Emergence in Random Noisy Environments

TL;DR

This paper studies how different dynamical systems exhibit emergent behavior when subjected to random environmental noise, providing conditions for emergence and building on flocking models by Cucker and Smale.

Contribution

It introduces new sufficient conditions for emergence in dynamical systems under stochastic perturbations, inspired by foundational flocking models.

Findings

Conditions for nearly-emergence identified

Analysis applicable to multiple dynamical systems

Builds on and extends flocking model theories

Abstract

We investigate the emergent behavior of four types of generic dynamical systems under random environmental perturbations. Sufficient conditions for nearly-emergence in various scenarios are presented. Recent fundamental works of F. Cucker and S. Smale on the construction and analysis of flocking models directly inspired our present work.

Equations

$$I(D):\qquad\Big{\{}\begin{array}[]{l}x(t+h)=x(t)+hJ(x(t),y(t))\\ y(t+h)=S(x(t))y(t)+hH(t)\end{array}$$

$$||J(x,y)||\le C(1+||x||{)}^{\gamma }||y|{|}^{\delta }$$

$$||S(x)||\le 1-\frac{hG}{(1+||x||{)}^{\beta }}$$

$$II(D):\qquad\Big{\{}\begin{array}[]{l}x(t_{1}+h_{1})=S_{1}(y(t_{2}))x(t_{1})+h_{1}H_{1}(t_{1})\\ y(t_{2}+h_{2})=S_{2}(x(t_{1}))y(t_{2})+h_{2}H_{2}(t_{2})\end{array}$$

$$\qquad\Big{\{}\begin{array}[]{l}x[t+1]=S_{1}(y[t])x[t]+h_{1}H_{1}[t]\\ y[t+1]=S_{2}(x[t])y[t]+h_{2}H_{2}[t]\end{array}$$

$$||S_1(y)||\le 1-\frac{h_1{G}_1}{(1+||y||{)}^{{\beta }_1}},||S_2(x)||\le 1-\frac{h_2{G}_2}{(1+||x||{)}^{{\beta }_2}}$$

$$Q(\delta )=1\vee \frac{1}{\delta },a=\frac{2C}{G}Q(\delta )||y(0)|{|}^{\delta },b=1+||x(0)||$$

$$\quad B_{0}=U_{0}-1,\quad\mathcal{H}_{0}=\frac{2^{-\beta-1}G}{U_{0}^{\beta}},\quad U_{0}=\left\{\begin{array}[]{ll}\max\{(2a)^{\frac{1}{1-\gamma-\beta}},2b\},&\mathrm{if}\ \beta+\gamma<1\\ \frac{b}{1-a},&\mathrm{if}\ \beta+\gamma=1\\ \frac{(\beta+\gamma)b}{\beta+\gamma-1},&\mathrm{if}\ \beta+\gamma>1\end{array}\right.$$

$${\mathcal{H}}_1=\frac{G_1}{2(1+||y(0)||{)}^{{\beta }_1}},{\mathcal{H}}_2=\frac{G_2}{2(1+||x(0)||{)}^{{\beta }_2}}$$

$$||y[t]||=||S(x[t-1])y[t-1]+hH[t-1]||\leq\Big{(}1-\frac{hG}{(1+||x[t-1]||)^{\beta}}+h\mathcal{H}_{0}\Big{)}||y[t-1]||.$$

$$\frac{G}{(1+||x[i]||{)}^{\beta }}-{\mathcal{H}}_0\ge \frac{G}{(1+||x[t^{\ast }]||{)}^{\beta }}-{\mathcal{H}}_0\ge \frac{G}{2(1+||x{[}^{\ast }]||{)}^{\beta }}:=R(t^{\ast }).$$

$$||x[\tau ]||$$

$$1+||x[t^{\ast }]||\le 1+||x(0)||+\frac{2CQ(\delta )}{G}||y(0)|{|}^{\delta }(1+||x[t^{\ast }]||{)}^{\gamma +\beta }.$$

$$(1+||x[t]||)^{\gamma+\beta}\leq(1+||x[t^{*}]||)^{\gamma+\beta}\leq U_{0}^{\gamma+\beta}=\Big{(}\frac{2^{-\beta}G}{2\mathcal{H}_{0}}\Big{)}^{\frac{\gamma}{\beta}+1},$$

$$1+||x[\hat{t}]||$$

$$||y[t]||\leq||y(0)||\prod_{i=0}^{t-1}\big{(}1-\frac{hG}{(1+||x[i]||)^{\beta}}+h\mathcal{H}_{0}\big{)}\leq||v(0)||(1-hJ_{0})^{t}.$$

$$||x[\tau ]-x[t]||$$

$$||x[t]-\hat{x}||\leq CU_{0}^{\gamma}||v(0)||^{\delta}\frac{Q(\delta)}{J_{0}}(1-hJ_{0})^{t\delta}\leq aU_{0}^{\beta+\gamma}\Big{(}1-\frac{hG}{2U_{0}^{\beta}}\Big{)}^{\delta t}.$$

$$(1+||x[t^{*}]||)\Big{(}1-\frac{2CQ(\delta)}{G}||y(0)||^{\delta}\Big{)}-(1+||x(0)||)\leq 0.$$

$$(1+||x[t]||)^{\gamma+\beta}\leq 1+||x[t^{*}]||\leq\frac{G(1+||x(0)||)}{G-2CQ(\delta)||y(0)||^{\delta}}=\Big{(}\frac{2^{-\beta}G}{2\mathcal{H}_{0}}\Big{)}^{\frac{1}{\beta}}.$$

$$||x[t]||\le z_l-1\le z_0-1=B_0.$$

$$||x[r]-x[r-1]||\le hC(1+||x[r-1]||{)}^{\gamma }||y[r-1]|{|}^{\delta }\le hC(1+B_0{)}^{\gamma }||y(0)|{|}^{\delta },$$

$$(1+||x[t]||)^{\gamma+\beta}\leq z_{0}^{\gamma+\beta}=\Big{(}\frac{(1+||x(0)||)(\gamma+\beta)}{\gamma+\beta-1}\Big{)}^{\gamma+\beta}=\Big{(}\frac{2^{-\beta}G}{2\mathcal{H}_{0}}\Big{)}^{\frac{\gamma}{\beta}+1}$$

$$P(||H[t]||\le {\mathcal{H}}_0||y[t]||)\ge P(||H[t]||\le {\mathcal{H}}_0\nu )=F({\mathcal{H}}_0\nu ).$$

$$P(||H[t]||\le {\mathcal{H}}_0||y[t]||\mathrm{for}t=0,\cdots ,T_0-1)\ge F({\mathcal{H}}_0\nu {)}^{T_0},$$

$$P(||H[t]||\le {\mathcal{H}}_0||y[t]||\mathrm{for}t=0,\cdots ,T_0\vee T_1-1)\ge F({\mathcal{H}}_0\nu {)}^{T_0\vee T_1}.$$

$$\frac{G_1}{(1+||y[i]||{)}^{{\beta }_1}}-{\mathcal{H}}_1\ge \frac{G_1}{(1+||y[t_1^{\ast }]||{)}^{{\beta }_1}}-{\mathcal{H}}_1\ge \frac{G_1}{2(1+||y[t_1^{\ast }]||{)}^{{\beta }_1}}:=R_1(t_1^{\ast }).$$

$$||x[\tau ]||$$

$$\frac{G_2}{(1+||x[j]||{)}^{{\beta }_2}}-{\mathcal{H}}_2\ge \frac{G_2}{(1+||x[t_2^{\ast }]||{)}^{{\beta }_2}}-{\mathcal{H}}_2\ge \frac{G_2}{2(1+||x[t_2^{\ast }]||{)}^{{\beta }_2}}:=R_2(t_2^{\ast }).$$

$$||y[\tau]||\leq||y(0)||\prod_{j=0}^{\tau-1}\big{(}1-\frac{h_{2}G_{2}}{(1+||x[j]||)^{\beta_{2}}}+h_{2}\mathcal{H}_{2}\big{)}\leq||y(0)||(1-h_{2}R_{2}(t_{2}^{*}))^{\tau}.$$

Full text

Keywords: emergence; Cucker-Smale model; dynamical system; flocking; multi-agent system; consensus problems; random noise; linguistic evolution.

