2.1. Main model

The number of particles in the system is given by $n\in\mathbb{N}_+$ , where the process (i.e. path) of each $\mathbb{R}$ -valued particle $X^{(i,n)}$ over time is denoted by

(2.2) \begin{align}&\{X^{(i,n)}_t\}_{t\in\mathbb{T}} = \{X^{(i,n)}_t \colon \forall t\in\mathbb{T} \} \quad \text{with $ X^{(i,n)}_0 = x\in\mathbb{R}$.}\end{align}

We shall note that the initial points can be generalised to random variables, and each particle can be extended to take values in $\mathbb{R}^d$ for some $d\in\mathbb{N}_+$ , which we skip in this paper for parsimony. For the rest of the paper, $\{A_t^{(n)}\}_{t\in\mathbb{T}}$ models the ensemble average

(2.3) \begin{align}A_t^{(n)} = \dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)} X^{(j,n)}_t \quad \text{for all $t\in\mathbb{T}$,}\end{align}

with the conditions

\begin{align*}|\beta^{(i,n)}| <\infty \quad \text{and} \quad \sum_{i=1}^n \beta^{(i,n)} = n,\end{align*}

which dynamically quantifies the weighted empirical average state of the particle system. If $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ , then $A_t^{(n)}$ is the standard simple ensemble average for any $t\in\mathbb{T}$ . We work on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t \leq \infty},\mathbb{P})$ , where a time-shifted $(\mathbb{P},\{\mathcal{F}_{t}\})$ -Brownian motion is denoted by

(2.4) \begin{align}&\{W^{(i,k)}_t\}_{ t \in\mathbb{T}} = \{W^{(i,k)}_t \colon \forall t \in \mathbb{T} \} \quad \text{with} \ W^{(i,k)}_s = 0 \quad \text{for all $s \leq T^{(k)}_{\mathrm{start}}$,}\end{align}

for all $i\in\mathcal{I}=\{1,\ldots,n\}$ . We can collect every $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ in the $\mathbb{R}^{n\times m}$ -valued $\{\boldsymbol{W}_t\}_{t\in\mathbb{T}}$ given by

\begin{equation*}\boldsymbol{W}_t =\begin{bmatrix}W^{(1,1)}_t, \ldots, W^{(1,k)}_t, \ldots, W^{(1,m)}_t \\\vdots \hspace{0.5in} \vdots \hspace{0.5in} \vdots \\W^{(i,1)}_t, \ldots, W^{(i,k)}_t, \ldots, W^{(i,m)}_t \\\vdots \hspace{0.5in} \vdots \hspace{0.5in} \vdots \\W^{(n,1)}_t, \ldots, W^{(n,k)}_t, \ldots, W^{(n,m)}_t\end{bmatrix}\!.\end{equation*}

We assume that each $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ is mutually independent across $i\in\mathcal{I}$ for a fixed $k\in\mathcal{K}$ , i.e. across the rows of $\{\boldsymbol{W}_t\}_{t\in\mathbb{T}}$ . On the other hand, unless stated otherwise, $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ does not have to be mutually independent across $k\in\mathcal{K}$ for a fixed $i\in\mathcal{I}$ , i.e. across the columns of $\{\boldsymbol{W}_t\}_{t\in\mathbb{T}}$ . We are now in a position to construct a stochastic system of interacting particles pinned sequentially to $A_t^{(n)}$ at every $t\in\mathbb{T}^{\mathrm{end}}$ . We shall first propose the following system of SDEs and later specify certain conditions on it for our purposes:

(2.5) \begin{align}{\mathrm{d}} X^{(i,n)}_t = f^{(k)}(t)\bigl(A^{(n)}_t - X^{(i,n)}_t \bigr)\,{\mathrm{d}} t + \sigma^{(k)}_t\Bigl(\rho^{(k)} \,{\mathrm{d}} B_t^{(k)} + \sqrt{1 -(\rho^{(k)})^2}\,{\mathrm{d}} W^{(i,k)}_t\Bigr),\end{align}

for $t\in\mathbb{T}^{(k)}$ , $ k\in\mathcal{K}$ , and

(2.6) \begin{align}X^{(i,n)}_{e} = A_{T^{(k-1)}_{\mathrm{end}}}^{(n)} \quad \text{for all}\ e\in\bigl[T^{(k-1)}_{\mathrm{end}},T^{(k)}_{\mathrm{start}}\bigr],\ k\in\mathcal{K},\end{align}

for all $i\in\mathcal{I}$ with $T^{(0)}_{\mathrm{end}}=T^{(1)}_{\mathrm{start}}=0$ , where

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\int_{T^{(k)}_{\mathrm{start}}}^t\bigl(\sigma^{(k)}_s\bigr)^2\mathrm{d} s \lt \infty.\end{align*}

Here $\rho^{(k)}\in[-1,1]$ , $\{B_t^{(k)}\}_{t\in\mathbb{T}}$ is a mutually independent time-shifted $(\mathbb{P},\{\mathcal{F}_{t}\})$ -Brownian motion (as in (2.4)) that represents common noise in the system, and $f^{(k)}\colon \mathbb{T}^{(k)} \rightarrow\mathbb{R}_{+}$ is a measurable map that is continuous. Note that we have

\begin{align*}&\int_{T^{(k)}_{\mathrm{start}}}^t\exp\biggl( -\int_{s}^t f^{(k)}(u)\,{\mathrm{d}} u\biggr)\,{\mathrm{d}} s < \infty \quad \text{for all $t\in\mathbb{T}^{(k)}, \ k\in\mathcal{K}$,} \end{align*}

since $f^{(k)}$ is non-negative. For parsimony, we shall set $\sigma^{(k)}_t = \sigma^{(k)}\neq 0$ for the rest of this work.

From (2.5)–(2.6), we see that each particle takes the value $A_e^{(n)}$ at each $e\in\mathbb{T}^{\mathrm{end}}\setminus\{T^{(m)}_{\mathrm{end}}\}$ , which are unknown prior to any of these time-points. Hence (2.6) may appear ill-defined at first since (2.5) defines the dynamics of $X_t^{(i,n)}$ only up to some t strictly smaller than $T^{(k)}_{\mathrm{end}}$ for $k\in\mathcal{K}$ . However, the system is in fact well-defined since we can show (see (2.10) below) that

(2.7) \begin{align}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} A^{(n)}_{t} = A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}, \end{align}

which is well-defined. This property of being able to write the limit of the ensemble average exogenously through $B^{(k)}_{T^{(k)}_{\mathrm{end}}}$ and $W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}$ is a key component for (2.5)–(2.6) to be well-defined for every $t\in\mathbb{T}$ . On the other hand, even if we can achieve a well-defined system, these processes are a priori not necessarily time-continuous $\mathbb{P}\otimes\mathrm{d} t$ everywhere for an arbitrary choice of $f^{(k)}$ , which possibly makes the system jump to $A_e^{(n)}$ at $e\in\mathbb{T}^{\mathrm{end}}\setminus\{T^{(m)}_{\mathrm{end}}\}$ in an ad hoc manner, rather than converge to these values in a continuous manner. To address this, we shall impose sufficiency conditions on $f^{(k)}$ to achieve time-continuity $\mathbb{P}\otimes\mathrm{d} t$ everywhere for the entire system with convergence to $A_e^{(n)}$ at every $e\in\mathbb{T}^{\mathrm{end}}$ (including $T^{(m)}_{\mathrm{end}}$ ) given by (2.7), which will make (2.5)–(2.6) a well-defined continuous system with consistent limits for each $t\rightarrow T^{(k)}_{\mathrm{end}}$ .

2.2. Temporal analysis

As part of Definition 2.1 below, we are now in a position to highlight the sufficiency conditions on $f^{(k)}$ to achieve time-continuity for the system; for example, $f^{(k)}$ cannot be constant – an uncommon restriction one would typically not be concerned about in the classical mean-field setting.

1. (i) $\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^{t} f^{(k)}(s)\,{\mathrm{d}} s = \infty$ ,

2. (ii) $\int_{T^{(k)}_{\mathrm{start}}}^{\tau} f^{(k)}(s)\,{\mathrm{d}} s < \infty$ for any $\tau\in\mathbb{T}^{(k)}$ .

Unless stated otherwise, $t\rightarrow T^{(k)}_{\mathrm{end}}$ should be understood as the limit from the left for $t\in\mathbb{T}^{(k)}$ .

Proposition 2.1. Let $f^{(k)}\in F^{(k)}$ for every $k\in\mathcal{K}$ . Then each $\{X^{(i,n)}_t\}_{t\in\mathbb{T}}$ is time-continuous $\mathbb{P}\otimes\mathrm{d} t$ everywhere, such that

(2.8) \begin{align}X^{(i,n)}_{e}=A^{(n)}_{e} \quad for\ all \text{ $e\in\mathbb{T}^{\mathrm{end}},\ i\in\mathcal{I}$.}\end{align}

Proof. If we write (2.5) in the integral form and use (2.2) and (2.3), we have

(2.9) \begin{align}X^{(i,n)}_t &= A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \int_{T^{(k)}_{\mathrm{start}}}^t f^{(k)}(s)\bigl(A^{(n)}_s - X^{(i,n)}_s \bigr)\,{\mathrm{d}} s + \sigma^{(k)}\Bigl(\rho^{(k)} B_t^{(k)} + \sqrt{1 -(\rho^{(k)})^2} W^{(i,k)}_t\Bigr)\end{align}

for every $t\in\mathbb{T}^{(k)}$ and $k\in\mathcal{K}$ . Then, using $\sum_{j=1}^n \beta^{(j,n)} = n$ and (2.9), we can rewrite the weighted ensemble average as

(2.10) \begin{align}A^{(n)}_t = A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_t\end{align}

for $t\in\mathbb{T}^{(k)}$ , where the time-shifted Brownian motions in (2.10) satisfy

\begin{align*}B_{T^{(k)}_{\mathrm{start}}} = 0 \quad \text{and} \quad W^{(i,k)}_{T^{(k)}_{\mathrm{start}}} = 0\end{align*}

for all $i\in\mathcal{I}$ . Hence, in solving the system of SDEs in (2.5)–(2.6), we generalise the solution in [Reference Mengütürk26, Proposition 2.1] through the ansatz maps

\begin{align*}\dfrac{1}{n}\sum_{j=1}^n W^{(j)}_t & \mapsto \dfrac{1}{\sum_{j=1}^n \beta^{(j,n)}}\sum_{j=1}^n \beta^{(j,n)}W^{(j,k)}_t, \quad \text{$t\in\mathbb{T}^{(k)}$} \\\dfrac{1}{n}\sum_{j=1}^n \int_{0}^t \dfrac{\gamma(t)}{\gamma(s)}\,{\mathrm{d}} W^{(j)}_s & \mapsto \dfrac{1}{\sum_{j=1}^n \beta^{(j,n)}}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(j,n)}\,{\mathrm{d}} W^{(j,k)}_s, \quad \text{$t\in\mathbb{T}^{(k)}$},\end{align*}

which, by verifying against (2.5)–(2.6) using Itô’s lemma, gives us the integral representation

