Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communications

TL;DR

This paper studies the collective behavior of the Cucker-Smale model under uncertain communication channels, introducing a stochastic kinetic equation with noise, and proves well-posedness and flocking behavior.

Contribution

It introduces a stochastic kinetic Cucker-Smale model with multiplicative noise and establishes global well-posedness and flocking dynamics under certain conditions.

Findings

Proves global well-posedness of the stochastic model.

Demonstrates asymptotic flocking behavior.

Establishes conditions for regular initial data and positive lower bounds on communication weights.

Abstract

We present collective dynamics of the Cucker-Smale (C-S) ensemble under random communications. As an effective modeling of the C-S ensemble, we introduce a stochastic kinetic C-S equation with a multiplicative white noise. For the proposed stochastic kinetic model with a multiplicative noise, we present a global well-posedness of strong solutions and their asymptotic flocking dynamics, when initial datum is sufficiently regular, and communication weight function has a positive lower bound.

Equations

$$\begin{array}{rl}& {\partial }_{t}f+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot ({\tilde{F}}_a[f]f)=0,(t,x,v)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^{2d},\\ & F_a[f](t,x,v):={\int }_{{\mathbb{R}}^{2d}}\varphi (x_{\ast }-x)(v_{\ast }-v)f(t,x_{\ast },v_{\ast })d{v}_{\ast }d{x}_{\ast }.\end{array}$$

$$\varphi (x)=\bar{\varphi }(|x|)\ge 0,\forall x\in {\mathbb{R}}^d,$$

$$\begin{array}{rl}& 0\le \bar{\varphi }(r)\le \bar{\varphi }(0)=:{\varphi }_M,(\bar{\varphi }(r)-\bar{\varphi }(s))(r-s)\le 0,\forall r,s\in [0,\infty ),\\ & \text{mbox}and[\varphi {]}_{Lip}:=\underset{r\ne s}{\sup }\frac{|\bar{\varphi }(r)-\bar{\varphi }(s)|}{|r-s|}<\infty .\end{array}$$

$$F_a[f]⟹F_a[f]+\sigma (v_c-v)\circ {\dot{W}}_t.$$

$${\partial }_t{f}_t+v\cdot {\nabla }_x{f}_t+{\nabla }_v\cdot (F_a[f_t]f_t)=\sigma {\nabla }_v\cdot ((v-v_c)f_t)\circ {\dot{W}}_t.$$

$$\mathcal{P}_{p}(\mathbb{R}^{2d}):=\{\mu\ :\ \mbox{probability measure on $\mathbb{R}^{2d}$ such that }\int_{\mathbb{R}^{2d}}|(x,v)|^{p}d\mu(x,v)<\infty\},$$

$$W_p(\mu ,\nu ):={\left(\underset{\gamma \in \prod (\mu ,\nu )}{\inf }{\int }_{{\mathbb{R}}^{4d}}|(x,v)-(y,w){|}^{p}d\gamma \right)}^{1/p},$$

$$\begin{array}{rl}& d{x}_t^i=v_t^{i}dt,t>0,1\le i\le N,\\ & d{v}_t^i=F_a[{\mu }_t^N](x_t^i,v_t^i)dt,{\mu }_t^N:=\frac{1}{N}\sum _{i=1}^N{\delta }_{(x_t^i,v_t^i)},\end{array}$$

$$\begin{array}{rl}& d{x}_t^i=v_t^{i}dt,t>0,1\le i\le N,\\ & d{v}_t^i=F_a[{\mu }_t^N](x_t^i,v_t^i)dt+\sigma ({\bar{v}}_t-v_t^i)\circ d{W}_t^i,{\bar{v}}_t:=\frac{1}{N}\sum _{i=1}^N{v}_t^i.\end{array}$$

$$\begin{array}{rl}& d{x}_t^{i,\epsilon }=v_t^{i,\epsilon }dt,t>0,1\le i\le N,\\ & d{v}_t^{i,\epsilon }=F_a[{\mu }_t^{N,\epsilon }](x_t^{i,\epsilon },v_t^{i,\epsilon })dt+\sigma \left({{\bar{v}}^{\epsilon }}_t-v_t^{i,\epsilon }\right)d{W}_t^{i,\epsilon },\end{array}$$

$$\begin{array}{rl}& d{x}_t^i=v_t^{i}dt,\\ & d{v}_t^i=\left[F_a[{\mu }_t^N](x_t^i,v_t^i)-\frac{1}{2}{\sigma }^2({\bar{v}}_t-v_t^i)\right]dt+\sigma ({\bar{v}}_t-v_t^i)d{W}_t^i.\end{array}$$

$${\partial }_{t}f+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot \left[\left(F_a[f]-\frac{1}{2}{\sigma }^2(v_c-v)\right)f\right]=\sigma {\Delta }_v(|v-v_c{|}^{2}f),x,v\in {\mathbb{R}}^d,t>0,$$

$$W_{2}(\mu_{t},\tilde{\mu}_{t})\leq CW_{2}(\mu_{0},\tilde{\mu}_{0})e^{C(1+W_{2}(\mu_{0},\tilde{\mu}_{0})},\quad\mbox{for a.s. $t\in[0,T]$},$$

$$W_{2}(\mu_{0},\mu_{0}^{N})\to 0\quad\mbox{as $N\to\infty$},$$

$$\underset{0\le t\le T}{\sup }{W}_2({\mu }_t,{\mu }_t^N)\le C{W}_2({\mu }_0,{\mu }_0^N)e^{C(1+W_2({\mu }_0,{\mu }_0^N))}\to 0,\text{mbox}asN\to \infty ,$$