AMS subject classification: 34F05, 37N35, 39A11, 93C15, 93C73, 92D50, 92D40, 91C20

1. Introduction

The emergent behaviors of a large number of autonomous interacting agents such as flocking/schooling/swarming/herding in birds/fishes/bacterica/animals [20, 13, 24], multi-agent cooperative coordination in mobile networks [1, 3] and emergence of a common language in primitive societies [9, 15, 19, 14] have been attracting great research attentions since the last two decades from biologists, computer scientists, physicists, sociologists, engineers and mathematicians.

Recently, Cucker and Smale [8] have proposed a remarkable model aiming to exploring the flocking phenomenon and mathematical analysis is performed to show the convergence results only depend on some initial states of the population. This notable feature is in contrast with the previous models (e.g. the so called Vicsek model [24, 13]) where convergence relies on the global behavior of the agents’ trajectories (or on the neighborhood graphs of the underlying dynamical systems), which are quite hard to verify in general. The same authors [7] extend the model later to a more general setting beyond flocking. [21] further develops a hierarchical leadership architecture in the Cucker-Smale flocking model. The work in [5] focuses on a situation where uniform or Gaussian noises are involved in the environments. A hydrodynamic description and the mean-field limit of this very model are also provided in [10].

The starting point of our present work is directly motivated by the aforementioned series work. Primarily, we want to refine the rudimental results (in the noisy environment) in [5] and extend them to more general scenarios such as those discussed by [7]; and try to shed some light on the understanding of various emergence behaviors observed in diverse natural, social and man-made complex systems [12, 22]. To do so, we first introduce four types of non-autonomous, nonlinear dynamical systems; two ( I(D) and II(D) ) for discrete time and two ( I(C) and II(C) ) for continuous time. In each case, we provide a convergence analysis. Systems I(D) and I(C) are adapted from [7, 8] and the underlying idea stems from the birds flocking in a noisy environment. Whereas the original idea behind systems II(D) and II(C) is the linguistic evolution with some possible fluctuations in a primitive society. The random noises considered here may reflect the change of the environment which is usually unclear to the objects. Moreover, information interaction among agents may be contaminated or corrupted by errors. Hence, it becomes significant to analyze systems in the presence of random noises. We mention that the systems tackled in this paper are quintessential in the sense of reflecting some typical mechanisms behind emergence (see Remark 1 in Section 2.1), but by no means limited to flocking or language evolution since we will treat them in a quite general manner with emphasis on the methodology. Concrete examples will follow and illustrate the strength of our generic frameworks. Some other related work about emergent behaviors under random environmental perturbation can be found in e.g. [17, 16, 18, 23] and references therein.

The rest of this paper is organized as follows. In Section 2, we will study the discrete time models I(D), II(D) and the statement of main results in this section appears at Section 2.2. Section 3 is devoted to the continuous counterparts I(C), II(C) and see Section 3.2 for the statement of main results in the continuous case. We then draw our conclusion and discuss future direction in Section 4.

2. Discrete-time emergence

Let $k\in\mathbb{N}$. We assume the population under consideration consists of $k$ agents throughout the paper.

2.1 Models setup (I(D), II(D))

We shall first introduce the dynamical system I(D), which is developed similarly with that considered in [7].

Suppose $X$ and $Y$ are two given inner product spaces whose elements are denoted as $x$ and $y$, respectively. Let $x(t)=(x_{1}(t),\cdots,x_{k}(t))\in X$ and $(y_{1}(t),\cdots,y_{k}(t))\in Y^{k}$ represent two kinds of characteristics of the agents at time instant $t$. Convergence of $x\in X$ (or $y\in Y^{k}$) is naturally understood as entrywise convergence as $t$ approaches infinity. Let $\triangle$ signify the diagonal of $Y^{k}$, that is, $\triangle:=\{(y,\cdots,y)|\ y\in Y\}$. Denote $\widetilde{Y}:=Y^{k}/\triangle$ and fix an inner product $\langle\cdot,\cdot\rangle$ in $\widetilde{Y}$, which induces a norm $||\cdot||$. (Here in the discrete case, we do not really need an inner; what we want is $\widetilde{Y}$ should be a normed space. The same remark applies to $X$ and $Y$.) Since $\widetilde{Y}$ is a finite dimensional space, $\hat{y}:=(y_{1}(t),\cdots,y_{k}(t))\rightarrow(y_{0},\cdots,y_{0})$ for some $y_{0}\in Y$ if and only if $||\bar{\hat{y}}-\bar{0}||\rightarrow 0$, where $\bar{a}:=a+\triangle\in\widetilde{Y}$ for $a\in Y^{k}$. In what follows, we denote norms in all different spaces as $||\cdot||$ with some ambiguity, but the proper meaning will be clear in the context.

For $x\in X$, $y\in\widetilde{Y}$, consider the following dynamical system:

[TABLE]

Here $h$ is the time step and we shall denote in the sequel $x[t]:=x(th)$, $y[t]:=y(th)$ and $H[t]:=H(th)$ for brevity. Take $t\in\mathbb{N}$ herein. We now explain the notations in system I(D). Let $J:X\times\widetilde{Y}\rightarrow X$ be a Lipschitz or $C^{1}$ operator satisfying, for some $C,\delta>0$, $0\leq\gamma<1$, that

[TABLE]

for all $x\in X$, $y\in\widetilde{Y}$. Let $S:X\rightarrow\mathrm{End}(\widetilde{Y})$ be an operator satisfying, for some $G>0$, $\beta\geq 0$, that

[TABLE]

for all $x\in X$. The operator norm in (2) is defined as $||S(x)||=\sup_{y\neq 0\atop y\in\widetilde{Y}}\frac{||S(x)y||}{||y||}$. Let $H:(\Omega,\mathcal{F},P)\rightarrow(\widetilde{Y},\mathcal{B}(\widetilde{Y}))$ be a random element. $(\Omega,\mathcal{F},P)$ is some probability space and $\mathcal{B}(\widetilde{Y})$ is the Borel $\sigma$-algebra on $\widetilde{Y}$. We assume $H[t]$ is independent and identically distributed for different $t\in\mathbb{N}$. Notice that $||H||:(\Omega,\mathcal{F},P)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a random variable and let $F(x):=P(||H||\leq x)$ for $x\in\mathbb{R}$ be the distribution function of $||H||$.

Next, we present our dynamical system II(D) as follows. The spaces $X,Y,\widetilde{Y}$ are defined as before. For $x\in X$, $y\in\widetilde{Y}$, consider the following dynamical system:

[TABLE]

Here $h_{1},h_{2}$ are the time steps w.r.t. $x$ and $y$. Take $t_{1},t_{2}\in\mathbb{N}$ and we denote $x[t_{1}]=x(t_{1}h_{1})$, $y[t_{2}]=y(t_{2}h_{2})$, $H_{1}[t_{1}]=H_{1}(t_{1}h_{1})$ and $H_{2}[t_{2}]=H_{2}(t_{2}h_{2})$. In light of these notations, the system II(D) can be rewritten as follows

[TABLE]

In analogy with the system I(D), suppose $S_{1}:\widetilde{Y}\rightarrow\mathrm{End}(X)$ and $S_{2}:X\rightarrow\mathrm{End}(\widetilde{Y})$ are two operators satisfying, for some $G_{1},G_{2}>0$, $\beta_{1},\beta_{2}\geq 0$, that

[TABLE]

for all $y\in\widetilde{Y}$, $x\in X$. Let $H_{1}:(\Omega_{1},\mathcal{F}_{1},P_{1})\rightarrow(X,\mathcal{B}(X))$ and $H_{2}:(\Omega_{2},\mathcal{F}_{2},P_{2})\rightarrow(\widetilde{Y},\mathcal{B}(\widetilde{Y}))$ be two random elements as before. We assume $H_{1}[t]$ is independent and identically distributed for different $t\in\mathbb{N}$ and so is $H_{2}[t]$. Furthermore, $H_{1}$ is assumed to be independent with $H_{2}$. The distribution functions of random variables $||H_{1}||$ and $||H_{2}||$ are defined as $F_{1}(x):=P_{1}(||H_{1}||\leq x)$ and $F_{2}(y):=P_{2}(||H_{2}||\leq y)$ for $x,y\in\mathbb{R}$, respectively. It is worth noting that we do not ask the time scales $h_{1}$, $h_{2}$ to be the same; and the coupled system may thus work in a kind of asynchronous way.