(2.11) \begin{align}X^{(i,n)}_t &= A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}W^{(j,k)}_t\Biggr) \notag \\&\quad + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(j,n)}\,{\mathrm{d}} W^{(j,k)}_s \Biggr) \notag \\&= A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}W^{(j,k)}_t\Biggr) \notag \\&\quad + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\gamma^{(k)}(t)Y^{(i,k)}_t - \dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}\gamma^{(k)}(t)Y^{(j,k)}_t \Biggr)\end{align}

for $t\in\mathbb{T}^{(k)}$ , where $\gamma^{(k)}\colon \mathbb{T}^{(k)}\rightarrow\mathbb{R}_+$ and $\{Y^{(i,k)}_t\}_{t\in\mathbb{T}^{(k)}}$ are given by

\begin{align*}\gamma^{(k)}(t) = \exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t f^{(k)}(u)\,{\mathrm{d}} u \biggr), \quad Y^{(i,k)}_t = \int_{T^{(k)}_{\mathrm{start}}}^t \gamma^{(k)}(s)^{-1}\,{\mathrm{d}} W^{(i,k)}_s,\end{align*}

given $\sum_{j=1}^n \beta^{(j,n)} = n$ , and having the time-shift property of $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ from (2.4). Denoting the first derivative by

\begin{align*}(\gamma^{(k)})^{'}(t) = \dfrac{\,\mathrm{d} \gamma^{(k)}(t) }{\,\mathrm{d} t},\end{align*}

and applying Itô’s integration-by-parts formula as in [Reference Hildebrandt and Roelly14, Reference Li20], we obtain

\begin{align*}\,\mathrm{d}\bigl(\gamma^{(k)}(t)^{-1} W^{(i,k)}_t\bigr) = \mathrm{d} Y^{(i,k)}_t - W^{(i,k)}_t\dfrac{(\gamma^{(k)})^{'}(t)}{(\gamma^{(k)})^{2}(t)}\,\mathrm{d} t\end{align*}

for $t\in\mathbb{T}^{(k)}$ , where we have

\begin{align*}(\gamma^{(k)})^{'}(t) = -f^{(k)}(t)\exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t f^{(k)}(u)\,{\mathrm{d}} u \biggr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ . By integration, it thus follows that

(2.12) \begin{align}\gamma^{(k)}(t)^{-1} W^{(i,k)}_t &= Y^{(i,k)}_t - \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_s\dfrac{(\gamma^{(k)})^{'}(s)}{(\gamma^{(k)})^{2}(s)}\,{\mathrm{d}} s \notag \\&= Y^{(i,k)}_t + \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_s\dfrac{f^{(k)}(s)}{\gamma^{(k)}(s)}\,{\mathrm{d}} s. \end{align}

For the following, define the map $U^{(k)}\colon \mathbb{T}^{(k)}\times \mathbb{T}^{(k)}\rightarrow\mathbb{R}$ by

(2.13) \begin{align}U^{(k)}(s,t)=f^{(k)}(s)\dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}. \end{align}

If we rearrange (2.12) and multiply both sides by $\gamma(t)$ , we can write

(2.14) \begin{align}Y^{(i,k)}_t &= \gamma^{(k)}(t)^{-1}W^{(i,k)}_t - \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_s\dfrac{f^{(k)}(s)}{\gamma^{(k)}(s)}\,{\mathrm{d}} s \notag \\&\Rightarrow \quad \gamma^{(k)}(t)Y^{(i,k)}_t = W^{(i,k)}_t - \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_sU^{(k)}(s,t)\,{\mathrm{d}} s. \end{align}

From (2.13), we have

(2.15) \begin{align}\int_{T^{(k)}_{\mathrm{start}}}^\tau U^{(k)}(s,t) \,{\mathrm{d}} s &= \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(\tau)} - \gamma^{(k)}(t), \end{align}

and since $f^{(k)}\in F^{(k)}$ for every $k\in\mathcal{K}$ , we have

(2.16) \begin{align}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \gamma^{(k)}(t) = \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t f^{(k)}(u)\,{\mathrm{d}} u \biggr) = 0 , \end{align}

which, using (2.15), implies

(2.17) \begin{align}&\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^\tau U^{(k)}(s,t) \,{\mathrm{d}} s = 0 \end{align}

for $\tau\in\mathbb{T}^{(k)}$ , given that $t_{-}\rightarrow T^{(k)}_{\mathrm{end}}$ indicates the limit from the left with $t\in\mathbb{T}^{(k)}$ ; we specify the directional notation of the limit for the purpose of this proof rather than just using $t\rightarrow T^{(k)}_{\mathrm{end}}$ . In addition,

(2.18) \begin{align}\int_{T^{(k)}_{\mathrm{start}}}^t U^{(k)}(s,t) \,{\mathrm{d}} s &= 1 - \gamma^{(k)}(t), \end{align}

and hence, using (2.16) and (2.18), we have

(2.19) \begin{align}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^t U^{(k)}(s,t) \,{\mathrm{d}} s = 1. \end{align}

From (2.17) and (2.19), the map $U^{(k)}$ is an approximation to the identity in the sense of [Reference Li20]. Accordingly, using [Reference Li20], if there is a continuous function $\lambda\colon [\rho_1,\rho_2] \rightarrow \mathbb{R}$ , with $0 \leq \rho_1 < \rho_2 < \infty$ , and a function $\Lambda\colon [\rho_1,\rho_2) \times [\rho_1,\rho_2) \rightarrow \mathbb{R}_+$ that satisfy

\begin{align*}&\lim_{t\rightarrow \rho_2} \int_{\rho_1}^t \Lambda(s,t) \,{\mathrm{d}} s = 1 \quad \text{and} \quad \lim_{t\rightarrow \rho_2} \int_{\rho_1}^{\rho^*} \Lambda(s,t) \,{\mathrm{d}} s = 0, \quad \rho^*\in[\rho_1,\rho_2),\end{align*}

it is an approximation to the identity, so that

(2.20) \begin{align}\lim_{t\rightarrow \rho_2} \biggl( \lambda(t) - \int_{\rho_1}^t \lambda(s) \Lambda(s,t) \,{\mathrm{d}} s \biggr) = 0.\end{align}

Since the time-shifted Brownian motion $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ has continuous sample paths $\mathbb{P}$ -a.s. on $\mathbb{T}$ , using (2.20), we thus have

(2.21) \begin{align}\mathbb{P}\biggl(\, \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \biggl( W^{(i,k)}_t - \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_sU^{(k)}(s,t) \,{\mathrm{d}} s \biggr) = 0 \biggr) = 1.\end{align}

Note that, using (2.14), we have

\begin{align*}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \gamma^{(k)}(t)Y^{(i,k)}_t = \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} W^{(i,k)}_t - \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^t W^{(i,k)}_sU^{(k)}(s,t)\,{\mathrm{d}} s,\end{align*}

which, by using (2.21), implies

\begin{align*}\mathbb{P}\biggl(\, \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \gamma^{(k)}(t)Y^{(i,k)}_t = 0 \biggr) = 1, \quad \mathbb{P}\Biggl( \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \dfrac{1}{n}\sum_{j=1}^n \gamma^{(k)}(t)Y^{(j,k)}_t = 0 \Biggr) = 1,\end{align*}

which further means

(2.22) \begin{align}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \Biggl( \gamma^{(k)}(t)Y^{(i,k)}_t - \dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}\gamma^{(k)}(t)Y^{(j,k)}_t\Biggr) = 0, \quad \text{$\mathbb{P}$-a.s.} \end{align}

Since affine transformations of Gaussian processes are Gaussian, $X^{(i,n)}_t$ is Gaussian for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ . Taking the limit from the left $t_{-}\rightarrow T^{(k)}_{\mathrm{end}}$ of $X^{(i,n)}_t$ in (2.11), and using (2.22),

(2.23) \begin{align}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t &= A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}\Biggr), \quad \text{$\mathbb{P}$-a.s.} \notag \\&= A^{(n)}_{T^{(k)}_{\mathrm{end}}},\end{align}

which provides $L^1$ -convergence due to the Gaussian property. On the other hand, since the limit from the right $t_{+}\rightarrow T^{(k)}_{\mathrm{end}}$ with $t > T^{(k)}_{\mathrm{end}}$ gives

\begin{align*}\lim_{t_{+}\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t = A^{(n)}_{T^{(k)}_{\mathrm{end}}},\end{align*}

using the SDE in (2.5)–(2.6), each $\{X^{(i,n)}_t\}_{t\in\mathbb{T}}$ must be time-continuous $\mathbb{P}\otimes\mathrm{d} t$ everywhere. Since the above arguments hold for every $k\in\mathcal{K}$ and $i\in\mathcal{I}$ , the equality in (2.8) holds.

Proposition 2.1 proves that solution to (2.5)–(2.6) exists for all $t\in\mathbb{T}$ , and in turn provides us with a system of particles with continuous paths, where each unit converges to the random ensemble average of the system at every $t\in\mathbb{T}^{\mathrm{end}}$ . We shall now make sense of the distribution of the ensemble average at $t\in\mathbb{T}^{\mathrm{end}}$ recursively, as we pass along $k\in\mathcal{K}$ . Let

\begin{align*}\|\boldsymbol{\beta}^{(n)}\|^{2}_{L^2} = \sum_{j=1}^n (\beta^{(j,n)})^2,\end{align*}

as the squared $L^2$ -norm of the averaging weights, and define, for every $k\in\mathcal{K}$ ,

(2.24) \begin{align}V^{(k,n)} = \sum_{l=1}^k (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)\biggl((\rho^{(l)})^2 + \dfrac{1 - (\rho^{(l)})^2}{n^2}\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}\biggr).\end{align}

The results for the rest of this section require the Brownian motions in (2.5) to be mutually independent.

Proposition 2.2. Let every time-shifted Brownian motion $\{B^{(k)}_t\}_{t\in\mathbb{T}}$ and $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ in the system be mutually independent for every $k\in\mathcal{K}$ and $i\in\mathcal{I}$ . Then

\begin{align*}A^{(n)}_{T^{(k)}_{\mathrm{end}}} \sim \mathcal{N}(x, V^{(k,n)} ),\end{align*}

given that $\mathcal{N}(.)$ stands for the Gaussian distribution.

Proof. We prove this result recursively. As a start, fix $k=1$ as the base case. Using (2.10), we have

(2.25)

where $Z^{(1)}_{T^{(1)}_{\mathrm{end}}}$ is a Gaussian random variable such that

\begin{align*}Z^{(1)}_{T^{(1)}_{\mathrm{end}}} &\sim \mathcal{N}\bigl(0, T^{(1)}_{\mathrm{end}} - T^{(1)}_{\mathrm{start}}\bigr) = \mathcal{N}\bigl(0, T^{(1)}_{\mathrm{end}} \bigr).\end{align*}

Now fix $k=2$ . From (2.5)–(2.6), we know that

(2.26) \begin{align}A^{(n)}_{T^{(1)}_{\mathrm{end}}}=A^{(n)}_{T^{(2)}_{\mathrm{start}}}, \end{align}

even when $T^{(1)}_{\mathrm{end}} \neq T^{(2)}_{\mathrm{start}}$ . We can now evaluate the distribution of $A^{(n)}_{T^{(2)}_{\mathrm{end}}}$ using (2.10), (2.25), and (2.26) as follows:

(2.27)

since all Brownian motions in the system are mutually independent, including the ones across the columns of $\{\boldsymbol{W}_t\}_{t\in\mathbb{T}}$ . We can see that for any $k>2$ the additive structure to the variance of the Gaussian distribution continues, and the result follows.

The following result provides the covariance structure of the system over every time segment $\mathbb{T}^{(k)}$ . For the statement below, we shall denote

\begin{align*}\mathbf{1}_{i,j}\triangleq\mathbf{1}(i=j) \quad \text{for $i,j\in\mathcal{I}$}\end{align*}

as the indicator function. Moreover, we denote the covariance between $X^{(i,n)}_t$ and $X^{(j,n)}_t$ as

\begin{align*}C^{(i,j,n)}_t&=\mathbb{E}\bigl[X^{(i,n)}_tX^{(j,n)}_t\bigr]-\mathbb{E}\bigl[X^{(i,n)}_t\bigr]\mathbb{E}\bigl[X^{(j,n)}_t\bigr]\end{align*}

for $t\in\mathbb{T}^{(k)}$ . We are now in a position to state the following.