$$0<\phi_{m}\leq\phi(x)\leq\phi_{M}\quad\mbox{for $x\in\mathbb{R}^{d}$},$$

$$\mathbb{E}[E_0]e^{-2({\psi }_M-{\sigma }^2)t}\le \mathbb{E}[E_t]\le \mathbb{E}[E_0]e^{-2({\psi }_m-{\sigma }^2)t},$$

$$E_t:={\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}|{\bar{v}}_0-v{|}^2{\mu }_t(dx,dv),{\bar{v}}_0:={\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}v{\mu }_0(dx,dv).$$

$$\begin{array}{rl}& {\partial }_t{f}_t+v\cdot {\nabla }_x{f}_t+{\nabla }_v\cdot (F_a[f_t]f_t)=\sigma {\nabla }_v\cdot ((v-v_c)f_t)\circ {\dot{W}}_t,(t,x,v)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^{2d},\\ & f_0(x,v)=f^{in}(x,v),\end{array}$$

$$\displaystyle\begin{aligned} \int_{\mathbb{R}^{2d}}f_{t}\psi\ dvdx&=\int_{\mathbb{R}^{2d}}f^{in}\psi\ dvdx+\int_{0}^{t}\int_{\mathbb{R}^{2d}}f_{s}\left(v\cdot\nabla_{x}\psi+F_{a}[f_{s}]\cdot\nabla_{v}\psi\right)\ dvdxds\\ &\hskip 5.69046pt-\sigma\int_{0}^{t}\Big{(}\int_{\mathbb{R}^{2d}}[(v-v_{c})f_{s}]\cdot\nabla_{v}\psi\ dvdx\Big{)}\circ dW_{s},\quad\mbox{a.s.~{} $\omega\in\Omega$}.\end{aligned}$$

$${\int }_{{\mathbb{R}}^{2d}}{f}^{in}dvdx=1,{\int }_{{\mathbb{R}}^{2d}}v{f}^{in}dvdx=0.$$

$$\mathbb{E}\Vert f_t{\Vert }_{L^{\infty }}\le \Vert f^{in}{\Vert }_{L^{\infty }}\exp \left\{\left(d{\varphi }_M+\frac{(\sigma d{)}^2}{2}\right)t\right\},\mathbb{E}[M_2](t)\le M_2(0)\exp (2{\sigma }^{2}t),t\in [0,T).$$

$$\mathbb{E}[M_2](t)\le M_2(0)\exp (-2({\varphi }_m-{\sigma }^2)t),t>0.$$

$$\displaystyle\begin{aligned} \int_{\mathbb{R}^{2d}}f_{t}\psi\ dvdx&=\int_{\mathbb{R}^{2d}}f^{in}\psi\ dvdx+\int_{0}^{t}\int_{\mathbb{R}^{2d}}f_{s}\left(v\cdot\nabla_{x}\psi+F_{a}[f_{s}]\cdot\nabla_{v}\psi\right)\ dvdxds\\ &\hskip 5.69046pt-\sigma\int_{0}^{t}\Big{(}\int_{\mathbb{R}^{2d}}[(v-v_{c})f_{s}]\cdot\nabla_{v}\psi\ dvdx\Big{)}dW_{s}\\ &\hskip 5.69046pt+\frac{\sigma^{2}}{2}\int_{0}^{t}\int_{\mathbb{R}^{2d}}(v-v_{c})f_{s}\cdot\Big{[}\nabla_{v}\big{(}(v-v_{c})\cdot\nabla_{v}\psi\big{)}\Big{]}dvdxds\quad\mbox{a.s.~{} $\omega\in\Omega$}.\end{aligned}$$

$${\int }_0^t{h}_s\circ d{W}_s={\int }_0^t{h}_{s}d{W}_s+\frac{1}{2}⟨h,W{⟩}_t,$$

$$\Big{\langle}\int_{\mathbb{R}^{2d}}[(v-v_{c})f_{\Large\cdot}]\cdot\nabla_{v}\psi dvdx,W\Big{\rangle}_{t}=-\sigma\int_{0}^{t}\int_{\mathbb{R}^{2d}}[(v-v_{c})f_{s}]\cdot\nabla_{v}[(v-v_{c})\cdot\nabla_{v}\psi]dvdxds,$$

$$\displaystyle\begin{aligned} f_{t}&(x,v)\\ &=f^{in}(x,v)-\int_{0}^{t}\Big{(}v\cdot\nabla_{x}f_{s}+\nabla_{v}\cdot(F_{a}[f_{s}]f_{s})\Big{)}ds+\sigma\int_{0}^{t}\Big{[}\nabla_{v}\cdot\big{(}(v-v_{c})f_{s}\big{)}\Big{]}dW_{s}\\ &\quad+\frac{\sigma^{2}}{2}\int_{0}^{t}\nabla_{v}\cdot\Big{[}(v-v_{c})\nabla_{v}\cdot\big{(}(v-v_{c})f_{s}\big{)}\Big{]}ds,\quad\mathbb{P}\otimes dx\otimes dv\mbox{-a.s.}\end{aligned}$$

$$\displaystyle\begin{aligned} \int_{\mathbb{R}^{2d}}f_{t}\psi\ dvdx&=\int_{\mathbb{R}^{2d}}f^{in}\psi\ dvdx-\int_{0}^{t}\int_{\mathbb{R}^{2d}}\Big{[}v\cdot\nabla_{x}f_{s}+\nabla_{v}\cdot(F_{a}[f_{s}]f_{s})\Big{]}\psi\ dvdxds\\ &\hskip 5.69046pt+\sigma\int_{0}^{t}\Big{(}\int_{\mathbb{R}^{2d}}\nabla_{v}\cdot[(v-v_{c})f_{s}]\psi\ dvdx\Big{)}dW_{s}\\ &\hskip 5.69046pt+\frac{\sigma^{2}}{2}\int_{0}^{t}\int_{\mathbb{R}^{2d}}\nabla_{v}\cdot\Big{[}(v-v_{c})\nabla_{v}\cdot\big{(}(v-v_{c})f_{s}\big{)}\Big{]}dvdxds\quad\mbox{a.s.~{} $\omega\in\Omega$}.\end{aligned}$$

