varphi_{\varepsilon}$ instead of $\varphi$. Then for any $n\in\mathbb{N}$ and for any $u_{1},\ldots,u_{n}\in\mathbb{R}$ the weak convergence

[TABLE]

takes place in the space $C([0;+\infty),\mathbb{R}^{n})$.

Note that in this case for the covariance function $\Gamma_{\varepsilon}$ of the Harris flow $x_{\varepsilon}$ we have

[TABLE]

where $b:=a_{2}-a_{1}$, and also

[TABLE]

in the sense of generalized functions.

Here we show that the proof presented in [8] can be extended to the case when the right-hand side of (4) is replaced by a discrete probability measure on the real line satisfying some mild conditions. To be more precise, let $\nu$ be an arbitrary finite singular measure on the real line having at least one atom, i. e. such that

[TABLE]

where

[TABLE]

Suppose that the function $\varphi$ considered above is additionally non-decreasing on $(-\infty;0]$ and is non-increasing on $[0;+\infty)$ and that $\varphi_{\varepsilon}$ is defined by (2). Let us set

[TABLE]

where the constant $c_{\varepsilon}>0$ is chosen to be such that

[TABLE]

It is clear that

[TABLE]

and also

[TABLE]

For $\varepsilon>0$ let $\{x_{\varepsilon}(u,t),\;u\in\mathbb{R},\;t\geqslant 0\}$ be the Harris flow formed by the solutions of the stochastic integral equation (1) with $\psi_{\varepsilon}$ instead of $\varphi$. The covariance functions of these Harris flows are given by

[TABLE]

The main result of this note is the following theorem.

[TABLE]

takes place in the space $C([0;+\infty),\mathbb{R}^{n})$.

Following [8] we divide the proof into several lemmas. We repeat the considerations of [8], where necessary, as concisely as possible and omit the proofs which are similar to those of that paper. The main difference lies in the proof of Lemma 10, since the idea used in the proof of its analogue [8, Lemma 6] cannot be applied to our case. Our proof of Lemma 10 is based on additional Lemmas 6 and 9.

Before proceeding to the proof of the main result, however, we prove an analogue of relations (3) and (4). To formulate it, let $\nu_{0}$ be a discrete probability measure on the real line defined by

[TABLE]

where $\{a_{k},\;k\geqslant 1\}$ are the atoms of the measure $\nu$ and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-field of the real line.

[TABLE]

[TABLE]

However, by the dominated convergence theorem

[TABLE]

Moreover, since for any $q_{1},q_{2}\in\mathbb{R}$ we have

[TABLE]

where

[TABLE]

by the same theorem

[TABLE]

It remains to note that

[TABLE]

Now let us set

[TABLE]

Then it is easy to see that for every $z\in\mathbb{R}$ we have

[TABLE]

where the function $\Gamma_{0}$ is given by

[TABLE]

Moreover, the set

[TABLE]

is countable, since the family $\{\Delta_{z},\;z\in\mathbb{R}\}$ is a partition of $\mathbb{R}^{2}$ and $\nu^{2}(\mathbb{R}^{2})<+\infty$.

[TABLE]

[TABLE]

and that by the dominated convergence theorem the last expression converges to zero as $\left|z\right|\to+\infty$.

Now suppose that (7) is false, i. e. that there exists some $\delta_{0}>0$ such that

[TABLE]

It means, in particular, that we can find some $z_{1}\geqslant\delta_{0}$ such that

[TABLE]

Since the function $\Gamma_{0}$ is non-negative definite and $\Gamma_{0}(0)=1$, we have (e. g., see [6, p. 22])

[TABLE]

and, in particular, for any $z\in\mathbb{R}$

[TABLE]

Using (8) and (9) and the symmetry of $\Gamma_{0}$ we can choose $z_{2}\geqslant\delta_{0}$ such that

[TABLE]

Proceeding further in this way we obtain a sequence $\{z_{n}\}_{n=1}^{\infty}$ such that

[TABLE]

which contradicts (6). ∎

Now fix arbitrary $n\in\mathbb{N}$ and $u_{1},\ldots,u_{n}\in\mathbb{R}$, $u_{1}<\ldots<u_{n}$, and consider the family

[TABLE]

of random elements in the space $C([0;+\infty),\mathbb{R}^{n})$ endowed with the distance

[TABLE]

Since all stochastic processes $\{x_{\varepsilon}(u_{i},t),\;t\geqslant 0\}$, $1\leqslant i\leqslant n$, are Wiener processes, thus having the same distribution in the complete separable metric space $C([0;+\infty),\mathbb{R})$, using Prohorov’s theorem one can easily show that the family $\{\vec{x}_{\varepsilon},\;\varepsilon>0\}$ is weakly relatively compact. Let $\vec{x}=(x(u_{1},\cdot),\ldots,x(u_{n},\cdot))$ be one of its limit points (as $\varepsilon\to 0+$).