Proposition 2.3. Keep the conditions of Proposition 2.1 and let every time-shifted Brownian motion $\{B^{(k)}_t\}_{t\in\mathbb{T}}$ and $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ in the system be mutually independent for every $k\in\mathcal{K}$ and $i\in\mathcal{I}$ . Then

(2.28) \begin{align}& C^{(i,j,n)}_t \notag \\& = (\sigma^{(k)})^2\bigl(t - T^{(k)}_{\mathrm{start}}\bigr)\biggl((\rho^{(k)})^2 + \dfrac{(1-(\rho^{(k)})^2)\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n^2}\biggr) \notag \\& \quad + (\sigma^{(k)})^2(1-(\rho^{(k)})^2)\mathbf{1}_{i,j}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s+ V^{(k-1,n)} \notag \\& \quad + \dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)}{n}\biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s - 2\dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s\biggr) \notag \\& \quad -\dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)}{n}\biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s - \dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s\biggr)\end{align}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ , where $V^{(k,n)}$ is defined as in (2.24) and $V^{(0,n)}=0$ .

Proof. First of all, for $\mathbb{E}\bigl[X^{(i,n)}_t X^{(j,n)}_t\bigr]$ , we define the following functions:

\begin{align*}\phi^{(i,j)}_t &= \Bigl(A^{(n)}_{T^{(k)}_{\mathrm{start}}}\Bigr)^2 + 2A^{(n)}_{T^{(k)}_{\mathrm{start}}}\sigma^{(k)}\rho^{(k)} B^{(k)}_t + A^{(n)}_{T^{(k)}_{\mathrm{start}}}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{p=1}^n \beta^{(p,n)}W^{(p,k)}_t\Biggr) \\&\quad + A^{(n)}_{T^{(k)}_{\mathrm{start}}}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{p=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(p,n)}\,{\mathrm{d}} W^{(p,k)}_s \Biggr) \\&\quad +A^{(n)}_{T^{(k)}_{\mathrm{start}}}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{r=1}^n \beta^{(r,n)}W^{(r,k)}_t\Biggr) \\&\quad +A^{(n)}_{T^{(k)}_{\mathrm{start}}}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(j,k)}_s - \dfrac{1}{n}\sum_{r=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(r,n)}\,{\mathrm{d}} W^{(r,k)}_s \Biggr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ , $i,j\in\mathcal{I}$ , and

\begin{align*}&\psi^{(i,j)}_t \\ & = \bigl(\sigma^{(k)}\rho^{(k)}B^{(k)}_t\bigr)^2 + \sigma^{(k)}\rho^{(k)} B^{(k)}_t\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl( \dfrac{1}{n}\sum_{p=1}^n\beta^{(p,n)} W^{(p,k)}_t\Biggr) \\& \quad + \sigma^{(k)}\rho^{(k)} B^{(k)}_t\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{p=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \beta^{(p,n)}\dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(p,k)}_s \Biggr) \\& \quad + \sigma^{(k)}\rho^{(k)} B^{(k)}_t\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl( \dfrac{1}{n}\sum_{r=1}^n \beta^{(r,n)}W^{(r,k)}_t\Biggr) \\& \quad + \sigma^{(k)}\rho^{(k)} B^{(k)}_t\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(j,k)}_s - \dfrac{1}{n}\sum_{r=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \beta^{(r,n)}\dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(r,k)}_s \Biggr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ , $i,j\in\mathcal{I}$ , and finally

(2.29)

for $t\in\mathbb{T}^{(k)}$ , $i,j\in\mathcal{I}$ . Therefore we have

(2.30) \begin{align}X^{(i,n)}_t X^{(j,n)}_t = \phi^{(i,j)}_t + \psi^{(i,j)}_t + \kappa^{(i,j)}_t \quad \Rightarrow \quad \mathbb{E}\bigl[X^{(i,n)}_t X^{(j,n)}_t\bigr] = \mathbb{E}\bigl[\phi^{(i,j)}_t\bigr] + \mathbb{E}\bigl[\psi^{(i,j)}_t\bigr] + \mathbb{E}\bigl[\kappa^{(i,j)}_t\bigr].\end{align}

We shall now compute each expectation in (2.30). First, note that

(2.31) \begin{align}A^{(n)}_{T^{(k)}_{\mathrm{start}}} = A^{(n)}_{T^{(k-1)}_{\mathrm{start}}} + \sigma^{(k-1)}\rho^{(k-1)} B^{(k-1)}_{T^{(k-1)}_{\mathrm{end}}} + \dfrac{\sigma^{(k-1)}\sqrt{1 - (\rho^{(k-1)})^2}}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k-1)}_{T^{(k-1)}_{\mathrm{end}}}. \end{align}

Since each time-shifted Brownian motion $\{B^{(k)}_t\}_{t\in\mathbb{T}}$ and $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ in the system is mutually independent for every $k\in\mathcal{K}$ and $i\in\mathcal{I}$ , we have

(2.32)

Now note that (2.32) itself collapses to zero recursively at $\mathbb{E}\bigl[ x B^{(1)}_t\bigr] = 0$ , and hence

\begin{align*}&\mathbb{E}\Bigl[A^{(n)}_{T^{(k)}_{\mathrm{start}}}B^{(k)}_t\Bigr] = 0.\end{align*}

Following similar steps, we also have

\begin{align*}&\mathbb{E}\Biggl[A^{(n)}_{T^{(k)}_{\mathrm{start}}}\Biggl(\dfrac{1}{n}\sum_{p=1}^n \beta^{(p,n)}W^{(p,k)}_t\Biggr)\Biggr] = \mathbb{E}\Biggl[A^{(n)}_{T^{(k-1)}_{\mathrm{end}}}\Biggl(\dfrac{1}{n}\sum_{p=1}^n \beta^{(p,n)}W^{(p,k)}_t\Biggr)\Biggr] = 0.\end{align*}

Therefore we have

which follows recursively from (2.31), which can also be verified using Proposition 2.2. In addition, since

\begin{align*}\mathbb{E}\Biggl[B^{(k)}_t\Biggl( \dfrac{1}{n}\sum_{p=1}^n\beta^{(p,n)} W^{(p,k)}_t\Biggr)\Biggr] = \mathbb{E}\bigl[B^{(k)}_t\bigr]\mathbb{E}\Biggl[\Biggl( \dfrac{1}{n}\sum_{p=1}^n\beta^{(p,n)} W^{(p,k)}_t\Biggr)\Biggr] = 0,\end{align*}

and the same for similar terms in $\psi^{(i,j)}_t$ , we have

\begin{align*}\mathbb{E}\bigl[\psi^{(i,j)}_t\bigr] &= \mathbb{E}\bigl[(\sigma^{(k)}\rho^{(k)} B^{(k)}_t)^2\bigr] \\&=\bigl(\sigma^{(k)}\rho^{(k)}\bigr)^2 \bigl(t - T^{(k)}_{\mathrm{start}}\bigr).\end{align*}

To compute the third term in (2.30), we use the mutual independence of Brownian motions and Itô isometry to get the following:

\begin{align*}\sum_{p=1}^n\beta^{(p,n)}\mathbb{E}\biggl[\int_{T^{(k)}_{\mathrm{start}}}^t \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} W^{(p,k)}_s\,{\mathrm{d}} W^{(j,k)}_s\biggr]& = \beta^{(j,n)}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s , \\\sum_{p=1}^n\sum_{r=1}^n\beta^{(p,n)}\beta^{(r,n)}\mathbb{E}\biggl[ \int_{T^{(k)}_{\mathrm{start}}}^t\int_{T^{(k)}_{\mathrm{start}}}^t\dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(p,k)}_s\,{\mathrm{d}} W^{(r,k)}_s\biggr]& = \| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s.\end{align*}

We also get

\begin{align*}&\mathbb{E}\Biggl[\sum_{r=1}^n \beta^{(r,n)}\int_{T^{(k)}_{\mathrm{start}}}^t\int_{T^{(k)}_{\mathrm{start}}}^t\dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} W^{(i,k)}_s\,{\mathrm{d}} W^{(r,k)}_s\Biggr] = \beta^{(i,n)}\int_{T^{(k)}_{\mathrm{start}}}^t\dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s .\end{align*}

Thus, taking the expectation of (2.29) provides

\begin{align*}\mathbb{E}\bigl[\kappa^{(i,j)}_t\bigr] &= \dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)}{n^2}\sum_{p=1}^n (\beta^{(p,n)})^2\mathbb{E}\bigl[W^{(p,k)}_t W^{(p,k)}_t\bigr] \\&\quad + \dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)}{n}(\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s \\&\quad - 2\dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n^2}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s \\&\quad + (\sigma^{(k)})^2(1-(\rho^{(k)})^2)\mathbf{1}(i=j)\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s \\&\quad -\dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)}{n}(\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s \\&\quad + \dfrac{(\sigma^{(k)})^2(1-(\rho^{(k)})^2)\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n^2}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s,\end{align*}

where we have the following:

\begin{align*}\mathbb{E}\bigl[W^{(p,k)}_t W^{(p,k)}_t\bigr] = t - T^{(k)}_{\mathrm{start}}.\end{align*}

Finally, from (2.5)–(2.6), (2.10) and (2.11), we get

\begin{align*}A^{(n)}_{T^{(1)}_{\mathrm{start}}} = x \quad \text{and} \quad A^{(n)}_{T^{(k-1)}_{\mathrm{end}}}=A^{(n)}_{T^{(k)}_{\mathrm{start}}} \forall k\in\mathcal{K}\setminus\{1\}\quad \Rightarrow \quad \mathbb{E}\bigl[X^{(i,n)}_t\bigr]\mathbb{E}\bigl[X^{(j,n)}_t\bigr]=x^2\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ , which can also be verified from (2.27). Putting all terms together under

\begin{align*}C^{(i,j,n)}_t&=\mathbb{E}\bigl[X^{(i,n)}_tX^{(j,n)}_t\bigr]-\mathbb{E}\bigl[X^{(i,n)}_t\bigr]\mathbb{E}\bigl[X^{(j,n)}_t\bigr] =\mathbb{E}\bigl[\phi^{(i,j)}_t\bigr] + \mathbb{E}\bigl[\psi^{(i,j)}_t\bigr] + \mathbb{E}\bigl[\kappa^{(i,j)}_t\bigr] - x^2\end{align*}

provides the result for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ .

We defined $V^{(0,n)}=0$ . For compact representation, we shall always allow expressions of the form

\begin{align*}\sum_{l=1}^{0} z^{(l)} \triangleq 0 \quad \text{for any $z^{(l)}\in\mathbb{R}$.}\end{align*}

Now, as an example, where $\sigma^{(k)}=1$ , $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ and $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ , we get an interacting particle system with no common noise component, where the ensemble average is given by the equally weighted simple average. Accordingly, using Proposition 2.3, we have the covariance structure

\begin{align*}C^{(i,j,n)}_t &= \dfrac{1}{n}\sum_{l=1}^{k-1} \bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr) + \dfrac{t - T^{(k)}_{\mathrm{start}}}{n} + \mathbf{1}_{i,j}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s- \dfrac{1}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ ; we shall provide simulations of such a system in the next section due to its fundamental standing in our framework.