$$\displaystyle\begin{aligned} &\mathscr{L}[f_{t}](x,v):=f_{t}-f^{in}+\int_{0}^{t}\Big{(}v\cdot\nabla_{x}f_{s}+\nabla_{v}\cdot(F_{a}[f_{s}]f_{s})\Big{)}ds\\ &\hskip 71.13188pt-\frac{\sigma^{2}}{2}\int_{0}^{t}\nabla_{v}\cdot\Big{[}(v-v_{c})\nabla_{v}\cdot\big{(}(v-v_{c})f_{s}\big{)}\Big{]}ds,\\ &\mathscr{M}[f_{t}]:=\nabla_{v}\cdot[(v-v_{c})f_{t}].\end{aligned}$$

$$\Big{|}(\rho_{i}*\mathscr{L}[f_{t}])(x^{*},v^{*})-\mathscr{L}[f_{t}](x^{*},v^{*})\Big{|}+\int_{0}^{t}\Big{|}(\rho_{i}*\mathscr{M}[f_{s}])(x^{*},v^{*})-\mathscr{M}[f_{s}](x^{*},v^{*})\Big{|}^{2}ds\leq\frac{1}{2^{i+1}},$$

Full text

Acknowledgment. The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2017R1A2B2001864), and the work of J. Jung is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) : NRF-2016K2A9A2A13003815. The work of M. Röckner is supported by the German Research Foundation (DFG) under the project number IRTG2235.

1. Introduction

Collective behaviors of self-propelled particles are ubiquitous in many biological systems in our nature, to name a few, flocking of birds, herding of sheep and swarming of fish, etc. Throughout the paper, we will use a terminology “flocking” to denote aforementioned coherent collective motions. More precisely, flocking phenomenon denotes a situation in which self-propelled particles adjust their motions into a self-organized ordered motion using only the limited environmental information based on simple rules [14, 15, 29, 30, 39, 41]. After Reynolds and Vicsek et al’s pioneering works in [34, 42], several mechanical models were introduced in literature [5, 7, 11, 12, 27, 29, 31, 32, 38, 41] to model such coherent collective motions. Among others, our main interest lies on the mean-field kinetic model, namely “kinetic C-S model [22, 24, 25]”. Let $f:=f(t,x,v)$ be the one-particle distribution function for the C-S ensemble at position $x$ with microscopic velocity $v$ at time $t$. Then, the dynamics of $f$ is governed by the kinetic C-S equation:

[TABLE]

Here $F_{a}[f]$ is a non-local operator measuring the attractive interactions between particles, and $\phi$ is a communication weight function which is nonnegative and radially symmetric:

[TABLE]

where $\bar{\phi}:[0,\infty)\to\mathbb{R}_{+}$ is Lipschitz continuous, bounded and monotonically decreasing:

[TABLE]

In the last decade, the mean-field equation (LABEL:A-1) and its variants have been extensively studied from various perspectives, e.g., well-posedness and emergent dynamics [4, 25], Fokker-Planck perturbation [21], local sensitivity analysis [20], etc. For more detailed discussion, we refer to a recent survey article [7]. In this paper, we are interested in the quantitative effects on the flocking dynamics of (LABEL:A-1) due to the uncertain communication weights. Recently, the local sensitivity analysis for (LABEL:A-1) has been discussed in an abstract and general framework in [20] and some quantitative pathwise estimates for the variations in $f$ and its derivatives in random space were studied. However, authors in [20] could not provide interesting probabilistic estimates in relation with the emergent dynamics (see [2, 6, 13, 19, 28] for a related local sensitivity analysis in uncertainty quantification (UQ)). Thus, our primary goal of this paper is to address some probability estimate for (LABEL:A-1) with uncertain communication.

To fix the idea, we incorporate a stochastic noise into the communication weight, i.e. $\phi\to\phi+\sigma\circ\dot{W}_{t}$, where ${\dot{W}}_{t}$ is a one-dimensional white noise on the probability space $(\Omega,\mathcal{F},\mathbb{P})$, $\sigma$ denotes the strength of the noise and $\circ$ denotes the stochastic integral in Stratonovich’s sense. Then formally, under the unit mass assumption $\int_{\mathbb{R}^{2d}}f(t,x,v)dxdv=1,$ the non-local operator $F_{a}[f]$ is replaced by a combination of the deterministic part $F_{a}[f]$ and stochastic part involving with ${\dot{W}}_{t}$:

[TABLE]

Now, we combine (LABEL:A-1) and (1.2) to derive the stochastic kinetic C-S equation:

[TABLE]

For a rigorous derivation, we refer to Section 2.1 and here, we use the standard notation for random probability density function $f_{t}(x,v):=f(t,x,v)$.

At the particle level, the effects of white noise perturbations were discussed in [1, 16, 23]. Moreover, a rigorous derivation of equation (1.3) as a mean-field limit of the C-S systems with multiplicative noises was recently discussed in [8] based on the propagation of chaos in [10], and a mean-field limit of the C-S systems with another type of stochastic perturbations was also addressed in [35]. However, as far as the authors know, the kinetic C-S equation (1.2) perturbed by a multiplicative white noise has only been addressed in measure spaces such as $\mathcal{P}_{2}$, not in other function spaces (e.g. Sobolev spaces). For other types of stochastic kinetic equations, we refer to [18, 33]. In this paper, we address the following two questions:

• •

(Well-posedness): Is the stochastic kinetic C-S equation (1.3) well posed in a suitable function space such as Sobolev spaces?