Before going further, we give a definition for nearly-emergence that we adopt in this paper.

Definition 1.* Let $\mu,\nu>0$, $x\in X$, $y\in\widetilde{Y}$, $\nu$(or $\mu$)-nearly-emergence occurs for the population $\{1,\cdots,k\}$ if $||y||\leq\nu$(or $||x||\leq\mu$).*

Clearly, the exact emergence is no longer possible due to the random perturbation.

Remark 1.* The two features $x$ and $y$ of agents in the system I(D) are asymmetric and $y$ is the object whose emergence behavior is of interest. In the system II(D), the status of $x$ and $y$ is symmetric and both emergence behaviors may be of interest. The same can be said for the continuous case in Section 3 below.*

2.2 Main results

We define several constants that are only related with the initial state $(x(0),y(0))$ of the population.

For system I(D):

[TABLE]

[TABLE]

For system II(D):

[TABLE]

The main results in this section are stated as follows.

Theorem 1.* For dynamical system I(D), we assume h<\min\Big{\{}\frac{1}{G},\ \frac{1}{2^{1-\gamma}C||y(0)||^{\delta}}\big{(}\frac{G}{2\mathcal{H}_{0}}\big{)}^{\frac{1-\gamma}{\beta}}\Big{\}}, and one of the following hypotheses holds:

(i) $\beta+\gamma<1$,

(ii)$\beta+\gamma=1$, and ||y(0)||<\big{(}\frac{G}{2CQ(\delta)}\big{)}^{\frac{1}{\delta}},

(iii)$\beta+\gamma>1$, and \big{(}\frac{1}{a(\beta+\gamma)}\big{)}^{\frac{1}{\beta+\gamma-1}}\frac{\beta+\gamma-1}{\beta+\gamma}>b+h\big{(}(\frac{\beta+\gamma}{\beta+\gamma-1})b\big{)}^{\gamma}\frac{aG}{2Q(\delta)}.

Then, for $\nu<||y(0)||$, $\nu$-nearly-emergence occurs in a number of iterations bounded by T_{0}:=\frac{2U_{0}^{\beta}}{hG}\ln\big{(}\frac{||y(0)||}{\nu}\big{)} with probability at least $F(\mathcal{H}_{0}\nu)^{T_{0}}$. In addition, if $\mu<aU_{0}^{\beta+\gamma}$, let T_{1}:=\frac{2U_{0}^{\beta}}{\delta hG}\ln\big{(}\frac{aU_{0}^{\beta+\gamma}}{\mu}\big{)}, then the events $\{||x[t]-x[\tau]||\leq\mu,\ for\ \tau>t\geq T_{0}\vee T_{1}\}$ and $\{\nu$-nearly-emergence occurs in a number of iterations bounded by $T_{0}\vee T_{1}\}$ hold simultaneously with probability at least $F(\mathcal{H}_{0}\nu)^{T_{0}\vee T_{1}}$.*

Theorem 2.* For dynamical system II(D), we assume $h_{1}<\frac{1}{G_{1}}$ and $h_{2}<\frac{1}{G_{2}}$. Then, for $\mu<||x(0)||$, $\nu<||y(0)||$ with T_{2}:=\frac{1}{h_{1}\mathcal{H}_{1}}\ln\big{(}\frac{||x(0)||}{\mu}\big{)} and T_{3}:=\frac{1}{h_{2}\mathcal{H}_{2}}\ln\big{(}\frac{||y(0)||}{\nu}\big{)}, $\mu$-nearly-emergence and $\nu$-nearly-emergence both occur in a number of iterations bounded by $T_{2}\vee T_{3}$ with probability at least $(F_{1}(\mathcal{H}_{1}\mu)F_{2}(\mathcal{H}_{2}\nu))^{T_{2}\vee T_{3}}$.*

We now give some concrete substances to illustrate emergence behaviors of the general models I(D) and II(D).

For the system I(D), take $Y=\mathbb{R}^{3}$ with standard inner product $\langle\cdot,\cdot\rangle$, and $X=\widetilde{Y}=(\mathbb{R}^{3})^{k}/\triangle\cong\triangle^{\perp}$. For $u=(u_{1},\cdots,u_{k}),v=(v_{1},\cdots,v_{k})\in\widetilde{Y}$, define the inner product on $\widetilde{Y}$ as $\langle u,v\rangle_{\widetilde{Y}}=\frac{1}{2}\sum_{i,j=1}^{k}\langle u_{i}-u_{j},v_{i}-v_{j}\rangle$. Here $x\in X$ represents the spatial positions of agents (e.g. birds, fishes, robots,…) and $y\in\widetilde{Y}$ their velocities both projected to the subspace $\triangle^{\perp}$ [8]. Given $x\in X$, let the $k\times k$ matrix $A_{x}$ has entries $a_{ij}\geq\frac{K}{(1+||x_{i}-x_{j}||)^{\beta}}$. Let $D_{x}$ be the $k\times k$ diagonal matrix whose $i$th diagonal element is $d_{i}=\sum_{j\leq k}a_{ij}$ and $L_{x}:=D_{x}-A_{x}$. Then $L_{x}$ is the Laplacian of $A_{x}$ [4]. For $t\in\mathbb{N}$, take $J(x[t],y[t])=y[t]$, $S(x[t])=I_{k}-hL_{x}$, here $I_{k}$ is the identity matrix of order $k$, and let the noise term $H$ has the uniform distribution $U_{3k}(0,r)$ for some $r>0$ or the Gaussian distribution $N(0,\sigma^{2}I_{3k})$ in the model I(D), and then we recover the situations encountered in [5]. Thm.1 in [5] is clearly a special case of Theorem 1 (and note that we really said more). Other kinds of flocking scenarios such as flocking with unrelated pairs and flocking with leader-follower schemes can also be dealt with under our present framework (c.f. [7] Sect. 3). We mention here that the asymmetric conclusions of $x$ and $y$ in Theorem 1 indeed give what we desire in a flocking phenomenon; see [8] (Rem. 2).

For the system II(D), let $\triangle_{X}$ be the diagonal of $(\mathbb{R}^{3})^{k}$ and take $X=(\mathbb{R}^{3})^{k}/\triangle_{X}$ with inner product defined as $\langle\cdot,\cdot\rangle_{\widetilde{Y}}$ above. Let $Y$ be the space of languages with some appropriate distance defined on it (c.f. [9]); and the metric of $\widetilde{Y}$ is inherited from that of $Y$. Given $x\in X,y\in\widetilde{Y}$, let $A_{x}=(a_{ij})$, $B_{y}=(b_{ij})$ be the $k\times k$ matrices with entries $a_{ij}=f(||x_{i}-x_{j}||)$ and $b_{ij}=g(||y_{i}-y_{j}||)$. $f,g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ are some bounded non-increasing functions. For $t\in\mathbb{N}$, take $S_{1}(y[t])=I_{k}-h_{1}L_{1}y$ and $S_{2}(x[t])=I_{k}-h_{2}L_{2}x$ with $L_{1}y:=D_{y}-B_{y}$, $L_{2}x:=D_{x}-A_{x}$ in the model II(D). Here $D_{x}$ and $D_{y}$ are $k\times k$ diagonal matrices defined similarly as above. Computation of the distributions of $||H_{1}||$ and $||H_{2}||$ from some proper random noises $H_{1}$, $H_{2}$ is a routine [2]. Here $x\in X$ is interpreted as the geographical positions of agents projected to $\triangle_{X}^{\perp}$ and $y\in\widetilde{Y}$ as the space of languages projected to $\triangle^{\perp}$. This specification of system II(D) can be used to model emergence behavior in linguistic evolution, since each agent tends to move to others using similar languages and meanwhile the influence from other agents’ languages decreases according to distances [8] (Sect. 6).