Proposition 2.4. Keep the conditions of Proposition 2.3. Then

\begin{align*}&\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}C^{(i,j,n)}_t = V^{(k,n)}\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ .

Proof. Since $f^{(k)}\in\mathcal{F}^{(k)}$ for all $k\in\mathcal{K}$ , we must also have

(2.33) \begin{align}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\int_{T^{(k)}_{\mathrm{start}}}^t yf^{(k)}(s) \,{\mathrm{d}} s = \infty \quad \text{and} \quad \int_{T^{(k)}_{\mathrm{start}}}^\tau yf^{(k)}(s) \,{\mathrm{d}} s < \infty \end{align}

for any $\tau\in\mathbb{T}^{(k)}$ and $1 \leq y <\infty$ . Accordingly, we introduce a scaled map g by

(2.34) \begin{align}g^{(k)}_y(t) = yf^{(k)}(t)\end{align}

for all $t\in\mathbb{T}^{(k)}$ , and define the following:

\begin{align*}\Psi^{(k)}_y(t) &\triangleq \gamma^{(k)}_y(t) = \exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t g^{(k)}_y(u)\,{\mathrm{d}} u \biggr), \\Z_t &\triangleq T^{(k)}_{\mathrm{start}} + \int_{T^{(k)}_{\mathrm{start}}}^t \Psi^{(k)}_y(s)^{-1}\,{\mathrm{d}} s.\end{align*}

We denote

\begin{align*}{\bigl(\Psi^{(k)}_y\bigr)^{'}(t) = \dfrac{\,\mathrm{d} \Psi^{(k)}_y(t) }{\,\mathrm{d} t}}\end{align*}

as the first derivative. If we apply integration by parts, then

(2.35) \begin{align}\,\mathrm{d}\biggl(\dfrac{t}{\Psi^{(k)}_y(t)}\biggr) = \mathrm{d} Z_t - t\dfrac{(\Psi^{(k)}_y)^{'}(t)}{(\Psi^{(k)}_y(t))^{2}}\,\mathrm{d} t \quad \Rightarrow \quad \dfrac{t}{\Psi^{(k)}_y(t)} = Z_t - \int_{T^{(k)}_{\mathrm{start}}}^t s\dfrac{(\Psi^{(k)}_y)^{'}(s)}{(\Psi^{(k)}_y(s))^{2}}\,\mathrm{d} s, \end{align}

which, by using (2.34), implies

\begin{align*}\dfrac{t}{\Psi^{(k)}_y(t)} = Z_t + \int_{T^{(k)}_{\mathrm{start}}}^t s\dfrac{g^{(k)}_y(s)}{\Psi^{(k)}_y(s)}\,{\mathrm{d}} s \quad \Rightarrow \quad \Psi^{(k)}_y(t)Z_t = t - \int_{T^{(k)}_{\mathrm{start}}}^t sQ^{(k)}_y(s,t)\,{\mathrm{d}} s,\end{align*}

where we defined the function $Q^{(k)}_y\colon \mathbb{T}^{(k)}\times\mathbb{T}^{(k)}\rightarrow\mathbb{R}$ as follows:

\begin{align*}Q^{(k)}_y(s,t)=g^{(k)}_y(s)\dfrac{\Psi^{(k)}_y(t)}{\Psi^{(k)}_y(s)}.\end{align*}

Taking similar steps as in Proposition 2.1, we have

(2.36) \begin{align}\int_{T^{(k)}_{\mathrm{start}}}^\tau Q^{(k)}_y(s,t) \,{\mathrm{d}} s = \dfrac{\Psi^{(k)}_y(t)}{\Psi^{(k)}_y(\tau)} - \Psi^{(k)}_y(t) \quad \Rightarrow \quad \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^\tau Q^{(k)}_y(s,t) \,{\mathrm{d}} s = 0 \end{align}

for $\tau\in\mathbb{T}^{(k)}$ , and

(2.37) \begin{align}\int_{T^{(k)}_{\mathrm{start}}}^t Q^{(k)}_y(s,t) \,{\mathrm{d}} s = 1 - \Psi^{(k)}_y(t) \quad \Rightarrow \quad \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \int_{T^{(k)}_{\mathrm{start}}}^t Q^{(k)}_y(s,t) \,{\mathrm{d}} s = 1, \end{align}

since, using (2.33)–(2.34), we have

\begin{align*}\lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \Psi^{(k)}_y(t) = \lim_{t_{-}\rightarrow T^{(k)}_{\mathrm{end}}} \exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t g^{(k)}_y(u)\,{\mathrm{d}} u \biggr) = 0.\end{align*}

From (2.36)–(2.37), $Q^{(k)}_y$ approximates the identity as in [Reference Li20], and hence, using the continuity of t and (2.35), we get the following:

(2.38) \begin{align}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \biggl( t - \int_{T^{(k)}_{\mathrm{start}}}^t sQ^{(k)}_y(s,t) \,{\mathrm{d}} s \biggr) = \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \bigl( \Psi^{(k)}_y(t)Z_t \bigr) = 0.\end{align}

Using (2.38) and setting $y=1$ , we have

(2.39) \begin{align}&\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\Psi^{(k)}_1(t)}{\Psi^{(k)}_1(s)} \,{\mathrm{d}} s - 2\dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\Psi^{(k)}_1(t)}{\Psi^{(k)}_1(s)} \,{\mathrm{d}} s\biggr) \notag \\&\ \ =\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s - 2\dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)} \,{\mathrm{d}} s\biggr) \notag \\&\ \ = 0, \end{align}

and by setting $y=2$ , we have

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \mathbf{1}_{i,j}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\Psi^{(k)}_2(t)}{\Psi^{(k)}_2(s)}\,{\mathrm{d}} s = \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \mathbf{1}_{i,j}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s = 0 \end{align*}

as well as

(2.40) \begin{align}&\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\Psi^{(k)}_2(t)}{\Psi^{(k)}_2(s)} \,{\mathrm{d}} s - \dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\Psi^{(k)}_2(t)}{\Psi^{(k)}_2(s)} \,{\mathrm{d}} s\biggr) \notag \\&\quad \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \biggl((\beta^{(i,n)}+\beta^{(j,n)})\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s - \dfrac{\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2} \,{\mathrm{d}} s\biggr) = 0. \end{align}

Using (2.39)–(2.40), and (2.28) in Proposition 2.3, we have

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}C^{(i,j,n)}_t &= (\sigma^{(k)})^2\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\bigl(t - T^{(k)}_{\mathrm{start}}\bigr)\biggl((\rho^{(k)})^2 + \dfrac{(1-(\rho^{(k)})^2)\| \boldsymbol{\beta}^{(n)}\|^{2}_{L^2}}{n^2}\biggr) + V^{(k-1,n)},\end{align*}

and the result follows.

The following corollary provides a fundamental scenario with simple ensemble average and no common noise. We omit its proof as it follows directly from Proposition 2.4.

Corollary 2.1. Keep the conditions of Proposition 2.3 with (2.28), where $\sigma^{(k)}=1$ , $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ and $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ . Then

(2.41) \begin{align}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} C^{(i,j,n)}_t &= \dfrac{1}{n}\sum_{l=1}^{k} \bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr) \end{align}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ .

From (2.41), if $n=k$ , the limiting covariance as $t\rightarrow T^{(k)}_{\mathrm{end}}$ becomes the average of the distance between the termination and revival time-points at each time segment until $T^{(k)}_{\mathrm{end}}$ .

Regarding the effect of $f^{(k)}$ on the convergence rate, we shall leave a detailed study for future research.

2.3. Spatial limits

Thus far, we have considered the behaviour of the interacting particle system over time limits. We shall also study the limit as $n\rightarrow\infty$ , that is, as the number of particles in the system grows asymptotically.

Assumption 2.1. Kolmogorov’s strong law property holds:

(2.42) \begin{align}\mathcal{K} = \sum_{j=1}^{\infty}\dfrac{(\beta^{(j)})^2}{j^2} < \infty, \quad \text{where $\beta^{(j)}=\beta^{(j,n)}$ for all $j\leq n$.}\end{align}

As an example, for the simple ensemble average in (2.3) with $\beta^{(j)}=1$ , (2.42) holds with $\mathcal{K}=\pi^2/6$ .

Proposition 2.5. Keep the conditions of Proposition 2.1 where (2.42) holds. Then

(2.43) \begin{align}\lim_{n\rightarrow\infty} X^{(i,n)}_t = \xi^{(i)}_t\end{align}

exists for every $t\in\mathbb{T}$ and $i\in\mathcal{I}$ , and $\{\xi^{(i)}_t\}_{t\in\mathbb{T}}$ is time-continuous $\mathbb{P}\otimes\mathrm{d} t$ everywhere such that

(2.44) \begin{align}\xi^{(i)}_t = x + \sum_{l=1}^{k-1}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s \biggr)\end{align}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ .

Proof. Using Kolmogorov’s strong law property in (2.42), the limit

(2.45) \begin{align}\lim_{n\rightarrow \infty}\dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_t= 0\end{align}

holds $\mathbb{P}$ -a.s., due to the strong law of large numbers. Then, using (2.5)–(2.6) and (2.10) recursively, and the $\mathbb{P}\otimes\mathrm{d} t$ everywhere time-continuity of $\{X^{(i)}_t\}_{t\in\mathbb{T}}$ , we get

(2.46) \begin{align}\lim_{n\rightarrow \infty} A^{(n)}_t &= A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t \notag \\&= A^{(n)}_{T^{(k-1)}_{\mathrm{start}}} + \sigma^{(k-1)}\rho^{(k-1)} B^{(k-1)}_{T^{(k-1)}_{\mathrm{end}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t \notag \\&\vdots \notag \\&=x + \sum_{l=1}^{k-1}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t \end{align}

for every $t\in\mathbb{T}^{(k)}$ for $k\in\mathcal{K}$ . In addition, employing Kolmogorov’s strong law property in (2.42), we further have

(2.47) \begin{align}\lim_{n\rightarrow \infty}\dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(j,n)}\,{\mathrm{d}} W^{(j,k)}_s = 0, \quad \text{$\mathbb{P}$-a.s.} \end{align}

From (2.11), (2.46), and (2.47), we have the representation in (2.44), which also proves the existence of $\{\xi^{(i)}_t\}_{t\in\mathbb{T}}$ . Finally, the $\mathbb{P}\otimes\mathrm{d} t$ everywhere time-continuity of $\{\xi^{(i)}_t\}_{t\in\mathbb{T}}$ is due to the $\mathbb{P}\otimes\mathrm{d} t$ everywhere time-continuity of $\{X^{(i)}_t\}_{t\in\mathbb{T}}$ .

The convergence in (2.43) is pointwise; we shall leave it for future research to study the convergence rate.

Proof. If $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ , we have from (2.44)

\begin{align*}\xi^{(i)}_t = x + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s \biggr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , where each $\{W^{(i)}_t\}_{t\in\mathbb{T}}$ is mutually independent across $i\in\mathcal{I}$ for $k\in\mathcal{K}$ .

We are now in a position to prove an important consistency property: the time and space limits are commutative. More specifically, the limits $t\rightarrow T^{(k)}_{\mathrm{end}}$ and $n\rightarrow \infty$ are exchangeable; that is, the limiting values are the same irrespective of the order we take these limits. In Section 4 we shall discuss the interrelation of the limits $n\rightarrow \infty$ vs. $m\rightarrow \infty$ , for which commutativity may break.

Proposition 2.6. Keep the conditions of Proposition 2.1 where (2.42) holds. Then

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} X^{(i,n)}_t &= \lim_{n\rightarrow \infty} \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t \\&= x + \sum_{l=1}^{k}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}}\end{align*}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , $\mathbb{P}$ -a.s.