• •

(Emergence of flocking):  If so, does the solution to (1.3) exhibit asymptotic flocking dynamics?

Our main results in this paper provide affirmative answers to the above posed questions. First, we introduce a concept of a strong solution to (1.3) and then provide a global well-posedness for strong solutions by employing a suitable regularization method and stopping time argument. Second, we provide a stochastic flocking estimate by showing that the expectation of the second velocity moment decays to zero exponentially fast, when the communication weight function $\phi$ has a positive infimum $\phi_{m}:=\inf_{x\in\mathbb{R}^{d}}\phi(x)$ and noise strength $\sigma$ is sufficiently small compared to $\phi_{m}$. The main difficulty in our analysis arises, when we prove the existence of a solution to the regularized equation. Here, we obtain $W^{m,\infty}$-estimates for the sequence of functions that approximates the regularized equation. Our $W^{m,\infty}$-estimates contain terms with infinite expectation. Hence, even though we can find a limit function of the sequence from the pathwise estimates, it is not certain that the limit function becomes a solution to the regularized equation (see Remark 4.4 for detailed discussion). To cope with this problem, we used stopping time argument to get a solution to the regularized equation.

The rest of this paper is organized as follows. In Section 2, we provide a rigorous derivation of equation (1.3) from the C-S system with a multiplicative noise, and then briefly discuss our main results on the global well-posedness for strong solutions and asymptotic flocking estimates of classical solutions. In Section 3, we provide several a priori estimates for classical solutions to (1.3). In Section 4, we show our global well-posedness and emergent dynamics for strong solutions to (1.3). Finally, Section 5 is devoted to a brief summary of our main results and discussion on future works.

Gallery of notation Throughout this paper, we denote $(\Omega,\mathcal{F},\mathbb{P})$ by a generic probability space. For $m,k\in\mathbb{N}$ and $p\in[1,\infty]$, we write $W^{m,p}(\mathbb{R}^{k})$ as $m$-th order $L^{p}$ Sobolev space on $\mathbb{R}^{k}$. For $(x,v)\in\mathbb{R}^{2d}$, $\delta_{(x,v)}$ denotes a point mass concentrated at $(x,v)$.

For each $p\in[1,\infty)$, we denote $\mathcal{P}_{p}(\mathbb{R}^{2d})$ by

[TABLE]

and we write $p$-Wasserstein distance on $\mathcal{P}_{p}(\mathbb{R}^{2d})$ as

[TABLE]

where $\prod(\mu,\nu)$ denotes the collection of all measures on $\mathbb{R}^{4d}$ whose marginals are $\mu$ and $\nu$.

2. Preliminaries

In this section, we provide a rigorous derivation of the equation (1.3) and present our main results on the global existence of strong solutions to (1.3) and emergent flocking dynamics.

2.1. Derivation of the SPDE

In this subsection, following [8], we present a derivation of (1.3) from the C-S system perturbed by a multiplicative noise. To be specific, we begin our discussion with the C-S model [14].

Let $(x_{t}^{i},v_{t}^{i})\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ be the position and velocity of the $i$-th particle at time $t\geq 0$, respectively. Then, the ensemble of C-S particles is governed by the following system:

[TABLE]

where the flocking force $F_{a}$ is given in $\eqref{A-1}_{2}$. However, in a real world situation, the communications among particles are subject to the neighboring environment, e.g. drag forces from the fluid, abrupt disconnection, etc, which can be regarded as an intrinsic randomness in the model. To reflect these effects in the communication, stochastic noises can be incorporated into the communication weight $\phi$ appearing in system (LABEL:C-S). To address a stochastic perturbation in system (LABEL:C-S), we replace $\phi$ by $\phi+\sigma\circ\dot{W}_{t}$ and yield the following system of stochastic differential equations:

[TABLE]

Let us compare (LABEL:SC-S) with the model presented in [1], where the authors replaced $\phi$ in (LABEL:C-S) by $\phi+\sigma\dot{W}_{t}$ to obtain the C-S system with a multiplicative noise in Itô’s sense. However, we adopt the integral in Stratonovich’s sense rather than Itô’s sense, since it enables us to use the method of stochastic characteristics once we derive a stochastic partial differential equation from system (LABEL:SC-S). Moreover, it is natural in the following sense: for each $1\leq i\leq N$, let $W_{t}^{i,\varepsilon}$ be a smooth approximation to the Wiener process $W_{t}^{i}$ (e.g. approximation by using a mollifier). Now, we consider the following system of deterministic equations:

[TABLE]

Then, the Wong-Zakai theorem [37, 43, 44] implies that the solution to system (LABEL:SC-S_reg) converges in probability to the solution to system (LABEL:SC-S). Here, we note that system (LABEL:SC-S) is equivalent to the following Itô equation [17]:

[TABLE]

When $W^{i}$’s are i.i.d Wiener processes, a similar analysis as in [21] yields the mean field limit of system (LABEL:SC-S2) as $N\to\infty$, which is the following Fokker-Planck type equation:

[TABLE]

where $v_{c}:=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}vfdxdv$. However, if each $W_{t}^{i}$ is identical to a single Wiener process $W$, i.e. $W^{i}\equiv W_{t}$, we can use a propagation of chaos result [8] to obtain that the empirical measure $\mu_{t}^{N}$ associated with system (LABEL:SC-S) converges to a measure-valued solution to (1.3). Let us summarize the results on the mean-field limit and asymptotic flocking estimates in [8] as follows.

Theorem 2.1**.**

Suppose that $T>0$ and consider a communication weight $\phi$ with $\bar{\phi}\in\mathcal{C}_{b}^{1}(\mathbb{R}_{+})$, and let $\mu_{0},\tilde{\mu}_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ be compactly supported in velocity. Then, the following assertions hold.