[TABLE]

[TABLE]

where

[TABLE]

As in the proof of [8, Lemma 3] one can show that the family $\{\vec{x}_{\varepsilon}^{(ij)},\;\varepsilon>0\}$ is weakly relatively compact and that, if

[TABLE]

in the space $C([0;+\infty),\mathbb{R}^{n+1})$ for some sequence $\{\varepsilon_{n}\}_{n=1}^{\infty}$ strictly decreasing to zero, with $\vec{x}^{(ij)}:=(x(u_{1},\cdot),\ldots,x(u_{n},\cdot),\theta^{(ij)})$, then

[TABLE]

Now, since the set

[TABLE]

where $0\leqslant s\leqslant t<+\infty$ and

[TABLE]

with $\delta>0$, is closed and

[TABLE]

for $\varepsilon<\delta$, we obtain that

[TABLE]

Thus, for every $\delta>0$ with probability one

[TABLE]

and so with probability one

[TABLE]

The lemma is proved. ∎

[TABLE]

[TABLE]

is similar to the proof of [7, Lemma 1]. Namely, we note (e. g., see [5, Theorem 18.4]) that with probability one the following representation takes place:

[TABLE]

where $\{\beta(t),\;t\geqslant 0\}$ is a standard Wiener process (maybe defined on an extended probability space) and

[TABLE]

Then

[TABLE]

implies that

[TABLE]

where

[TABLE]

Therefore, there exists the limit

[TABLE]

and so, due to the continuity of $\beta$,

[TABLE]

Now suppose that

[TABLE]

Then there exists $\delta_{0}>0$ (depending on $\omega$) such that

[TABLE]

So using Lemma 8 (with $s=0$) and Lemma 6 we obtain that

[TABLE]

which contradicts the almost sure finiteness of $\overline{\tau}$. ∎

[TABLE]

where $\lambda$ is the one-dimensional Lebesgue measure and $D$ is defined in (5).

[TABLE]

From Lemma 9 it follows that $\sigma_{z}$ is finite almost surely. Also let $\tau_{z}$ be the restriction of the (random) mapping $\tau\colon[0;+\infty)\rightarrow[0;+\infty)$ defined in (11) to the set $[0;\sigma_{z}]$ and $\tau_{z}^{-1}$ be its inverse. Then using (10) we get

[TABLE]

Moreover, since the stochastic process $\{x(u_{i},t)-x(u_{j},t),\;t\geqslant 0\}$ is a non-negative (continuous) martingale, we have

[TABLE]

and so by Lemma 6

[TABLE]

Thus, we obtain that for any $s,t\in[0;\sigma_{z}]$, $s<t$, we have

[TABLE]

Therefore, for any $s,t\in[0;\tau(\sigma_{z})]$, $s<t$,

[TABLE]

This implies that the function $\tau_{z}$ is absolutely continuous and so it maps the sets of zero Lebesgue measure to the sets with the same property. Thus, from

[TABLE]

it follows that

[TABLE]

Finally, since $z\in(0;u_{i}-u_{j})$ was arbitrary and $x(u_{i},\cdot)-x(u_{j},\cdot)\geqslant 0$, we can conclude that

[TABLE]

The assertion of the lemma now follows trivially. ∎

To finish the proof of Theorem 4 (obviously, it is enough to consider the case when $u_{1}<\ldots<u_{n}$) suppose that $(x(u_{1},\cdot),\ldots,x(u_{n},\cdot))$ is one of the weak limits (as $\varepsilon\to 0+$) of the family $\{\vec{x}_{\varepsilon}=(x_{\varepsilon}(u_{1},\cdot),\ldots,x_{\varepsilon}(u_{n},\cdot)),\;\varepsilon>0\}$. Then for any $i,j\in\{1,\ldots,n\}$, $i>j$, the stochastic process $\{x(u_{i},t)-x(u_{j},t),\;t\geqslant 0\}$ is a non-negative martingale and so does not leave zero after hitting it. Since both $x(u_{i},\cdot)$ and $x(u_{j},\cdot)$ are standard Brownian motions, this implies that

[TABLE]

However, from Lemma 8 (with $s=0$) and Lemma 10 it follows that

[TABLE]

Hence

[TABLE]

Thus, we conclude that any weak limit (as $\varepsilon\to 0+$) of the family $\{\vec{x}_{\varepsilon},\;\varepsilon>0\}$ coincides in distribution with the $n$-point motion of the Arratia flow, which means that this family converges weakly to the latter.

Acknowledgements. The author is grateful to the anonymous referee for the careful reading of the paper and useful comments.