Proof. Using Proposition 2.5 and (2.21), we have

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} X^{(i,n)}_t& = \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\xi^{(i)}_t \\& =x + \sum_{l=1}^{k-1}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}} + \sigma^{(k)}\rho^{(k)}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} B^{(k)}_t \\& \quad + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s \biggr) \\& = x + \sum_{l=1}^{k}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}} \end{align*}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , $\mathbb{P}$ -a.s. Exchanging the order of the limits and using (2.23) and the strong law of large numbers in (2.45) due to (2.42), we have the following:

\begin{align*}&\lim_{n\rightarrow \infty} \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t \\&\ = \lim_{n\rightarrow \infty} A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \lim_{n\rightarrow \infty}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}\Biggr) \\&\ = \lim_{n\rightarrow \infty} A^{(n)}_{T^{(k-1)}_{\mathrm{start}}} + \sigma^{(k-1)}\rho^{(k-1)} B^{(k-1)}_{T^{(k-1)}_{\mathrm{end}}} + \sigma^{(k-1)}\sqrt{1 - (\rho^{(k-1)})^2}\lim_{n\rightarrow \infty}\dfrac{1}{n}\Biggl(\sum_{j=1}^n \beta^{(j,n)} W^{(j,k-1)}_t\Biggr) \\&\ \quad +\sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \lim_{n\rightarrow \infty}\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}\Biggr) \\ &\ =x + \sum_{l=1}^{k}\sigma^{(l)}\rho^{(l)} B^{(l)}_{T^{(l)}_{\mathrm{end}}}\end{align*}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , $\mathbb{P}$ -a.s., which proves the commutativity of the double limits.

Proposition 2.7. Keep the conditions of Proposition 2.3 with (2.28), where $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ . Then the following holds:

(2.48) \begin{align}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} C^{(i,j,n)}_t &= \lim_{n\rightarrow \infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} C^{(i,j,n)}_t \notag \\&= \sum_{l=1}^{k} (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)(\rho^{(l)})^2 \end{align}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ .

Proof. First, $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ implies $\|\boldsymbol{\beta}^{(n)}\|^{2}_{L^2}=n$ . From Proposition 2.3, (2.38), and the strong law of large numbers, we have

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} C^{(i,j,n)}_t &= \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}(\sigma^{(l)})^2\bigl(t - T^{(l)}_{\mathrm{start}}\bigr)(\rho^{(l)})^2 + \sum_{l=1}^{k-1} (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)(\rho^{(l)})^2 \\&\quad + \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}(\sigma^{(k)})^2(1-(\rho^{(k)})^2)\mathbf{1}_{i,j}\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)^2}{\gamma^{(k)}(s)^2}\,{\mathrm{d}} s, \\&=\sum_{l=1}^{k} (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)(\rho^{(l)})^2\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ . Using Proposition 2.4 and the strong law of large numbers, we get

\begin{align*}\lim_{n\rightarrow \infty} \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} C^{(i,j,n)}_t &= \lim_{n\rightarrow \infty} V^{(k,n)} \\&= \sum_{l=1}^{k} (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)(\rho^{(l)})^2\end{align*}

for $t\in\mathbb{T}^{(k)}$ and $i,j\in\mathcal{I}$ .

We shall highlight an important subclass of the proposed family of interacting particles; the case $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ such that there is no common noise component in the system. The statements below follow directly from Propositions 2.6 and 2.7.

Corollary 2.3. Keep the conditions in Proposition 2.6 and let $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ . Then the following holds:

\begin{equation*}\rho^{(k)}=0 \quad \textit{for all}\ \text{$k\in\mathcal{K}$} \quad \Rightarrow \quad \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} X^{(i,n)}_t = \lim_{n\rightarrow \infty} \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t = x\end{equation*}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , $\mathbb{P}$ -a.s.

Corollary 2.4. Keep the conditions in Proposition 2.7 with (2.48), and let $\rho^{(k)}=0$ for every $k\in\mathcal{K}$ . Then the following holds:

\begin{equation*}\rho^{(k)}=0 \quad \textit{for all}\ \text{$k\in\mathcal{K}$} \quad \Rightarrow \quad \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \lim_{n\rightarrow \infty} C^{(i,j,n)}_t = \lim_{n\rightarrow \infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} C^{(i,j,n)}_t = 0\end{equation*}

for all $t\in\mathbb{T}^{(k)}$ and $i\in\mathcal{I}$ , $\mathbb{P}$ -a.s.

Corollaries 2.3 and 2.4 indicate a key property of the proposed system of interacting particles: when there is no common noise, as the size of the system grows asymptotically with $n\rightarrow \infty$ , each and every particle in the system converges to their initial value x at every termination time-point in $\mathbb{T}^{\mathrm{end}}$ . More precisely, each particle becomes an independent diffusion that converges to x at each $\mathbb{T}^{\mathrm{end}}$ , where the covariance of the system vanishes, only for the system to be reanimated back to its stochastic behaviour between $\mathbb{T}^{\mathrm{start}}$ and $\mathbb{T}^{\mathrm{end}}$ .

Thus far, we have kept our presentation fairly abstract; we shall now provide a concrete example of such a system. First, using (2.11), we combine the piecewise representation over the entire $\mathbb{T}$ using indicator functions as follows:

\begin{align*}&X^{(i,n)}_t \\&= \sum_{k\in\mathcal{K}}\Biggl(A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}W^{(j,k)}_t\Biggr)\Biggr)\mathbf{1}_k \\&\quad + \sum_{k\in\mathcal{K}}\Biggl(\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\gamma^{(k)}(t)}{\gamma^{(k)}(s)}\beta^{(j,n)}\,{\mathrm{d}} W^{(j,k)}_s \Biggr)\Biggr)\mathbf{1}_k \\&\triangleq \sum_{k\in\mathcal{K}}\Biggl(A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_t + \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\Biggl(\dfrac{1}{n}\sum_{j=1}^n \beta^{(j,n)}W^{(j,k)}_t\Biggr) + \mathcal{R}(i,k,t)\Biggr)\mathbf{1}_k \end{align*}

for $t\in\mathbb{T}$ , where $\mathbf{1}_k \triangleq \mathbf{1}(t\in\mathbb{T}^{(k)})$ is the indicator function.

Example 2.1. Let $f^{(k)}\in F^{(k)}$ be given by

(2.49) \begin{equation}f^{(k)}(t) = \dfrac{\alpha^{(k)}}{T^{(k)}_{\mathrm{end}} - t}, \quad t\in\mathbb{T}^{(k)},\end{equation}

for some $\alpha^{(k)}\in(0,\infty)$ . Then we have

\begin{align*}\gamma^{(k)}(t) &= \exp\biggl( - \int_{T^{(k)}_{\mathrm{start}}}^t \alpha^{(k)}\bigl(T^{(k)}_{\mathrm{end}} - u\bigr)^{-1}\,{\mathrm{d}} u \biggr)\\& =\exp\Biggl( -\alpha^{(k)}\log\Biggl( \dfrac{T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}} \Biggr) +\alpha^{(k)}\log\Biggl( \dfrac{T^{(k)}_{\mathrm{end}} - t}{T^{(k)}_{\mathrm{end}}} \Biggr) \Biggr) \\&= \Biggl( \dfrac{T^{(k)}_{\mathrm{end}} - t}{T^{(k)}_{\mathrm{end}}} \Biggr)^{\alpha^{(k)}}\Biggl( \dfrac{T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}} \Biggr)^{-\alpha^{(k)}}.\end{align*}

Therefore we have

\begin{align*}\mathcal{R}(i,k,t) & = \sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}\\&\quad \times\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\bigl( T^{(k)}_{\mathrm{end}} - t \bigr)^{\alpha^{(k)}}}{\bigl( T^{(k)}_{\mathrm{end}} - s \bigr)^{\alpha^{(k)}}}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\bigl( T^{(k)}_{\mathrm{end}} - t \bigr)^{\alpha^{(k)}}}{\bigl( T^{(k)}_{\mathrm{end}} - s \bigr)^{\alpha^{(k)}}}\beta^{(j,n)}\,{\mathrm{d}} W^{(j,k)}_s \Biggr).\end{align*}

In Example 2.1, if we take $n\rightarrow\infty$ and $\rho^{(k)}=0$ for all $k\in\mathcal{K}$ , each particle behaves as a collection of $\alpha$ -Wiener bridges that are continuously glued to each other across successive time segments. We shall clarify that the process $\{Q_t\}_{t\in\mathbb{T}}$ given by

\begin{align*}Q_t &= \int_0^t \dfrac{(T-t)^\alpha}{(T-s)^\alpha}\,{\mathrm{d}} W_s\end{align*}

is an $\alpha$ -Wiener bridge with initial and terminal values at zero, with $\{W_t\}_{t\in\mathbb{T}}$ a standard $(\mathbb{P},\{\mathcal{F}_{t}\})$ -Brownian motion and $\alpha\in(0,\infty)$ .

3.1. Numerical illustration

We shall provide a numerical exercise for Example 2.1 for demonstration purposes, where we choose $f^{(k)}\in F^{(k)}$ as given in (2.49) and use the Euler–Maruyama method to approximate (2.5)–(2.6) over the mesh $0 = t_0 \leq t_1 \leq \cdots \leq t_E \leq T$ for some $E\in\mathbb{N}_+$ . For parsimony, we set $\alpha^{(k)}=1$ , $\sigma^{(k)}=\sigma$ and $\rho^{(k)}=\rho$ for every $k\in\mathcal{K}$ . Finally, we let $\{\hat{X}^{(i,n)}_{t_d}\}_{t_d\in\mathbb{T}}$ represent the discretised version of $\{X^{(i,n)}_t\}_{t\in\mathbb{T}}$ for $i\in\mathcal{I}$ , where

\begin{align*}\hat{X}^{(i,n)}_{t_{d+1}} = \hat{X}^{(i,n)}_{t_d} + f(t_d)\bigl(A^{(n)}_{t_d} - \hat{X}^{(i,n)}_{t_d}\bigr)\delta + \sigma\Bigl(\rho\bigl(B_{t_{d+1}} - B_{t_d}\bigr) + \sqrt{1-\rho^2}\bigl(W^{(i,n)}_{t_{d+1}} - W^{(i,n)}_{t_d}\bigr)\Bigr)\end{align*}

for every $i\in\mathcal{I}$ , where we set $\hat{X}^{(i,n)}_{t_0} = x = 0$ , $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ , $\delta=T/E$ , $t_d = \delta d$ and $\mathcal{K} = \{1,2,3\}$ . For the simulations below, we set $\rho=0$ to exclude common noise; the case where $\rho \neq 0$ can equally be studied, but we shall omit it from this paper to focus on demonstrating how the system converges to $x = 0$ at each $\mathbb{T}^{\mathrm{end}}$ as we increase the number of particles n.

Figure 1. (a, b) $n=10$ and $n=50$ , (c, d) $n=100$ and $n=1000$ .