1. (1)

If $\mu$ and $\tilde{\mu}$ are two measure-valued solutions to (1.3) with compactly supported initial data $\mu_{0}$ and $\tilde{\mu}_{0}$ in velocity, then

[TABLE]

where the constant $C$ depends only on $\phi$, $T$, $\sigma$, $\sup_{t\in[0,T]}|B_{t}|$, and the support in velocity of $\mu_{0}$ and $\tilde{\mu}_{0}$. 2. (2)

If $\mu_{0}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(x_{0}^{i},v_{0}^{i})}$ is an initial atomic measure such that

[TABLE]

then the empirical measure $\mu_{t}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(x_{t}^{i},v_{t}^{i})}$ associated with system (LABEL:SC-S) is a measure-valued solution to (1.3) with initial data $\mu_{0}^{N}$. Moreover, it converges almost surely to the measure-valued solution $\mu_{t}$ corresponding to the initial measure $\mu_{0}$:

[TABLE]

Note that the stability estimate in Wasserstein metric implies the uniqueness of measure-valued solutions in $\mathcal{P}_{2}(\mathbb{R}^{d}\times\mathbb{R}^{d})$.

Theorem 2.2**.**

Suppose that the communication weight function $\phi$ satisfies

[TABLE]

and let $\mu_{t}$ be a measure-valued solution to (1.3). Then we have

[TABLE]

where $E_{t}$ is defined as

[TABLE]

Remark 2.1**.**

The results in [8] imply that the equation (1.3) can be derived as a mean-field limit of the particle system (LABEL:SC-S). Now, our objective is to establish solutions with higher regularity than measure-valued solutions.

2.2. Presentation of main results

In this subsection, we provide our main results on the global well-posedness of (1.3) and emergent flocking dynamics. First, we consider the Cauchy problem for (1.3):

[TABLE]

where the initial datum $f^{in}$ is assumed to be deterministic. Next, we provide a definition for a strong solution to the Cauchy problem (LABEL:B-0) as follows.

Definition 2.1**.**

For a given $T\in(0,\infty]$, $f_{t}=f_{t}(x,v)$ is a strong solution to (LABEL:B-0) on $[0,T]$ if it satisfies the following relations:

1. (1)

(Regularity): For $k\geq 1$, $f_{t}\in\mathcal{C}([0,T];W^{k,\infty}(\mathbb{R}^{2d}))$ a.s. $\omega\in\Omega$. 2. (2)

(Integral relation): $f_{t}$ satisfies the equation (1.3) in distribution sense: for $\psi\in{\mathcal{C}}^{\infty}_{c}([0,T]\times\mathbb{R}^{2d})$,

[TABLE]

Remark 2.2**.**

1. We say $f_{t}$ is a classical solution to (LABEL:B-0) if it is a $\mathcal{F}_{t}$-semimartingale satisfying relation (LABEL:B-0) pointwise and the regularity condition $f_{t}\in L^{\infty}(\Omega;\mathcal{C}([0,T];\mathcal{C}^{3,\delta}(\mathbb{R}^{2d})))$ for some $\delta\in(0,1)$. We require this regularity condition to use Itô’s formula and the relation between Itô and Stratonovich integration without any restriction.

2. As can be seen later, the representation of a classical solution to (LABEL:B-0) via the stochastic characteristics shows that $f_{t}$ can not satisfy the $L^{\infty}$-boundedness over $\Omega$ due to the exponential Wiener process. To handle this, we would use a suitable stopping time.

Next, we are ready to provide a framework $(\mathcal{F})$ and main results below:

• •

($\mathcal{F}1$): The initial datum $f^{in}$ is nonnegative, compactly supported in $x$ and $v$ and independent of $\omega$.

• •

($\mathcal{F}2$): For $k\geq 1$, $f^{in}$ and $\phi$ are assumed to be in $W^{k,\infty}(\mathbb{R}^{2d})$ and $\mathcal{C}^{\infty}(\mathbb{R}^{2d})$, respectively.

• •

($\mathcal{F}3$): The first two moments of $f^{in}$ are normalized as follows:

[TABLE]

Under the framework $({\mathcal{F}})$, our main results can be summarized as follows.

Theorem 2.3**.**

Let $T\in(0,\infty)$ and assume that $f^{in}$ and $\phi$ satisfies the framework $(\mathcal{F})$. Then, there exists a strong solution $f_{t}$ to (LABEL:B-0) on $[0,T]$ such that

[TABLE]

Moreover, if a strong solution $f_{t}$ exists on $(0,\infty)$ and $\phi_{m}:=\inf_{x\in\mathbb{R}^{N}}\phi(x)>\sigma^{2}$, then one obtains an asymptotic flocking estimate:

[TABLE]

Proof.

For a proof, we first regularize the initial datum using the standard mollification and then solve the linearized system for (LABEL:B-0) to get a sequence of approximate solutions. Then, we use the stopping time argument to get a strong solution for (LABEL:B-0) with the given initial datum. The detailed proof will be presented in Section 4. ∎

Remark 2.3**.**

Note that for $k>3$, a strong solution $f_{t}$ to (LABEL:B-0) can be shown to satisfy the equation (LABEL:B-0) pointwise within our framework.

3. A priori estimates for classical solutions

In this section, we study a priori estimates for classical solutions to (1.3). First, we study several equivalent relations to the weak formulation (2.7), when a strong solution satisfies suitable conditions.