2.3 Proof of Theorem 1

The proof closely follows that of [5], and we prove Theorem 1 through some intermediate steps.

Proposition 1.* Suppose $0<h<\frac{1}{G}$, and $||H||\leq\mathcal{H}_{0}||y[t]||$ for $0\leq t<T$. Then we have ||y[t]||\leq\big{(}1-\frac{hG}{(1+||x[t-1]||)^{\beta}}+h\mathcal{H}_{0}\big{)}||y[t-1]|| for $t\leq T$. Hence, for $1\leq t\leq T$, ||y[t]||\leq||y(0)||\prod_{i=0}^{t-1}\big{(}1-\frac{hG}{(1+||x[i]||)^{\beta}}+h\mathcal{H}_{0}\big{)}.*

Proof. By (2), we have

[TABLE]

$\Box$

The following proposition is taken from [6].

Proposition 2.([6]) Let $c_{1},c_{2}>0$ and $s>q>0$. Then the equation $M(z)=z^{s}-c_{1}z^{q}-c_{2}=0$ has a unique positive zero $z^{*}$. Moreover, $z^{*}\leq\max\{(2c_{1})^{\frac{1}{s-q}},(2c_{2})^{\frac{1}{s}}\}$ and $M(z)\leq 0$ for $0\leq z\leq z^{*}$.

Proposition 3.* Let $T\in\mathbb{N}$ or $T=\infty$. Suppose $||H||\leq\mathcal{H}_{0}||y[t]||$ for $0\leq t<T$ and h<\min\Big{\{}\frac{1}{G},\ \frac{1}{2^{1-\gamma}C||y(0)||^{\delta}}\big{(}\frac{G}{2\mathcal{H}_{0}}\big{)}^{\frac{1-\gamma}{\beta}}\Big{\}}, and one of the following hypotheses holds:

(i) $\beta+\gamma<1$,

(ii)$\beta+\gamma=1$, and ||y(0)||<\big{(}\frac{G}{2CQ(\delta)}\big{)}^{\frac{1}{\delta}},

(iii)$\beta+\gamma>1$, and \big{(}\frac{1}{a(\beta+\gamma)}\big{)}^{\frac{1}{\beta+\gamma-1}}\frac{\beta+\gamma-1}{\beta+\gamma}>b+h\big{(}(\frac{\beta+\gamma}{\beta+\gamma-1})b\big{)}^{\gamma}\frac{aG}{2Q(\delta)}.

Then $1-\frac{hG}{2U_{0}^{\beta}}\in(0,1)$, for $0\leq t<T$, $||x[t]||\leq B_{0}$ and ||y[t]||\leq||y(0)||\big{(}1-\frac{hG}{2U_{0}^{\beta}}\big{)}^{t}. If $T=\infty$, then $||y[t]||\rightarrow 0$ as $t\rightarrow\infty$, and moreover, there exists $\hat{x}\in X$ such that $x[t]\rightarrow\hat{x}$ as $t\rightarrow\infty$, and ||x[t]-\hat{x}||\leq aU_{0}^{\beta+\gamma}\big{(}1-\frac{hG}{2U_{0}^{\beta}}\big{)}^{\delta t} for $t\geq 0$.*

Proof. Let $\Gamma:=\{0\leq t\leq T-1|\ (1+||x[t]||)^{\beta}\leq\frac{G}{2\mathcal{H}_{0}}\}$. Assume $\Gamma\neq\{0,\cdots,T-1\}$. Since $0\in\Gamma$ by the assumptions, let $\hat{t}=\min\{\{0,\cdots,T-1\}\backslash\Gamma\}$. For $t<T$, let $t^{*}$ be the point maximizing $||x||$ in $\{0,\cdots,t\}$. Then by Proposition 1, for $t^{*}\leq t<\hat{t}$ and $0\leq i\leq t$ we have

[TABLE]

By Proposition 1, for $\tau\leq t$,

[TABLE]

Take $\tau=t^{*}$, and then we have

[TABLE]

Let $z=1+||x[t^{*}]||$, then the above inequality becomes $M(z):=z-az^{\gamma+\beta}-b\leq 0$.

(i) Assume $\beta+\gamma<1$. By Proposition 2 and the definition of $U_{0}$, we get $1+||x[t^{*}]||\leq U_{0}$. Thereby for $t<\hat{t}$, $||x[t]||\leq U_{0}-1=B_{0}$. This implies

[TABLE]

for $t<\hat{t}$. Hence by Proposition 1 and $||y[t]||$ is decreasing,

[TABLE]

This is in contradiction with the definition of $\hat{t}$, therefore $||x[t]||\leq B_{0}$ and $\frac{G}{(1+||x[t]||)^{\beta}}-\mathcal{H}_{0}\geq J_{0}:=\frac{G}{2(1+B_{0})^{\beta}}=\frac{G}{2U_{0}^{\beta}}$, for all $t<T$. It follows from Proposition 1 that for $t<T$,

[TABLE]

For $T\geq\tau>t$, by employing (1) we obtain

[TABLE]

When $T=\infty$, we take $t\rightarrow\infty$ to see that there exists $\hat{x}\in X$ which is the limit of $x[t]$. Let $\tau\rightarrow\infty$ in the above expression, we have

[TABLE]

(ii) Assume $\beta+\gamma=1$. Then the inequality $M(z)\leq 0$ becomes

[TABLE]

By the assumption and definition of $B_{0}$, we get $||x[t^{*}]||\leq\frac{G(1+||x(0)||)}{G-2CQ(\delta)||y(0)||^{\delta}}-1=B_{0}$. Thereby, for $t<\hat{t}$

[TABLE]

We now proceed as in case (i).

(iii) Assume $\beta+\gamma>1$. $M^{\prime}(z)=1-(\beta+\gamma)az^{\beta+\gamma-1}$ has a unique zero $z^{*}:=(\frac{1}{a(\gamma+\beta)})^{\frac{1}{\gamma+\beta-1}}$ and $M(z^{*})=(\frac{1}{a(\gamma+\beta)})^{\frac{1}{\gamma+\beta-1}}\frac{\gamma+\beta-1}{\gamma+\beta}-b>0$. Since $M(0)=-b<0,M^{\prime\prime}(z)<0$ for $z>0$, $M$ is a convex function in $(0,\infty)$ and has two positive zero $z_{l}$, $z_{r}$ satisfying $0<z_{l}<z^{*}<z_{r}$.

For $t\in\mathbb{N}$, let $z(t):=1+||x[t^{*}]||$. We then have $z(0)=1+||x(0)||=b<z^{*}$, which suggests $z(0)<z_{l}$. Now assume there exists $t<\hat{t}$ such that $z(t)\geq z_{r}$ and denote $r$ be the first such $t$. Hence, $r=r^{*}\geq 1$ and $1+||x[t]||\leq z(r-1)\leq z_{l}$ for $t<r$. Let $z_{0}$ be the intersection of the $z$ axis with the line joining $(0,-b)$ and $(z^{*},M(z^{*}))$, (c.f. Fig.1 in [5]). Straightforward calculation yields $z_{0}=\frac{b}{1-az^{*{\gamma+\beta-1}}}=\frac{(1+||x(0)||)(\gamma+\beta)}{\gamma+\beta-1}$.