For the scenarios below, we set $T=3$ , $E=3000$ , and $\sigma=1$ . In addition, we set $\mathbb{T}^{\mathrm{start}}=\{0, 1, 2\}$ and $\mathbb{T}^{\mathrm{end}}=\{1, 2, 3\}$ ; hence the termination time-points coincide with the revival time-points. In Figure 1 we show samples of the system for $n=10$ , $n=50$ , $n=100$ , and $n=1000$ . At every termination time-point, the system converges to the ensemble average; that is, we get $X^{(i,n)}_{e}=A^{(n)}_{e}$ for every $e\in\mathbb{T}^{\mathrm{end}}$ and every $i\in\mathcal{I}$ , as proved in Proposition 2.1. We also see that the convergence points tend to 0 as we increase the number of particles, since we have

(3.1) \begin{align}A^{(n)}_{T^{(k)}_{\mathrm{end}}} \sim \mathcal{N}(0, V^{(k,n)} ), \quad \text{where} \quad \lim_{n \rightarrow \infty} V^{(k,n)} \rightarrow 0\end{align}

from Proposition 2.2, which means that (3.1) converges to the Dirac measure centred at 0. With a slight modification on the terminal and revival time-points, we can produce what we call adjourned coalescent mean-field interacting particles, where the system gets suspended at its ensemble average over longer time periods before it is restored back to its stochastic nature. Accordingly, in Figure 2, we set $\mathbb{T}^{\mathrm{start}}=\{0, 1.1, 2.1\}$ and $\mathbb{T}^{\mathrm{end}}=\{1, 2, 3\}$ , again for $n=10$ , $n=50$ , $n=100$ , and $n=1000$ . The system sticks to its ensemble average between $T^{(1)}_{\mathrm{end}}=1$ until $T^{(2)}_{\mathrm{start}}=1.1$ as well as between $T^{(2)}_{\mathrm{end}}=2$ until $T^{(3)}_{\mathrm{start}}=2.1$ ; this suspension is centred at 0 as $n \rightarrow \infty$ .

Figure 2. (a, b) $n=10$ and $n=50$ , (c, d) $n=100$ and $n=1000$ .

These processes can be useful in modelling multivariate systems where each particle requires a non-zero amount of time before they get reanimated; for example, for interacting particle collision systems, the environment may need to accumulate enough energy after each collision time before the system can become active again.

3.2. Changing particles across time segments

We are interested in a scenario where we incrementally add particles into the system as we move forward from one time segment $\mathbb{T}^{(k)}$ to $\mathbb{T}^{(k+1)}$ , without ever surpassing a finite number of particle inventory n in total, while ensuring a fixed confidence of convergence within a given epsilon ball. Before clarifying this further, we shall first extend our framework to account for a transformation from n to n(k), where the latter specifies the number of particles that we have over the time segment $\mathbb{T}^{(k)}$ such that

\begin{align*}n = \sum_{k=1}^m n(k), \quad n(k)>0.\end{align*}

In this spirit, we rewrite the ensemble average process as

\begin{align*}A_t^{(n(k))} = \dfrac{1}{n(k)}\sum_{j=1}^{n(k)} \beta^{(j,n(k))} X^{(j,n(k))}_t\end{align*}

for $t\in\mathbb{T}^{(k)}$ , with $|\beta^{(i,n(k))}| <\infty$ and $\sum_{i=1}^{n(k)} \beta^{(i,n(k))} = n(k)$ , and the SDE in (2.5)–(2.6) as follows:

\begin{align*}\,{\mathrm{d}} X^{(i,n(k))}_t &= f^{(k)}(t)\bigl(A^{(n(k))}_t - X^{(i,n(k))}_t \bigr)\,{\mathrm{d}} t + \sigma^{(k)}\Bigl(\rho^{(k)} \,{\mathrm{d}} B_t^{(k)} + \sqrt{1 -(\rho^{(k)})^2}\,{\mathrm{d}} W^{(i,k)}_t\Bigr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ , $k\in\mathcal{K}$ , and

\begin{align*}X^{(i,n(k))}_{e} &= A_{T^{(k-1)}_{\mathrm{end}}}^{(n(k-1))} \quad \text{for all $e\in\bigl[T^{(k-1)}_{\mathrm{end}},T^{(k)}_{\mathrm{start}}\bigr]$, $k\in\mathcal{K}$,}\end{align*}

for all $i\in\mathcal{I}(k)=\{1,\ldots,n(k)\}$ , with $T^{(0)}_{\mathrm{end}}=T^{(1)}_{\mathrm{start}}=0$ and $n(0)=n(1)$ . We demonstrate how this interacting particle system behaves under different n(k) sequences, when we choose $f^{(k)}\in F^{(k)}$ as in (2.49) as before. We set the initial value $\hat{X}^{(i,n)}_{t_0} = x = 0$ as before. In Figure 3, we see examples of different particle numbers across different time segments. Figure 3(a) (resp. Figure 3(b)) represents an interacting system that scatters into more (resp. fewer) particles after each collision. Figures 3(c) and 3(d) model situations where the number of particles changes significantly from one regime to another.

Figure 3. (a, b) [Reference Barczy and Pap2, Reference Mengütürk25, 100] and [100, Reference Mengütürk25, Reference Barczy and Pap2], (c, d) [1000, Reference Barczy and Pap2, 1000] and [Reference Barczy and Pap2, 1000, Reference Barczy and Pap2].

The following result extends Proposition 2.1 with a similar proof that we shall omit to avoid repetition. We shall note that Proposition 3.1 below provides us with a universality statement for our proposed framework.

Proposition 3.1. Let $f^{(k)}\in F^{(k)}$ for every $k\in\mathcal{K}$ . Then each $\{X^{(i,n(k))}_t\}_{t\in\mathbb{T}}$ is time-continuous $\mathbb{P}\otimes\mathrm{d} t$ everywhere, such that

\begin{align*}X^{(i,n(k))}_{e}=A^{(n(k))}_{e} \quad \textit{for all}\ e\in\mathbb{T}^{\mathrm{end}}, \ i\in\mathcal{I}(k), \ k\in\mathcal{K} ,\end{align*}

where the ensemble average at each $e\in\mathbb{T}^{\mathrm{end}}$ is given by

\begin{align*}A^{(n(k))}_{T^{(k)}_{\mathrm{end}}} = A^{(n(k))}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n(k)}\sum_{j=1}^{n(k)} \beta^{(j,n(k))} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}.\end{align*}

By virtue of Proposition 3.1, we can now take the space limits $n(k)\rightarrow\infty$ only over chosen time segments $\mathbb{T}^{(k)}$ . Surely, if we take $n(k)\rightarrow\infty$ for every $k\in\mathcal{K}$ , we recover the limiting behaviours as proved in the previous section. In essence, every result in the previous section can be generalised to account for this extended framework. For example, Proposition 2.2 becomes the following.

Proposition 3.2. Let every time-shifted Brownian motion $\{B^{(k)}_t\}_{t\in\mathbb{T}}$ and $\{W^{(i,k)}_t\}_{t\in\mathbb{T}}$ in the system be mutually independent for every $i\in\mathcal{I}(k)$ and $k\in\mathcal{K}$ . Then

\begin{align*}A^{(n(k))}_{T^{(k)}_{\mathrm{end}}} \sim \mathcal{N}\bigl(x, V^{(k,n(k))} \bigr),\end{align*}

given that $\mathcal{N}(.)$ stands for the Gaussian distribution, where

\begin{align*}V^{(k,n(k))} = \sum_{l=1}^k (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)\biggl((\rho^{(l)})^2 + \dfrac{1 - (\rho^{(l)})^2}{n(l)^2}\| \boldsymbol{\beta}^{(n(l))}\|^{2}_{L^2}\biggr).\end{align*}

In this extended model, we are specifically interested in understanding if we can construct a system where we only need to add fewer additional particles $n(k+1) - n(k)$ into the system as we move forward from $\mathbb{T}^{(k)}$ to $\mathbb{T}^{(k+1)}$ , while sustaining a fixed confidence interval that all the particles will converge within a fixed epsilon ball around the initial value x at every $T^{(k)}_{\mathrm{end}}$ for $k\in\mathcal{K}$ . That is, for $B_{\epsilon}(x)=(-\epsilon, \epsilon)$ being the epsilon ball around x, with $\epsilon > 0$ , we would like to achieve

\begin{align*}n_*(k) = \mathrm{argmin}_{n(k)}\mathbb{P}\biggl(\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n(k))}_t \in B_{\epsilon}(x)\biggr) \geq \lambda \quad \text{for $t\in\mathbb{T}^{(k)}$}, \ \text{$\lambda\in(0,1)$}.\end{align*}

In addition, defining $\delta_*^{(k)} = n_*(k+1) - n_*(k)$ , we also aim to keep the following order:

\begin{align*}\delta_*^{(1)} \geq \cdots \geq \delta_*^{(m-1)},\end{align*}

which we shall show is possible when $\sigma^{(k)}$ is inversely proportional to n(k), such that

\begin{align*}\sigma^{(l)} = \dfrac{a}{n(k)^b} \quad \text{for $l=1,\ldots,k$ and $0 < a,b < \infty$}.\end{align*}

In the analysis below; $x=0$ , $\epsilon=0.01$ , $\lambda = 0.99$ , $a=50$ , $b=1/2$ , $\rho^{(k)}=0$ , $\beta^{(i,n(k))}=1$ , $\mathbb{T}^{\mathrm{start}}=\{0, 1, 2\}$ , and $\mathbb{T}^{\mathrm{end}}=\{1, 2, 3\}$ . In Figure 4(a), for each $\mathbb{T}^{(k)}$ , we see the probabilities that the system will converge to a value between $(-0.01, 0.01)$ across different numbers of particles within that time segment, and in Figure 4(b) we see the difference between these probability curves as we move from $\mathbb{T}^{(k)}$ to $\mathbb{T}^{(k+1)}$ . As expected, we see these probabilities increase as we increase the number of particles within that $\mathbb{T}^{(k)}$ , and the rate of convergence is slower as we move from $\mathbb{T}^{(k)}$ to $\mathbb{T}^{(k+1)}$ ; we need more particles in total for the system to converge to $(-\epsilon, \epsilon)$ as k increases. However, the additional number of particles we require to reach the confidence interval of $\lambda=0.99$ reduces as we move from $\mathbb{T}^{(k)}$ to $\mathbb{T}^{(k+1)}$ , such that $\delta_*^{(1)} > \cdots > \delta_*^{(6)}$ . Essentially, the more time segments we add to the system (as we increase k), the fewer additional particles $\delta_*^{(k)}$ are required in each consecutive time segment to achieve the same confidence of convergence within $B_{\epsilon}(x)$ .

Figure 4. (a) Probability curves for $(-\epsilon, \epsilon)$ ; (b) difference of probability curves.