Lemma 3.1**.**

Suppose that for every $\psi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{2d})$ and a random process $f_{t}\in L^{\infty}(\Omega\times[0,T]\times\mathbb{R}^{2d})$, $\int_{\mathbb{R}^{2d}}f_{t}\psi dvdx$ has a continuous $\mathcal{F}_{t}$-adapted modification, where $\{\mathcal{F}_{t}\}$ is a family of $\sigma$-field generated by the Wiener process. Then, $f_{t}$ is a $\mathcal{F}_{t}$-semimartingale satisfying relation (2.7) if and only if for every $\psi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{2d})$,

[TABLE]

Proof.

The proof is almost the same as in Lemma 13 from [18], but we provide a sketch for a proof for readers’ convenience. Note that the following relation between Itô and Stratonovich integrals holds:

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the joint quadratic variation (see [26]). In our case, $h_{s}$ corresponds to $\int_{\mathbb{R}^{2d}}[(v-v_{c})f_{s}]\cdot\nabla_{v}\psi dvdx$. Then, to deal with $\langle h,W\rangle_{t},$, one needs to specify the stochastic part of $h_{s}$. Here, if we replace $\psi$ in (4.3) by $(v-v_{c})\cdot\nabla_{v}\psi$, we can find out that the stochastic part of $h_{s}$ becomes -\sigma\int_{0}^{t}\Big{[}\int_{\mathbb{R}^{2d}}((v-v_{c})f_{s})\cdot\nabla_{v}(v\cdot\nabla_{v}\psi)dvdx\Big{]}dW_{s}. This means

[TABLE]

and we may conclude the proof here. ∎

Once we reformulate relation (2.7) to Itô form (3.1), we can show that the solution process $f_{t}$ satisfies the following pointwise relation under the regularity condition for $f$.

Lemma 3.2**.**

Suppose that $f_{t}\in L^{\infty}(\Omega;\mathcal{C}([0,T];\mathcal{C}^{2}(\mathbb{R}^{2d})))$ has a continuous $\mathcal{F}_{t}$-adapted modification and has a compact support in $x$ and $v$. Then, $f_{t}$ satisfies relation (3.1) if and only if $f_{t}$ satisfies the following relation:

[TABLE]

Proof.

First, we assume that $f$ satisfies (3.1). Since $f_{t}$ is smooth and compactly supported, we use Fubini’s theorem to show that (3.1) is equivalent to

[TABLE]

Note that for each $\psi\in\mathcal{D}(\mathbb{R}^{2d})$, it satisfies the relation (3.3) outside $\mathbb{P}$-zero set depending on the choice of $\psi$. We recall from standard functional analysis that $\mathcal{D}(\mathbb{R}^{2d})$ is separable, i.e. there exists $\{\psi_{i}\}_{i=1}^{\infty}\subseteq\mathcal{D}(\mathbb{R}^{2d})$ which is dense in $\mathcal{D}(\mathbb{R}^{2d})$. Here, we choose $\Omega_{i}\subset\Omega$ such that $\mathbb{P}(\Omega_{i})=1$ and (2.7) holds for $f_{t}$ and $\psi_{i}$ over $\Omega_{i}$. Let $\tilde{\Omega}:=\cap_{i=1}^{\infty}\Omega_{i}$. Then $\mathbb{P}(\tilde{\Omega})=1$ and (2.7) holds for any $\psi_{i}$ and $f_{t}$ over $\tilde{\Omega}$.

Now, we show $f_{t}$ satisfies the relation (3.2). For this, we define functionals $\mathscr{L}_{t}[f]$ and $\mathscr{M}[f_{t}]$ as follows:

[TABLE]

For a given $(x^{*},v^{*})\in\mathbb{R}^{2d}$, we can choose a sequence $\{\rho_{i}\}\subset\mathcal{D}(\mathbb{R}^{2d})$, using the standard mollifier technique or other tools, such that for any $i\in\mathbb{N}$,

[TABLE]

where the regularity and compact support of $f$ can be used to guarantee the above inequality. We also use the denseness of $\{\psi_{i}\}$ to obtain $\{\tilde{\psi}_{i}\}\subseteq\{\psi_{i}\}$ which satisfies, for any $i\in\mathbb{N}$,

[TABLE]

Thus, we have

[TABLE]

Moreover, we use Itô isometry to get

[TABLE]

Hence, we can obtain the convergence of (3.3) with $\psi=\tilde{\psi}_{i}(x^{*}-x,v^{*}-v)$ towards (3.2) at $(x^{*},v^{*})$ as $i\to\infty$, by combining (3.4) and (3.5). We perform this procedure to obtain that for every $(x^{*},v^{*})\in\mathbb{R}^{2d}$, $f$ satisfies relation (3.2) $\mathbb{P}$-a.s. and this gives

[TABLE]

for every $(x,v)\in\mathbb{R}^{2d}$. Thus, we use Fubini theorem to get

[TABLE]

This implies our first assertion.

Next, we assume that $f$ satisfies (3.2) $\mathbb{P}\otimes dx\otimes dv$-a.s. Then by (deterministic) Fubini’s theorem, the following relation is easily obtained: for every $\psi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{2d})$,

[TABLE]

Since $f_{t}$ is in $L^{\infty}(\Omega;\mathcal{C}([0,T];\mathcal{C}^{2}(\mathbb{R}^{2d})))$ and compactly supported, we have

[TABLE]

Then, we can use the stochastic Fubini theorem (see [40] and references therein) and deterministic Fubini’s theorem to get

[TABLE]

This implies our desired result. ∎

Remark 3.1**.**

*1. If a strong solution $f_{t}$ to (1.3) satisfies conditions in Lemma 3.2, then $f_{t}$ satisfies the relation (3.2).

2. If $f_{t}$ is a classical solution to (1.3), we may use Lemma 3.2 in [9] to obtain that the Itô relation (3.2) is equivalent to (LABEL:B-0).