By the definition of $B_{0}$, for $t<r$

[TABLE]

We get $||x[r-1]||\leq z_{0}-1$, $||x[r]||\geq z_{r}-1$ and accordingly, $||x[r]-x[r-1]||\geq z^{*}-z_{l}$ since $z^{*}<z_{r}$. By the Lagrange intermediate value theorem, there is $\xi\in[z_{l},z^{*}]$ such that $M(z^{*})=M^{\prime}(\xi)(z^{*}-z_{l})$. Thereby $z^{*}-z_{l}\geq M(z^{*})$ since $0\leq M^{\prime}(\xi)\leq 1$. Then $||x[r]-x[r-1]||\geq M(z^{*})$. On the other hand,

[TABLE]

which together with the above expression gives $M(z^{*})\leq hCU_{0}^{\gamma}||y(0)||^{\delta}$. This is, however, contrary to our hypothesis. Therefore, for all $t<\hat{t}$, $z(t)<z_{l}$ and $||x[t]||\leq B_{0}$ by noting inequality (4). We then obtain,

[TABLE]

for $t<\hat{t}$. Now we proceed as in case (i). $\Box$

Proof of Theorem 1. Suppose the conditions of Proposition 3 hold for some $T>0$. Then we have ||y[t]||\leq||y(0)||\big{(}1-\frac{hG}{2U_{0}^{\beta}}\big{)}^{t} for $t<T$ and ||x[\tau]-x[t]||\leq aU_{0}^{\beta+\gamma}\big{(}1-\frac{hG}{2U_{0}^{\beta}}\big{)}^{\delta t} for $t<\tau\leq T$. If $T=\infty$, ||x[t]-\hat{x}||\leq aU_{0}^{\beta+\gamma}\big{(}1-\frac{hG}{2U_{0}^{\beta}}\big{)}^{\delta t} for $t\geq 0$. By the proof of Proposition 3 and straightforward calculations, we have $||y[T]||\leq\nu$ when $T\geq T_{0}$; $||x[\tau]-x[t]||\leq\mu$ when $\tau>t\geq T_{1}$; and $||x[t]-\hat{x}||\leq\mu$ when $t\geq T_{1}$. If $\nu$-nearly-emergence has not occurred, then $||y[t]||\geq\nu$, wherefore by the definition of function $F$,

[TABLE]

Since $\{H[t]\}$ are i.i.d. for varying $t$, we get

[TABLE]

which yields the first part of the conclusions. Likewise, we have

[TABLE]

We then conclude the proof. $\Box$

2.4 Proof of Theorem 2

Proposition 4.* Suppose $0<h_{1}<\frac{1}{G_{1}}$, $0<h_{2}<\frac{1}{G_{2}}$, and $||H_{1}[t_{1}]||\leq\mathcal{H}_{1}||x[t_{1}]||$ for $0\leq t_{1}<T^{1}$; $||H_{2}[t_{2}]||\leq\mathcal{H}_{2}||y[t_{2}]||$ for $0\leq t_{2}<T^{2}$. Then we have ||x[t_{1}]||\leq\big{(}1-\frac{h_{1}G_{1}}{(1+||y[t_{1}-1]||)^{\beta_{1}}}+h_{1}\mathcal{H}_{1}\big{)}||x[t_{1}-1]|| for $t_{1}\leq T^{1}$ and ||y[t_{2}]||\leq\big{(}1-\frac{h_{2}G_{2}}{(1+||x[t_{2}-1]||)^{\beta_{2}}}+h_{2}\mathcal{H}_{2}\big{)}||y[t_{2}-1]|| for $t_{2}\leq T^{2}$. Consequently, for $t_{1}\leq T^{1}$, ||x[t_{1}]||\leq||x(0)||\prod_{i=0}^{t-1}\big{(}1-\frac{h_{1}G_{1}}{(1+||y[i]||)^{\beta_{1}}}+h_{1}\mathcal{H}_{1}\big{)}; and for $t_{2}\leq T^{2}$, ||y[t_{2}]||\leq||y(0)||\prod_{j=0}^{t-1}\big{(}1-\frac{h_{2}G_{2}}{(1+||x[j]||)^{\beta_{2}}}+h_{2}\mathcal{H}_{2}\big{)}.*

Proof. The proof readily follows by utilizing condition (3). $\Box$

The following proposition is critical to the proof of Theorem 2.

Proposition 5.* Let $T^{1},T^{2}\in\mathbb{N}\cup\{\infty\}$. Suppose $||H_{1}||\leq\mathcal{H}_{1}||x[t_{1}]||$ for $0\leq t_{1}<T^{1}$; $||H_{2}||\leq\mathcal{H}_{2}||y[t_{2}]||$ for $0\leq t_{2}<T^{2}$ and $h_{1}<\frac{1}{G_{1}}$, $h_{2}<\frac{1}{G_{2}}$. Then, for $0\leq t<T^{1}\wedge T^{2}$, $||x[t]||\leq||x(0)||(1-h_{1}\mathcal{H}_{1})^{t}$ and $||y[t]||\leq||y(0)||(1-h_{2}\mathcal{H}_{2})^{t}$.*

Proof. Denote $T:=T^{1}\wedge T^{2}$ for brevity. Let $\Gamma_{1}:=\{0\leq t_{1}\leq T-1|\ (1+||y[t_{1}]||)^{\beta_{1}}\leq\frac{G_{1}}{2\mathcal{H}_{1}}\}$. Assume $\Gamma_{1}\neq\{0,\cdots,T-1\}$. Since $0\in\Gamma_{1}$ by the assumptions, let $\hat{t}_{1}=\min\{\{0,\cdots,T-1\}\backslash\Gamma_{1}\}$. For $t_{1}<T$, let $t_{1}^{*}$ be the point maximizing $||y||$ in $\{0,\cdots,t_{1}\}$. Analogously, we define $\Gamma_{2}:=\{0\leq t_{2}\leq T-1|\ (1+||x[t_{2}]||)^{\beta_{2}}\leq\frac{G_{2}}{2\mathcal{H}_{2}}\}$ and assume $\Gamma_{2}\neq\{0,\cdots,T-1\}$. Let $\hat{t}_{2}=\min\{\{0,\cdots,T-1\}\backslash\Gamma_{2}\}$. For $t_{2}<T$, let $t_{2}^{*}$ be the point maximizing $||x||$ in $\{0,\cdots,t_{2}\}$.

Then by Proposition 4, for $t_{1}^{*}\leq t_{1}<\hat{t}_{1}$ and $0\leq i\leq t_{1}$ we have

[TABLE]

By exploiting Proposition 4, for $\tau\leq t_{1}$,

[TABLE]

Likewise, for $t_{2}^{*}\leq t_{2}<\hat{t}_{2}$ and $0\leq j\leq t_{2}$ we have

[TABLE]

And for $\tau\leq t_{2}$, we have

[TABLE]

Now we shall compare $\hat{t}_{1}$ with $\hat{t}_{2}$ to deduce contradictions. If $\hat{t}_{1}<\hat{t}_{2}$. We take $t_{2}=\hat{t}_{1}$ and get from (6), $||y[\hat{t}_{1}]||\leq||y(0)||(1-h_{2}R_{2}(t_{2}^{*}))^{\hat{t}_{1}}\leq||y(0)||$. This is in contradiction with definition of $\hat{t}_{1}$. Similarly, if $\hat{t}_{1}>\hat{t}_{2}$, we can also deduce a contradiction. Now let’s consider the case $\hat{t}_{1}=\hat{t}_{2}$. If so, take $\tau=t_{2}^{*}$ in (5) and $\tau=t_{1}^{*}$ in (6). We then have $||x[t_{2}^{*}]||\leq||x(0)||(1-h_{1}R_{1}(t_{2}^{*}))^{t_{2}^{*}}$ and $||y[t_{1}^{*}]||\leq||y(0)||(1-h_{2}R_{2}(t_{1}^{*}))^{t_{1}^{*}}$. These inequalities imply $t_{1}^{*}=t_{2}^{*}=0$. That is to say, for all $t<\hat{t}_{1}=\hat{t}_{2}$, $||x[t]||\leq||x(0)||$ and $||y[t]||\leq||y(0)||$. Combining these with Proposition 4, we derive

[TABLE]

Here is the contradiction as in the first case above. It is at this stage that we may conclude that $\hat{t}_{1}$ and $\hat{t}_{2}$ must not both exist.