3.3. Number of particles vs. time segments

Thus far, we have analysed double limits across the number of particles n vs. time t, which we have shown to commute. We shall now direct our attention to the relationship between the number of particles n vs. the number of time segments m that define the cardinality of $\mathcal{K}$ . It turns out that the double limits between n and m are not necessarily commutative, where we may have

(3.2) \begin{align}\lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t \neq \lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t\end{align}

for $t\in\mathbb{T}^{(k)}$ . To show this, we set $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ , $\rho^{(k)}=0$ , $\sigma^{(k)}=\sigma>0$ for all $k\in\mathcal{K}$ , and fix the distance between each termination and revival time-points:

\begin{align*}\bigl(T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}\bigr)=\Delta > 0\end{align*}

for any $k\in\mathcal{K}$ . Then, using Proposition 2.2, we observe

\begin{align*}&\lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty} V^{(k,n)} = \lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty} \sum_{l=1}^k \sigma^2\dfrac{\Delta}{n} = \text{{undefined}}, \\&\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty} V^{(k,n)} = \lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty} \sum_{l=1}^k \sigma^2\dfrac{\Delta}{n} = 0,\end{align*}

and since from Proposition 2.1 we know that

\begin{align*}X^{(i,n)}_t \rightarrow A^{(n)}_{T^{(k)}_{\mathrm{end}}} \quad \text{as $t\rightarrow T^{(k)}_{\mathrm{end}}$} \quad \text{for $t\in\mathbb{T}^{(k)}$},\end{align*}

we have constructed an example where the non-commutativity in (3.2) manifests. We can still maintain a commutative relationship between the limits across n vs. m; that is, if we do not fix $\sigma$ and $\Delta$ as above, and construct a sequence where

\begin{align*}\lim_{k\rightarrow\infty} \sum_{l=1}^k (\sigma^{(l)})^2\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr) = c < \infty,\end{align*}

while keeping $\beta^{(i,n)}=1$ for every $i\in\mathcal{I}$ and $\rho^{(k)}=0$ for all $k\in\mathcal{K}$ , we can still achieve

\begin{align*}\lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t = \lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} X^{(i,n)}_t = 0.\end{align*}

3.4. Covariance waves

We shall show how our proposed framework gives rise to what we call covariance waves, which are continuous curves across time generated by the covariance function in (2.28), whose trajectories resemble wave-like behaviour. First, using Proposition 2.3, we have

\begin{align*}C^{(i,j,n)}_{T^{(k)}_{\mathrm{start}}} = V^{(k-1,n)} \quad \text{with $V^{(0,n)}=0$}, \ k\in K,\end{align*}

and using Proposition 2.4, we also have

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}C^{(i,j,n)}_t = V^{(k,n)} \quad \text{for $t\in \mathbb{T}^{(k)}$}, \ k\in K.\end{align*}

These results confirm that the covariance function is continuous in time, where the dependence structure at any termination point becomes the dependence structure of the revival time-point of the next time segment. Therefore we can naturally reach covariance trajectories that are pinned at each $\mathbb{T}^{\mathrm{end}}$ , which can in turn provide wave-like covariance trajectories. We demonstrate this via Example 2.1 with $\sigma^{(k)}=\sigma$ , $\rho^{(k)}=0$ , $\alpha^{(k)}=\alpha$ for all $k\in\mathcal{K}$ and $\beta^{(i,n)}=1$ for all $i\in\mathcal{I}$ , so that

\begin{align*}X^{(i,n)}_t &= \sum_{k\in\mathcal{K}}\Biggl(A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \dfrac{\sigma}{n}\sum_{j=1}^n W^{(j,k)}_t \Biggr)\mathbf{1}(t\in\mathbb{T}^{(k)}) \\&\quad + \sum_{k\in\mathcal{K}}\sigma\Biggl(\int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\bigl( T^{(k)}_{\mathrm{end}} - t \bigr)^{\alpha}}{\bigl( T^{(k)}_{\mathrm{end}} - s \bigr)^{\alpha}}\,{\mathrm{d}} W^{(i,k)}_s - \dfrac{1}{n}\sum_{j=1}^n \int_{T^{(k)}_{\mathrm{start}}}^t \dfrac{\bigl( T^{(k)}_{\mathrm{end}} - t \bigr)^{\alpha}}{\bigl( T^{(k)}_{\mathrm{end}} - s \bigr)^{\alpha}}\,{\mathrm{d}} W^{(j,k)}_s \Biggr)\mathbf{1}(t\in\mathbb{T}^{(k)})\end{align*}

for $t\in\mathbb{T}$ for $i\in\mathcal{I}$ . Using Proposition 2.3, for $\alpha \neq \frac12$ we have

\begin{align*}&C^{(i,j,n)}_t \\[3pt]& =\dfrac{\sigma^2(t-T^{(k)}_{\mathrm{start}})}{n} \\[3pt]&\quad + \mathbf{1}_{i,j}\sigma^2\!\left(\dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - t}{2\alpha -1} - \dfrac{\bigl(T^{(k)}_{\mathrm{end}} - t\bigr)^{2\alpha}}{\bigl(T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}\bigr)^{2\alpha}}\dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - T^{(k)}_{\mathrm{start}}}{2\alpha -1}\right)\\[3pt]&\quad +V^{(k-1,n)} \\[3pt]&\quad - \dfrac{\sigma^2}{n}\!\left(\dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - t}{2\alpha -1} - \dfrac{\bigl(T^{(k)}_{\mathrm{end}} - t\bigr)^{2\alpha}}{\bigl(T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}\bigr)^{2\alpha}}\dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - T^{(k)}_{\mathrm{start}}}{2\alpha -1}\right)\!,\end{align*}

and for $\alpha = \frac12$ we have

\begin{align*}C^{(i,j,n)}_t & =\dfrac{\sigma^2(t-T^{(k)}_{\mathrm{start}})}{n} \\&\quad + \mathbf{1}_{i,j}\sigma^2\Biggl(\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)\log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}\Biggr) -\bigl(t-T^{(k)}_{\mathrm{end}}\bigr) \log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}\Biggr) \Biggr) \\&\quad +V^{(k-1,n)} - \dfrac{\sigma^2}{n}\Biggl(\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)\log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}\Biggr) -\bigl(t-T^{(k)}_{\mathrm{end}}\bigr) \log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}\Biggr) \Biggr)\end{align*}

for $t\in\mathbb{T}^{(k)}$ . We shall now check the properties we expect from $C^{(i,j,n)}_t$ given above. For the case where $\alpha\neq \frac{1}{2}$ , we have the following:

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - t}{2\alpha -1} &= \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \dfrac{T^{(k)}_{\mathrm{end}}-t}{2\alpha -1} - \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\dfrac{T^{(k)}_{\mathrm{end}}\Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}}{2\alpha -1} \\&= 0.\end{align*}

For the case where $\alpha = \frac{1}{2}$ , we use L’Hôpital’s rule to get

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \bigl(t-T^{(k)}_{\mathrm{end}}\bigr)\log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}\Biggr) &= \dfrac{\log\Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)}{\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)^{-1}} =\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \dfrac{{\frac{\partial}{\partial t}}\log\Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)}{{\frac{\partial}{\partial t}}\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)^{-1}} \\&= \lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \dfrac{\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)^{-1}}{-\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)^{-2}} =\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}} \bigl(T^{(k)}_{\mathrm{end}}-t\bigr) = 0.\end{align*}

Hence, for any $\alpha\in(0,\infty)$ , we have the limit

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}C^{(i,j,n)}_t = \sum_{l=1}^{k-1} \sigma^2\dfrac{\bigl(T^{(l)}_{\mathrm{end}} - T^{(l)}_{\mathrm{start}}\bigr)}{n} + \sigma^2\dfrac{\bigl(T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}\bigr)}{n} = V^{(k,n)},\end{align*}

as expected. In addition, for any $\alpha\in(0,\infty)$ ,

\begin{align*}C^{(i,j,n)}_{T^{(k)}_{\mathrm{start}}} = V^{(k-1,n)},\end{align*}

as expected. Note that we also have, for $\alpha\neq \frac{1}{2}$ ,

\begin{align*}&\lim_{n\rightarrow\infty}C^{(i,j,n)}_t\\&\ =\mathbf{1}_{i,j}\sigma^2\!\left(\rule{0pt}{21pt}\right.\! \dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - t}{2\alpha -1} - \dfrac{\bigl(T^{(k)}_{\mathrm{end}} - t\bigr)^{2\alpha}}{\bigl(T^{(k)}_{\mathrm{end}} - T^{(k)}_{\mathrm{start}}\bigr)^{2\alpha}}\dfrac{T^{(k)}_{\mathrm{end}}\Bigl(1 - \Bigl({\frac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}}\Bigr)^{2\alpha}\Bigr) - T^{(k)}_{\mathrm{start}}}{2\alpha -1}\!\left.\rule{0pt}{21pt}\right)\! \!,\end{align*}

and for $\alpha = \frac{1}{2}$ ,

\begin{equation*}\lim_{n\rightarrow\infty}C^{(i,j,n)}_t =\mathbf{1}_{i,j}\sigma^2\Biggl(\bigl(t-T^{(k)}_{\mathrm{end}}\bigr)\log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-t}{T^{(k)}_{\mathrm{end}}}\Biggr) -\bigl(t-T^{(k)}_{\mathrm{end}}\bigr) \log\Biggl(\dfrac{T^{(k)}_{\mathrm{end}}-T^{(k)}_{\mathrm{start}}}{T^{(k)}_{\mathrm{end}}}\Biggr) \Biggr) .\end{equation*}

Finally, this leads us to the commutativity relation

\begin{align*}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}\lim_{n\rightarrow\infty}C^{(i,j,n)}_t = \lim_{n\rightarrow\infty}\lim_{t\rightarrow T^{(k)}_{\mathrm{end}}}C^{(i,j,n)}_t = 0,\end{align*}

as Proposition 2.7 provides. We shall now demonstrate how the covariance process studied above produces wave-like behaviour across time. In Figure 5 we set $\sigma=1$ , $\mathbb{T}^{\mathrm{start}}=\{0, 1, 2\}$ , and $\mathbb{T}^{\mathrm{end}}=\{1, 2, 3\}$ .

Figure 5. (a) $n=10\,000$ across different $\alpha$ , (b) $\alpha=1$ across different n.

In Figure 5 we see wave-like behaviour of covariance trajectories due to the multi-pinning property of the interacting system. In Figure 5(a) we see the trajectories for a fixed $n=10\,000$ across different $\alpha$ values that change the shape of the waves, and in Figure 5(b) we see the influence of the number of particles for a fixed $\alpha$ . From Corollary 2.2, we know that as $n\rightarrow\infty$ , the limiting particles are mutually independent, and accordingly, from Figure 5(b), we see how the end points of the covariance waves approach zero as n increases (from 20 to $10\,000$ ), as expected; also, we know this directly from Corollary 2.4. Building up on the extension given in Proposition 3.1, we can also consider different n(k) per each time segment that would influence wave magnitudes.

3.5. A view from partial differential equations

Denoting $\boldsymbol{X}^{(n)}_t = \bigl[X^{(1,n)}_t, \ldots, X^{(n,n)}_t\bigr]^{\top}$ for every $t\in\mathbb{T}$ , one can work with a system of conditional expectation problems

\begin{align*}v^{(k)}(t, \boldsymbol{x}^{(n)}) = \mathbb{E}\biggl[ \exp\biggl(-\int_t^{T^{(k)}_{\mathrm{end}}}h^{(k)}(s)\,{\mathrm{d}} s\biggr) \varsigma^{(k)}\Bigl(A^{(n)}_{T^{(k)}_{\mathrm{end}}}\Bigr) \biggm| \boldsymbol{X}_t^{(n)} = \boldsymbol{x}^{(n)} \biggr], \quad t\in\mathbb{T}^{(k)} \bigcup T^{(k)}_{\mathrm{end}}\end{align*}

for some $\varsigma^{(k)}\colon \mathbb{R}\rightarrow\mathbb{R}$ and integrable function $h^{(k)}\colon \mathbb{T}^{(k)}\rightarrow\mathbb{R}$ , that can be computed by solving the sequence of partial differential equations given by

\begin{align*}&\dfrac{\partial v^{(k)}(t,\boldsymbol{x}^{(n)})}{\partial t} - h^{(k)}(t)v^{(k)}(t,\boldsymbol{x}^{(n)}) + \sum_{i\in\mathcal{I}}\dfrac{\partial v^{(k)}(t,\boldsymbol{x}^{(n)})}{\partial x^{(i,n)}}f^{(k)}(t)\Biggl(\dfrac{1}{n}\sum_{i\in\mathcal{I}}\beta^{(i,n)}x^{(i,n)} - x^{(i,n)} \Biggr) \\&\quad + \dfrac{1}{2}(\sigma^{(k)})^2\sum_{i\in\mathcal{I}}\dfrac{\partial^2 v^{(k)}(t,\boldsymbol{x}^{(n)})}{\partial x^{(i,n)}\partial x^{(i,n)}} + \dfrac{1}{2}(\sigma^{(k)})^2(\rho^{(k)})^2\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{I}, j\neq i}\dfrac{\partial^2 v^{(k)}(t,\boldsymbol{x}^{(n)})}{\partial x^{(i,n)}\partial x^{(j,n)}} = 0,\end{align*}

where the sequence of boundary conditions are given by

\begin{align*}v^{(k)}\bigl(T^{(k)}_{\mathrm{end}},\boldsymbol{x}^{(n)}\bigr) = \varsigma^{(k)}\Biggl(\dfrac{1}{n}\sum_{i\in\mathcal{I}}\beta^{(i,n)}x^{(i,n)}\Biggr)\end{align*}

for $k\in\mathcal{K}$ . Also note that we have

\begin{align*}v^{(k)}\bigl(T^{(k)}_{\mathrm{start}}, \,.\, \bigr) = \mathbb{E}\biggl[ \exp\biggl(-\int_{T^{(k)}_{\mathrm{start}}}^{T^{(k)}_{\mathrm{end}}}h^{(k)}(s)\,{\mathrm{d}} s\biggr) \varsigma^{(k)}\Bigl(A^{(n)}_{T^{(k)}_{\mathrm{end}}}\Bigr) \biggm| A_{T^{(k-1)}_{\mathrm{end}}}^{(n)} \biggr], \quad k\in\mathcal{K}.\end{align*}