3.1. Quantitative estimates for classical solutions

We provide several properties of classical solutions $f$ to (1.3). First, we study the propagation of velocity moments along the stochastic flow of (LABEL:B-0)1. For a random density function $f_{t}$, we set velocity moments:

[TABLE]

Consider the following stochastic characteristics $\varphi_{t}(x,v):=(X_{t}(x,v),V_{t}(x,v))$:

[TABLE]

subject to the initial data:

[TABLE]

Note that if $f_{t}$ is compactly supported in $x$ and $v$, and satisfies the regularity condition for classical solutions, the system (3.7) has a unique solution and the family $\{\varphi_{s,t}(x,v):=\varphi_{t}(\varphi^{-1}_{s}(x,v))\}$, $0\leq s\leq t\leq T$, forms a stochastic flow of smooth diffeomorphisms (we refer to Lemma 4.1 in Chapter 2 of [9] for details). Furthermore, we define the functionals that measure spatial and velocity supports of $f_{t}$, respectively:

[TABLE]

Lemma 3.3**.**

Let $f_{t}$ be a classical solution to (LABEL:B-0) which is compactly supported in $x$ and $v$ and satisfies

[TABLE]

Then, we have

[TABLE]

Proof.

$\bullet$ (Conservation of mass): It follows from Remark 3.1 that

[TABLE]

We integrate (3.8) over $(x,v)\in\mathbb{R}^{2d}$ to get

[TABLE]

Next, we show that the terms ${\mathcal{I}}_{1i}$ are zero using deterministic and stochastic Fubini’s theorems.

$\diamond$ (Estimate of ${\mathcal{I}}_{11}$ and $\mathcal{I}_{13}$): Since $f_{t}$ has a compact support in $(x,v)$, we can use deterministic Fubini’s theorem to see

[TABLE]

$\diamond$ (Estimate of ${\mathcal{I}}_{12}$): As in the proof of Lemma 3.2, we can use the stochastic Fubini theorem to get

[TABLE]

$\bullet$ (Conservation of momentum): In this case, we multiply $v$ to (3.8) and use the same argument for conservation of mass to derive

[TABLE]

$\bullet$ (Dissipation estimate): We multiply (3.8) by $|v|^{2}$ and use stochastic Fubini’s theorem to have

[TABLE]

where we used the relation $M_{1}(t)=0$.

We use (3.9) to get

[TABLE]

Then we use Lemma A.1 and Lemma A.2 to get

[TABLE]

∎

Remark 3.2**.**

In Lemma 3.3, we observe that the first momentum is preserved. Thus, without loss of generality, we may assume that $v_{c}(t)=0$.

Next, we discuss the size of spatial and velocity supports of $f_{t}$.

Lemma 3.4**.**

The support functionals $\mathcal{X}$ and $\mathcal{V}$ satisfy the following estimates:

[TABLE]

Moreover, if $\phi_{m}>0$, then

[TABLE]

Proof.

First, we consider the case when $\phi_{m}>0$ may not hold.

$\diamond$ (Estimate of ${\mathcal{V}}$): Note that the stochastic characteristics $(X_{t},V_{t})=\{(x^{i}_{t},v^{i}_{t})\}_{i=1}^{d}$ starting from $(x,v)\in\mbox{supp}f^{in}$ satisfy

[TABLE]

Now, we rewrite (3.10)2 to have

[TABLE]

Thus, we apply Lemma A.1 and Lemma 3.3 to (3.11) to get

[TABLE]

Hence, we have

[TABLE]

where we used Young’s inequality, and this yields

[TABLE]

This gives the desired estimate.

$\diamond$ (Estimate of ${\mathcal{X}}$): We use Itô’s formula and the Cauchy-Schwarz inequality to get

[TABLE]

This and the estimates for $\mathcal{V}(t)$ yield

[TABLE]

We integrate the above differential inequality to get

[TABLE]

and this implies our desired estimate for $\mathcal{X}(t)$.

When $\phi_{m}>0$, we can use $\phi_{m}\leq\phi(2\mathcal{X}(s))$ to get the desired results.

∎

Remark 3.3**.**

Here, we discuss the necessity of the lower bound condition $\phi_{m}>0$ for flocking estimates. As observed in [25], the equation (LABEL:B-0) without noise exhibits flocking without the condition $\phi_{m}>0$, but it was attainable since the sizes of $x$- and $v$-supports increase at most in an algebraic order, which is not the case for (LABEL:B-0) due to the exponential Wiener process. Here, it is well known that

[TABLE]

Thus, for the pathwise flocking estimate, we require

[TABLE]

However, as observed in Lemma 4.1, it becomes difficult to estimate $\mathcal{X}(t)$ without the lower bound assumption $\phi_{m}>0$. Accordingly, it is hard to find a condition weaker than $\phi_{m}>0$ which entails the estimate (3.12).

Now, we are ready to state the stability results for (1.3).

Theorem 3.1**.**

($L^{\infty}$-stability)* Let $f_{t}$ and $\tilde{f}_{t}$ be two classical solutions to (1.3) corresponding to regular initial data $f^{in}$ and $\tilde{f}^{in}$, respectively, which are compactly supported in $x$ and $v$. Moreover, let $\varphi_{t}=\varphi_{t}(x,v)$ and $\tilde{\varphi}_{t}=\tilde{\varphi}_{t}(x,v)$ be the stochastic characteristics associated to $f$ and $\tilde{f}$, respectively. Then, we have*

[TABLE]

where $\mathcal{D}_{t}$ is a non-negatvie process with continuous sample paths and

[TABLE]

Proof.