Without loss of generality, we assume $\hat{t}_{1}$ does not exist, i.e. $\Gamma_{1}=\{0,\cdots,T-1\}$. For $t<T$, $(1+||y[t]||)^{\beta_{1}}\leq\frac{G_{1}}{2\mathcal{H}_{1}}=(1+||y(0)||)^{\beta_{1}}$. Hence, $||y[t]||\leq||y(0)||$ and $R_{1}(t_{1}^{*})\geq\frac{G_{1}}{2(1+||y(0)||)^{\beta_{1}}}$. From (5), we have $||x[t]||\leq||x(0)||(1-h_{1}\mathcal{H}_{1})^{t}$. If $\hat{t}_{2}$ exists, we then take $t=\hat{t}_{2}$ in the above inequality to get $||x[\hat{t}_{2}]||\leq||x(0)||$. It is in contradiction with the definition of $\hat{t}_{2}$. Therefore, $\hat{t}_{2}$ does not exist either. Now we can proceed as above to get $||y[t]||\leq||y(0)||(1-h_{2}\mathcal{H}_{2})^{t}$ for $t<T$, by using (6). $\Box$

Proof of Theorem 2. Suppose the conditions of Proposition 5 hold for some $T^{1},T^{2}>0$. Then we have $||x[t]||\leq||x(0)||(1-h_{1}\mathcal{H}_{1})^{t}$ and $||y[t]||\leq||y(0)||(1-h_{2}\mathcal{H}_{2})^{t}$ for $t<T:=T^{1}\wedge T^{2}$. By direct calculations, we get $||x[T]||\leq\mu$ when $T\geq T_{2}$ and $||y[T]||\leq\nu$ when $T\geq T_{3}$. If $\mu$-nearly-emergence and $\nu$-nearly-emergence have not occurred at time $t_{1}h_{1}$ and $t_{2}h_{2}$ resp., then $||x[t_{1}]||\geq\mu$ and $||y[t_{2}]||\geq\nu$. Whence,

[TABLE]

and

[TABLE]

Since $\{H_{i}[t_{i}]\}$ are i.i.d. for varying $t_{i}$, $i=1,2$, and $H_{1}$ is independent with $H_{2}$, by letting $P=P_{1}\times P_{2}$ be the independent product of $P_{1}$ and $P_{2}$ (c.f. [2]), we get

[TABLE]

which concludes the proof. $\Box$

3. Continuous-time emergence

3.1 Models setup (I(C), II(C))

In principle, by letting the time steps $h$, $h_{i}$ approach zero, we may derive the continuous counterparts of systems I(D) and II(D). We shall, however, make some modifications for technical reason and it is at this time the inner product structures of spaces $X,Y,\widetilde{Y}$ take effect.

Let $t\in\mathbb{R}^{+}$, for $x\in X$, $y\in\widetilde{Y}$, consider the following dynamical system:

[TABLE]

Here, as in the system I(D), $J:X\times\widetilde{Y}\rightarrow X$ is a Lipschitz or $C^{1}$ operator. We now require, for some $C,\delta>0$, $0\leq\gamma<1$, that

[TABLE]

for all $x\in X$, $y\in\widetilde{Y}$. Denote $\mathbb{R}^{k\times k}$ as the space of $k\times k$ real matrices. Let $L:X\rightarrow\mathbb{R}^{k\times k}$ be a Lipschitz or $C^{1}$ operator with $L:x\mapsto L_{x}$. $L_{x}$ can be seen as a linear transformation on $Y^{k}$ by mapping $(y_{1},\cdots,y_{k})\in Y^{k}$ to $(L_{x}(i,1)y_{1}+\cdots+L_{x}(i,k)y_{k})_{i\leq k}$. Here, $L_{x}(i,j)$ is the $(i,j)$ entry of matrix $L_{x}$. For $x\in X$, define $\phi_{x}:=\min_{y\neq 0\atop y\in\widetilde{Y}}\frac{\langle L_{x}y,y\rangle}{||y||^{2}}$. We impose the following two hypotheses on $L$: (i) For $x\in X$, $y\in Y$, $L_{x}(y,\cdots,y)=0$. (ii) there exists $K>0,\beta\geq 0$ such that

[TABLE]

for all $x\in X$. It is easy to see from (i) that $L_{x}$ induces a linear transformation on $\widetilde{Y}$, which will also be denoted as $L_{x}$ for notational simplicity. Let $H(t)$ be a continuous time stochastic process defined on some probability space $(\Omega,\mathcal{F},P)$ taking value in $(\widetilde{Y},\mathcal{B}(\widetilde{Y}))$. Let $F:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow[0,1]$ be a real function (not necessarily a distribution function) such that $P(\max_{0\leq t\leq T}||H(t)||\leq x)\geq F(x,T)$, for $x,T\in\mathbb{R}^{+}$. We may observe that $F(x,T)$ is non-decreasing w.r.t. $x$ while non-increasing w.r.t. $T$.

Next, we introduce a continuous version of II(D). For $x\in X$, $y\in\widetilde{Y}$, consider the following dynamical system:

[TABLE]

Here operators $L_{1},L_{2}$ are given similarly as $L$ above with $L_{1}$ defined on $\widetilde{Y}$ and $L_{2}$ on $X$. For $x\in X$, $y\in\widetilde{Y}$, define $\xi_{x}:=\min_{y\neq 0\atop y\in\widetilde{Y}}\frac{\langle L_{2x}y,y\rangle}{||y||^{2}}$ and $\eta_{y}:=\min_{x\neq 0\atop x\in X}\frac{\langle L_{1y}x,x\rangle}{||x||^{2}}$. The corresponding hypothesis (ii) above becomes (ii*′*): there exist $K_{1},K_{2}>0$, $\beta_{1},\beta_{2}\geq 0$ such that

[TABLE]

for all $x\in X$, $y\in\widetilde{Y}$. Let $H_{1}(t):(\Omega_{1},\mathcal{F}_{1},P_{1})\rightarrow(X,\mathcal{B}(X))$ and $H_{2}(t):(\Omega_{2},\mathcal{F}_{2},P_{2})\rightarrow(\widetilde{Y},\mathcal{B}(\widetilde{Y}))$ be two continuous time stochastic processes, which are independent with each other. Let $i=1,2$, $F_{i}:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow[0,1]$ be real functions such that $P_{i}(\max_{0\leq t\leq T}||H_{i}(t)||\leq x)\geq F_{i}(x,T)$, for $x,T\in\mathbb{R}^{+}$.

For notational convenience we sometimes write $L_{t}:=L_{x(t)}$, $\phi_{t}:=\phi_{x(t)}$, $\xi_{t}:=\xi_{x(t)}$ and $\eta_{t}:=\eta_{y(t)}$.

3.2 Main results

As in the discrete case, we define several constants which are only dependent on the initial state $(x(0),y(0))$ of the population.

For system I(C):

[TABLE]

[TABLE]

[TABLE]

For system II(C):

[TABLE]

The main results in this section are stated as follows.

Theorem 3.* Let $x(0)\in X$ and $y(0)\in\widetilde{Y}$, then there exists a unique solution $(x(t),y(t))$ of the dynamical system I(C) for all $t\in\mathbb{R}$. Moreover, assume one of the following hypotheses holds:

(i) $2\beta+\gamma<1$,

(ii)$2\beta+\gamma=1$, and ||y(0)||<\big{(}\frac{(\delta K)^{2}}{2^{1+\gamma+2\beta}((1-\gamma)C)^{2}}\big{)}^{\frac{1}{2\delta}},

(iii)$2\beta+\gamma>1$, and \big{(}\frac{1}{a\alpha}\big{)}^{\frac{1}{\alpha-1}}\frac{\alpha-1}{\alpha}>b.

Then, for $\nu<||y(0)||$, $\nu$-nearly-emergence occurs before time T_{0}:=\frac{2B_{0}^{\beta}}{K}\ln\big{(}\frac{||y(0)||}{\nu}\big{)} with probability at least $F(\mathcal{H}_{0}\nu,T_{0})$. In addition, if $\mu<B_{1}$, let T_{1}:=\frac{2B_{0}^{\beta}}{K\delta}\ln\big{(}\frac{B_{1}}{\mu}\big{)}, then the events $\{||x[t]-x[\tau]||\leq\mu,\ for\ \tau>t\geq T_{0}\vee T_{1}\}$ and $\{\nu$-nearly-emergence occurs before time $T_{0}\vee T_{1}\}$ hold simultaneously with probability at least $F(\mathcal{H}_{0}\nu,T_{0}\vee T_{1})$.*

As in Section 2.2, we may readily recover the continuous-time result in [5] by letting $L_{x}$ be the Laplacian of $A_{x}$, for $x\in X$; and taking $J(x,y)=y$ and the coordinate processes of $H(t)$ as independent smoothed Wiener processes.