If we set $h^{(k)} = 0$ and $\varsigma^{(k)}(x) = x$ as the identity map, then

\begin{align*}v^{(k)}\bigl(T^{(k)}_{\mathrm{start}}, \,.\, \bigr) = \mathbb{E}\bigl[A^{(n)}\bigl(T^{(k)}_{\mathrm{end}}\bigr) \mid A^{(n)}\bigl(T^{(k-1)}_{\mathrm{end}}\bigr)\bigr]\end{align*}

for $k\in\mathcal{K}$ ; such computations are straightforward via the results presented in this paper.

3.6. Connection to random n-bridges

A wide family of stochastic processes called random n-bridges (RnBs) were introduced in [Reference Mengütürk and Mengütürk29] and have led to a programme of producing stochastic Schrödinger dynamics on a complex Hilbert space for explaining sequential quantum reduction of commutative observables. Although RnBs and the family of interacting particle systems studied in this paper are not the same, we shall nonetheless discuss an intriguing connection between these two, since both exhibit convergence behaviour to a set of random variables over a set of predetermined time-points. First, we recall the definition of RnBs (with notational adjustments to fit this paper) and leave any missing detail for the reader to fill in from [Reference Mengütürk and Mengütürk29].

Definition 3.1. Let $\{G^{(i)}_t\}_{t\geq0}$ be an $\mathbb{R}$ -valued càdlàg stochastic process and let

\begin{align*}\textbf{X} = [X^{(1)}, \ldots, X^{(m)}]^{\top}\end{align*}

be a vector of $\mathbb{R}$ -valued random variables $X^{(k)}\in\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$ with a joint probability $\boldsymbol{\nu}(\mathrm{d} \boldsymbol{x})$ , for $i\in\mathcal{I}$ and $k\in\mathcal{K}$ . A stochastic process $\{\xi^{(i)}_{t}\}_{t\in\mathbb{T}}$ is said to be a random n-bridge (RnB) to $\textbf{X}$ over $\mathbb{T}^{\mathrm{end}}$ if the following hold.

1. (i) for $k\in\mathcal{K}$ such that

   \begin{align*}\mathbb{P}\Bigl(\xi^{(i)}_{T^{(1)}_{\mathrm{end}}} \in \mathrm{d} x_1, \ldots, \xi^{(i)}_{T^{(k)}_{\mathrm{end}}} \in \mathrm{d} x_k, \ldots, \xi^{(i)}_{T^{(m)}_{\mathrm{end}}} \in \mathrm{d} x_m \Bigr) = \boldsymbol{\nu}(\mathrm{d} x_1,\ldots,\mathrm{d} x_k, \ldots, \mathrm{d} x_m).\end{align*}

2. (ii) For all $m(k)\in\mathbb{N}_+$ , every $T^{(k-1)}_{\mathrm{end}}<t_{k,1}<\cdots<t_{k,m(k)}<T^{(k)}_{\mathrm{end}}$ , every $(y_{k,1},\ldots,y_{k,m(k)})\in\mathbb{R}^{m(k)}$ for $k\in\mathcal{K}$ , and for all $\boldsymbol{x}$ such that $\boldsymbol{\nu}(\mathrm{d} \boldsymbol{x})>0$ , it holds that

   \begin{align*}&\mathbb{P}\Bigl(\bigl\{\xi^{(i)}_{t_{k,1}}\leq y_{k,1},\ldots,\xi^{(i)}_{t_{k,m(k)}}\leq y_{k,m(k)} \colon k\in\mathcal{K} \bigr\} \bigm| \\&\quad \quad \xi^{(i)}_{T^{(1)}_{\mathrm{end}}} =x_1, \ldots, \xi^{(i)}_{T^{(k)}_{\mathrm{end}}} =x_k, \ldots, \xi^{(i)}_{T^{(m)}_{\mathrm{end}}} =x_m \Bigr) \nonumber \\&\quad =\mathbb{P}\Bigl(\bigl\{G^{(i)}_{t_{k,1}}\leq y_{k,1},\ldots,G^{(i)}_{t_{k,m(k)}}\leq y_{k,m(k)} \colon k\in\mathcal{K} \bigr\} \bigm| \\&\quad \quad \qquad G^{(i)}_{T^{(1)}_{\mathrm{end}}} =x_1, \ldots, G^{(i)}_{T^{(k)}_{\mathrm{end}}} =x_k, \ldots, G^{(i)}_{T^{(m)}_{\mathrm{end}}} =x_m\Bigr). \nonumber\end{align*}

In this setup, $T^{(k)}_{\mathrm{end}}=T^{(k+1)}_{\mathrm{start}}$ . Note that there is no (mean-field) interaction between $\{\xi^{(i)}_{t}\}_{t\in\mathbb{T}}$ s for $i\in\mathcal{I}$ , but each $\{\xi^{(i)}_{t}\}_{t\in\mathbb{T}}$ is probabilistically conditioned to take the law of $\textbf{X}$ over $\mathbb{T}^{\mathrm{end}}$ . As an example, if we choose $\{G^{(i)}_t\}_{t\geq0}$ (and thus $\{\xi^{(i)}_{t}\}_{t\in\mathbb{T}}$ ) to be a purely continuous process and let $\mathbb{P}( X^{(k)} = x) = 1$ , we can interpret each $\{\xi^{(i)}_{t}\}_{t\in\mathbb{T}}$ to be continuously pinned to the value x at each $T^{(k)}_{\mathrm{end}}$ for $k\in\mathcal{K}$ . In fact, if we choose

\begin{align*}X^{(k)} \triangleq A^{(n)}_{T^{(k)}_{\mathrm{end}}} = A^{(n)}_{T^{(k)}_{\mathrm{start}}} + \sigma^{(k)}\rho^{(k)} B^{(k)}_{T^{(k)}_{\mathrm{end}}} + \dfrac{\sigma^{(k)}\sqrt{1 - (\rho^{(k)})^2}}{n}\sum_{j=1}^n \beta^{(j,n)} W^{(j,k)}_{T^{(k)}_{\mathrm{end}}}\end{align*}

and set $X^{(0)}=x$ , let $\{G^{(i)}_t\}_{t\geq0}$ be the càdlàg process given by

\begin{align*}G^{(i)}_t = \sum_{k\in\mathcal{K}} X^{(k-1)}\mathbf{1}\bigl(t=T^{(k)}_{\mathrm{start}}\bigr) + \sum_{k\in\mathcal{K}} W^{(i,k)}_t \mathbf{1}(t\in\mathbb{T}^{(k)}) + X^{(m)}\mathbf{1}\bigl(t=T^{(m)}_{\mathrm{end}}\bigr),\end{align*}

we reach a system of RnBs having certain similarities to the aforementioned family of processes studied in this paper – e.g. pinning points having the same distribution – but we still cannot recover the mean-field interaction property or the same system of SDEs as given in (2.5)–(2.6). However, the SDEs we would get from the above RnB would still have some functional similarities to (2.5), for which we shall leave a more detailed analysis for future research. Our foresight here stems from an observation made in [Reference Mengütürk26], where there is only a single time segment with $m=1$ , and where we see a discussion of an SDE of the form

(3.3) \begin{align}\,\mathrm{d} \xi^{(i)}_t &= \dfrac{1}{T^{(1)}_{\mathrm{end}}-t}\bigl(\mathbb{E}\bigl[ X^{(1)}\mid \mathcal{F}^{\xi^{(i)}}_t \bigr] - \xi^{(i)}_t \bigr)\,\mathrm{d} t + \,\mathrm{d} W^{(i,1)}_{t} \quad \text{for $t\in\mathbb{T}^{(1)}$} \notag \\&= \dfrac{1}{T^{(1)}_{\mathrm{end}}-t}\bigl(\mathbb{E}\bigl[X^{(1)} \mid \xi^{(i)}_t \bigr] - \xi^{(i)}_t \bigr)\,\mathrm{d} t + \,\mathrm{d} W^{(i,1)}_{t} \quad \text{for $t\in\mathbb{T}^{(1)}$}, \end{align}

with $\mathcal{F}^{\xi^{(i)}}_t = \sigma(\{\xi^{(i,n)}_s\}\colon 0\leq s \leq t)$ for the so-called Brownian random bridge $\{\xi^{(i)}_t\}_{t\in\mathbb{T}}$ for $i\in\mathcal{I}$ (see [Reference Brody and Hughston4–Reference Brody, Hughston and Macrina6]), satisfying the following anticipative representation:

(3.4) \begin{align}\xi^{(i)}_t = X^{(1)}\dfrac{t}{T^{(1)}_{\mathrm{end}}} + Z^{(i)}_{tT^{(1)}_{\mathrm{end}}},\end{align}

where each $\{Z^{(i)}_{tT}\}_{t\in\mathbb{T}}$ is a mutually independent standard Brownian bridge. From (3.4), we can see that the following hold:

\begin{align*}\xi^{(i)}_t \rightarrow X^{(1)} \quad \text{as $t\rightarrow T^{(1)}_{\mathrm{end}}$}, \quad \text{since} \quad Z^{(i)}_{tT^{(1)}_{\mathrm{end}}} \rightarrow 0 \quad \text{as $t\rightarrow T^{(1)}_{\mathrm{end}}$}.\end{align*}

The SDE in (3.3) has similarities to the SDE in (2.5) when we choose $m=1$ , $x=0$ , $\rho^{(1)}=0$ , $\sigma^{(1)}=1$ , and $f^{(1)}(t)=1/\bigl(T^{(1)}_{\mathrm{end}}-t\bigr)$ . In this specific case, the main difference between (2.5) and (3.3) remains the presence of the ensemble average $\{A_t^{(n)}\}_{t\in\mathbb{T}}$ in (2.5) that is replaced by a marginal conditional expectation process $\{\mathbb{E}[X^{(1)} \mid \xi^{(i)}_t ]\}_{t\in\mathbb{T}}$ in (3.3), which essentially loses the mean-field interaction component. As for the case of $m>1$ , any such connection will be of a more complicated nature, which deserves a separate study.