Theorem 4.* Let $x(0)\in X$ and $y(0)\in\widetilde{Y}$, then there exists a unique solution $(x(t),y(t))$ of the dynamical system II(C) for all $t\in\mathbb{R}$. Furthermore, for $\mu<||x(0)||$, $\nu<||y(0)||$ with T_{2}:=\frac{2(1+||y(0)||^{2})^{\beta_{2}}}{K_{2}}\ln\big{(}\frac{||x(0)||}{\mu}\big{)} and T_{3}:=\frac{2(1+||x(0)||^{2})^{\beta_{1}}}{K_{1}}\ln\big{(}\frac{||y(0)||}{\nu}\big{)}, either $\mu$-nearly-emergence or $\nu$-nearly-emergence occurs before time $T_{2}\vee T_{3}$ with probability at least

$F_{1}(\mathcal{H}_{1}\mu,T_{2}\vee T_{3})F_{2}(\mathcal{H}_{2}\nu,T_{2}\vee T_{3})$.*

Compared with Theorem 2, the last result is weaker due to the fact that in the continuous case the stochastic processes $H_{i}(t)$ do not possess “independence property” among different “time steps”. However, if the noise does not impose on both equations of system II(C), we have the following corollary.

Corollary 1.* Suppose $H_{1}(t)\equiv 0$. Under the assumptions of Theorem 4, the events $\{\mu$-nearly-emergence occurs before time $T_{2}\}$ and $\{\nu$-nearly-emergence occurs before time $T_{3}\}$ hold simultaneously with probability at least $F_{2}(\mathcal{H}_{2}\nu,T_{3})$. An analogous result holds for the case $H_{2}(t)\equiv 0$.*

We mention that it is possible to have results similar with Corollary 1 when $H_{1}(t)$ is small enough or possesses independent increments.

3.3 Proof of Theorem 3

The main procedure of the proof follows that of [5]. Let $\theta_{t}:=\min_{\tau\in[0,t]}\phi_{\tau}$, $\Gamma(t):=\Gamma(x(t)):=||x(t)||^{2}$ and $\Lambda(t):=\Lambda(y(t)):=||y(t)||^{2}$.

Proposition 6.* Suppose for $0\leq t<T$, $||H(t)||\leq||y(t)||\mathcal{H}_{0}$. Then we have $\Lambda(t)\leq\Lambda(0)e^{-2t(\theta_{t}-\mathcal{H}_{0})}$ for $0\leq t<T$.*

Proof. The proof parallels with that of [5] (Prop. 4), by using assumption (9). Hence we omit it. $\Box$

Proposition 7.* Suppose for $0\leq t<T$, $\theta_{t}>\mathcal{H}_{0}$. Then we have*

[TABLE]

for $0\leq t<T$.

Proof. Mimicking the proof of [7] (Prop. 6) by employing Proposition 6. We leave this proof as an exercise for the reader. $\Box$

Proposition 8.* Suppose for $0\leq t<T$, $||H(t)||\leq||y(t)||\mathcal{H}_{0}$. Assume one of the following hypotheses holds:

(i) $2\beta+\gamma<1$,

(ii)$2\beta+\gamma=1$, and \Lambda(0)<\big{(}\frac{(\delta K)^{2}}{2^{1+\gamma+2\beta}((1-\gamma)C)^{2}}\big{)}^{\frac{1}{\delta}},

(iii)$2\beta+\gamma>1$, and \big{(}\frac{1}{a\alpha}\big{)}^{\frac{1}{\alpha-1}}\frac{\alpha-1}{\alpha}>b.

Then for $0\leq t<T$, $\Gamma(t)\leq B_{0}$ and $\Lambda(t)\leq\Lambda(0)e^{-\frac{Kt}{B_{0}^{\beta}}}$. If $T=\infty$, then $\Lambda(t)\rightarrow 0$ as $t\rightarrow\infty$ and there exists $\hat{x}\in X$ such that $x(t)\rightarrow\hat{x}$ as $t\rightarrow\infty$ and $||x(t)-\hat{x}||\leq B_{1}e^{-\frac{\delta Kt}{2B_{0}^{\beta}}}$ for $t\geq 0$.*

Proof. Let $\Xi:=\{t\in[0,T)|\ (1+||x(t)||^{2})^{\beta}\leq\frac{K}{2\mathcal{H}_{0}}\}$. Assume $\Xi\neq[0,T)$. Since $0\in\Xi$ by the assumptions, let $\hat{t}=\inf\{[0,T)\backslash\ \Xi\}$. Let $t<\hat{t}$ and $t^{*}\in[0,t]$ be the point maximizing $||x||$ in $[0,t]$. Then by Proposition 6,

[TABLE]

Since $t^{*}\leq t<\hat{t}$, $t^{*}\in\Xi$, we deduce

[TABLE]

Thereby Proposition 7 implies,

[TABLE]

Take $t=t^{*}$ in the above expression, and then we have

[TABLE]

Let $z=1+\Gamma(t^{*})$. Then inequality (11) becomes $M(z)\leq 0$ with $M(z):=z-az^{\alpha}-b$. To complete the proof, we may argue as in Proposition 3 dividing into three cases and consult the proof of Thm. 2 of [7]. $\Box$

Proof of Theorem 3. The existence of a unique solution for each $\omega\in\Omega$ follows from [11] (Ch. 8).

By assumption, we have for $\varepsilon>0$, $P(\max_{0\leq t\leq T}||H(t)||\leq\varepsilon)\geq F(\varepsilon,T)$. Define $T(\omega):=\inf\{t\geq 0|\ ||y(t)||\leq\nu\}$ and take $\varepsilon=\mathcal{H}_{0}\nu$. To prove the first claim in Theorem 3, it suffices to proof

[TABLE]

Take $\omega\in\{\omega\in\Omega|\ \max_{0\leq t\leq T_{0}}||H(t)||\leq\mathcal{H}_{0}\nu\}$. Assume $T(\omega)>T_{0}$. Then there is some $\zeta>0$ such that $T(\omega)>T_{0}+\zeta$. Wherefore, $||y(t)||>\nu$ on the interval $[0,T_{0}+\zeta]$. In particular, $||y(T_{0})||>\nu$. Since $\max_{0\leq t\leq T_{0}}||H(t)||\leq\mathcal{H}_{0}\nu<||y(t)||\mathcal{H}_{0}$, Proposition 8 holds for $T=T_{0}$. Then for $0\leq t<T_{0}$, we have $||y(t)||\leq||y(0)||e^{-\frac{Kt}{2B_{0}^{\beta}}}$. As $y(t)$ is continuous, we obtain $||y(T_{0})||\leq||y(0)||e^{-\frac{KT_{0}}{2B_{0}^{\beta}}}=\nu$. This is a contradiction, which then finishes the proof of (12).

Next, by the proof of Proposition 8 and straightforward calculations (much the same way as in the proof of Theorem 1), we have $||x(\tau)-x(t)||\leq\mu$ when $\tau>t\geq T_{1}$; and $||x(t)-\hat{x}||\leq\mu$ when $t\geq T_{1}$. We then conclude the proof by replace $T_{0}$ with $T_{0}\vee T_{1}$ in the above argument. $\Box$

3.4 Proof of Theorem 4

Denote $\Lambda_{1}(x(t)):=\langle x(t),x(t)\rangle$ and $\Lambda_{2}(y(t)):=\langle y(t),y(t)\rangle$. In the sequel, we will suppress the subscript $1,2$ of $\Lambda$ for brevity.

Proposition 9.* Suppose $||H_{1}(t_{1})||\leq||x(t_{1})||\mathcal{H}_{1}$ for $0\leq t_{1}<T^{1}$ and $||H_{2}(t_{2})||\leq||y(t_{2})||\mathcal{H}_{2}$ for $0\leq t_{2}<T^{2}$. Then for $0\leq t_{1}<T^{1}$, $\Lambda(x(t_{1}))\leq\Lambda(x(0))e^{-2\int_{0}^{t_{1}}\eta_{\tau}-\mathcal{H}_{1}\mathrm{d}\tau}$; and for $0\leq t_{2}<T^{2}$, $\Lambda(y(t_{2}))\leq\Lambda(y(0))e^{-2\int_{0}^{t_{2}}\xi_{\tau}-\mathcal{H}_{2}\mathrm{d}\tau}$.*

Proof. Let $\tau\in[0,t_{1}]$. Therefore, by using (10) we have

[TABLE]

Integrating the above inequality from [math] to $t_{1}$ deduces the first result. The other can be proved likewise. $\Box$

Proposition 10.* Suppose $||H_{1}(t_{1})||\leq\mathcal{H}_{1}||x(t_{1})||$ for $0\leq t_{1}<T^{1}$; and $||H_{2}(t_{2})||\leq\mathcal{H}_{2}||y(t_{2})||$ for $0\leq t_{2}<T^{2}$. Then we obtain for $0\leq t<T^{1}\wedge T^{2}$, $\Lambda(x(t))\leq\Lambda(x(0))e^{-2t\mathcal{H}_{1}}$ and $\Lambda(y(t))\leq\Lambda(y(0))e^{-2t\mathcal{H}_{2}}$.*

Proof. Denote $T:=T^{1}\wedge T^{2}$. Let $\Gamma_{1}:=\{t_{1}\in[0,T)|\ (1+\Lambda(y(t_{1})))^{\beta_{2}}\leq\frac{K_{2}}{2\mathcal{H}_{1}}\}$. Assume $\Gamma_{1}\neq[0,T)$. Since $0\in\Gamma_{1}$ by the assumptions, let $\hat{t}_{1}=\inf\{[0,T)\backslash\Gamma_{1}\}$. Let $t_{1}<\hat{t}_{1}$ and $t_{1}^{*}\in[0,t_{1}]$ be the point maximizing $\Lambda(y(t))$ in $[0,t_{1}]$. Analogously, we define $\Gamma_{2}:=\{t_{2}\in[0,T)|\ (1+\Lambda(x(t_{2})))^{\beta_{1}}\leq\frac{K_{1}}{2\mathcal{H}_{2}}\}$ and assume $\Gamma_{2}\neq[0,T)$. Let $\hat{t}_{2}=\inf\{[0,T)\backslash\Gamma_{2}\}$. Let $t_{2}<\hat{t}_{2}$ and $t_{2}^{*}\in[0,t_{2}]$ be the point maximizing $\Lambda(x(t))$ in $[0,t_{2}]$.

Since $t_{1}^{*}\leq t_{1}<\hat{t}_{1}$, $t_{1}^{*}\in\Gamma_{1}$, and then we have

[TABLE]

Likewise, since $t_{2}^{*}\leq t_{2}<\hat{t}_{2}$, $t_{2}^{*}\in\Gamma_{2}$. We then get

[TABLE]

By Proposition 9, for $s\leq t_{1}$

[TABLE]

and similarly, for $s\leq t_{2}$, $\Lambda(y(s))\leq\Lambda(y(0))e^{-\frac{sK_{1}}{(1+\Lambda(x(t_{2}^{*})))^{\beta_{1}}}}.$ Now we can proceed as in Proposition 5 to show neither $\hat{t}_{1}$ nor $\hat{t}_{2}$ exists. Hence, for all $t<T$, $(1+\Lambda(y(t)))^{\beta_{2}}\leq\frac{K_{2}}{2\mathcal{H}_{1}}=(1+\Lambda(y(0)))^{\beta_{2}}$, i.e. $\Lambda(y(t))\leq\Lambda(y(0))$. Combining this with expression (13), we obtain $\Lambda(x(t))\leq\Lambda(x(0))e^{-2t\mathcal{H}_{1}}$, which concludes the first part of the proposition. The other inequality can be derived similarly. $\Box$

Proof of Theorem 4. The existence of a unique solution for each $\omega_{i}\in\Omega_{i}$, $i=1,2$ follows from [11] (Ch. 8).

By the assumptions, we have for $\varepsilon_{i}>0$, $i=1,2$, $P_{i}(\max_{0\leq t\leq T_{2}\vee T_{3}}||H_{i}(t)||\leq\varepsilon_{i})\geq F_{i}(\varepsilon_{i},T_{2}\vee T_{3})$. Take $\varepsilon_{1}=\mathcal{H}_{1}\mu$ and $\varepsilon_{2}=\mathcal{H}_{2}\nu$ in the above expression, respectively. By the independence of $H_{1}$ and $H_{2}$ (and let $P:=P_{1}\times P_{2}$ as in the proof of Theorem 2), we obtain

[TABLE]

Define $T^{1}(\omega_{1}):=\inf\{t\geq 0|\ ||x(t)||\leq\mu\}$ and $T^{2}(\omega_{2}):=\inf\{t\geq 0|\ ||y(t)||\leq\nu\}$. It now suffices to proof

[TABLE]

Take $\omega_{1}\in\{\omega_{1}\in\Omega_{1}|\ \max_{0\leq t\leq T_{2}\vee T_{3}}||H_{1}(t)||\leq\mathcal{H}_{1}\mu\}$, and

$\omega_{2}\in\{\omega_{2}\in\Omega_{2}|\ \max_{0\leq t\leq T_{2}\vee T_{3}}||H_{2}(t)||\leq\mathcal{H}_{2}\nu\}$. Assume $\min\{T^{1}(\omega_{1}),T^{2}(\omega_{2})\}>T_{2}\vee T_{3}$. Then there are some $\zeta>0$ such that $\min\{T^{1}(\omega_{1}),T^{2}(\omega_{2})\}>T_{2}\vee T_{3}+\zeta$. Accordingly, $||x(t)||>\mu$, $||y(t)||>\nu$ on the interval $[0,T_{2}\vee T_{3}+\zeta]$. In particular, $||x(T_{2}\vee T_{3})||>\mu$ and $||y(T_{2}\vee T_{3})||>\nu$. Since $\max_{0\leq t\leq T_{2}\vee T_{3}}||H_{1}(t)||\leq\mathcal{H}_{1}\mu<||x(t)||\mathcal{H}_{1}$ and $\max_{0\leq t\leq T_{2}\vee T_{3}}||H_{2}(t)||\leq\mathcal{H}_{2}\nu<||y(t)||\mathcal{H}_{2}$, Proposition 10 holds for $T=T_{2}\vee T_{3}$. Then for $0\leq t<T_{2}\vee T_{3}$, we have $||x(t)||\leq||x(0)||e^{-\frac{K_{2}t}{2(1+\Lambda(y(0)))^{\beta_{2}}}}$ and $||y(t)||\leq||y(0)||e^{-\frac{K_{1}t}{2(1+\Lambda(x(0)))^{\beta_{1}}}}$. As $x(t),y(t)$ are continuous, we obtain $||x(T_{2}\vee T_{3})||\leq||x(0)||e^{-\frac{K_{2}(T_{2}\vee T_{3})}{2(1+\Lambda(y(0)))^{\beta_{2}}}}\leq\mu$ and $||y(T_{2}\vee T_{3})||\leq||y(0)||e^{-\frac{K_{1}(T_{2}\vee T_{3})}{2(1+\Lambda(x(0)))^{\beta_{1}}}}\leq\nu$. Now we obtain a contradiction, which then completes the proof of (15). $\Box$

4. Conclusion

In this paper, we have studied the emergent behavior of four dynamical systems (I(D), I(C), II(D), II(C)) in the presence of random fluctuation contained in the environments. In all these cases, “nearly-emergence” phenomena of interested objectives are shown under certain conditions on the systems and the noises. Our results are presented in a quite general setting and reveal some intrinsic mechanisms of emergence which come up in a variety of disciplines [12]. We will extend the results herein onto other dynamical systems and different kinds of random environment will be treated in future work.