Since the proof is rather lengthy, we postpone it to Appendix B. ∎

4. Global existence and asymptotic dynamics of strong solutions

In this section, we provide global existence and asymptotic flocking estimates for strong solutions to (1.3). Here, we show our desired estimates for (LABEL:B-0) corresponding to regularized initial data. Then, based on the stability estimates for classical solutions that we obtained in the previous section, we conclude that solutions to (LABEL:B-0) with regularized initial data converge to a strong solution to (LABEL:B-0). Moreover, we show that a strong solution obtained as above satisfies the asymptotic flocking estimates.

Let $f^{in,\varepsilon}$ be a smooth mollification of the given initial datum $f^{in}$ satisfying the framework $({\mathcal{F}})$. Then, consider the Cauchy problem (LABEL:B-0) with these regularized initial data:

[TABLE]

Note that due to the framework $(\mathcal{F})$, the initial datum $f^{in}$ and its partial derivatives up to order $k$ are uniformly continuous on $\mathbb{R}^{2d}$ and there exists a constant $R_{0}>0$, such that

[TABLE]

where $B_{R_{0}}(0)$ is a ball of radius $R_{0}$ centered at $0\in\mathbb{R}^{2d}$. As mentioned above, we use a mollifier to obtain a family of regularized initial data $f^{in,\varepsilon}\in\mathcal{C}^{\infty}(\mathbb{R}^{2d})$, $\varepsilon\in(0,1)$, so that the regularized datum satisfies the following conditions:

• •

($\mathcal{F}^{\varepsilon}1$): $\{f^{in,\varepsilon}\}$ are nonnegative, compactly supported, uniformly converge to $f^{in}$ in $\mathcal{C}^{0}(\mathbb{R}^{2d})$ and

[TABLE]

• •

($\mathcal{F}^{\varepsilon}2$): $\{M_{2}^{\varepsilon}\}(0)$ is uniformly bounded with respect to $\varepsilon$ and converges to $M_{2}(0)$ as $\varepsilon\to 0$.

• •

($\mathcal{F}^{\varepsilon}3$): The zeroth and first moment of $f^{in,\varepsilon}$ are initially constrained:

[TABLE]

• •

($\mathcal{F}^{\varepsilon}4$): $f^{in,\varepsilon}$ has a compact support in $x$ and $v$, and satisfy

[TABLE]

In the following three subsections, we will provide a global existence for system (LABEL:C-0).

4.1. Construction of approximate solutions

In this subsection, we provide a sequence of approximate solutions to (LABEL:C-0) using successive approximations.

First, the zeroth iterate $f_{t}^{0,\varepsilon}$ is simply defined as the mollified initial datum:

[TABLE]

For $n\geq 1$, suppose that the $(n-1)$-th iterate $f_{t}^{n-1,\varepsilon}$ is given. Then, the $n$-th iterate is defined as the solution to the linear equation with fixed initial datum:

[TABLE]

The linear system (4.2) can be solved by the method of stochastic characteristics. Let $\varphi_{t}^{n,\varepsilon}(x,v):=(X_{t}^{n,\varepsilon}(x,v),V_{t}^{n,\varepsilon}(x,v))$ be the forward stochastic characteristics, which is a solution to the following SDE:

[TABLE]

Note that the SDE (4.3) is equivalent to the following Itô SDE [17]:

[TABLE]

Here, we can deduce from Lemma 3.1 and Theorem 3.2 in [9] and our framework that for any $m\geq 3$, (4.3) has a unique solution $f_{t}^{n,\varepsilon}$ which is a $\mathcal{C}^{m}$-semimartingale for every $n\geq 0$ and the characteristics (4.3) becomes a $\mathcal{C}^{m}$-diffeomorphism. Then, $f_{t}^{n,\varepsilon}$ can also be represented by the following integral formula:

[TABLE]

Note that if $f^{in,\varepsilon}$ is nonnegative, then surely $f_{t}^{n,\varepsilon}$ is also nonnegative as well. Before we finish this subsection, we also remark that the linear, first-order Stratonovich equation (4.2) is equivalent to the following parabolic Itô equation (see Corollary 3.3. in [9]):

[TABLE]

4.2. Estimates on approximate solutions

In this subsection, we provide several estimates for the approximate solutions for (4.2). To be more precise, we would try to obtain $n$ and $\varepsilon$-independent estimates for the later sections. Before we move on, we define $p$-th velocity moments $M^{n,\varepsilon}_{p}(t)$, $p=0,1,2$:

[TABLE]

Before we provide the uniform estimates for the $p$-th ($p=0,1,2$) moments, we set

[TABLE]

We also present a technical lemma from [3] for a later discussion.

Lemma 4.1**.**

[3]** Let $T\in(0,\infty]$ and $(a_{n})_{n\in\mathbb{N}}$ be a sequence of nonnegative continuous functions on $[0,T]$ satisfying

[TABLE]

*where $A$, $B$ and $C$ are nonnegative constants.

1. (1)

If $A=0$, there exists a constant $\Lambda\geq 0$ depending on $B$, $C$ and $\displaystyle\sup_{t\in[0,T]}a_{0}(t)$ such that

[TABLE] 2. (2)

If $A>0$ and $C=0$, there exists a constant $\Lambda\geq 0$ depending on $A$, $B$ and $\displaystyle\sup_{t\in[0,T]}a_{0}(t)$ such that

[TABLE]

Remark 4.1**.**

1. In (2) of Lemma 4.1, $\Lambda$ can be explicitly written as

[TABLE]

2. We can also use the similar argument to obtain the following estimate for (2):

[TABLE]

where $\Lambda:=\max\{A,B\}$ and $\kappa_{t}:=\sup_{0\leq s\leq t}a_{0}(s)$.

Proposition 4.1**.**

For every $n\in\mathbb{N}$ and $T\in(0,\infty)$, let $f_{t}^{n,\varepsilon}$ be a solution to (4.2). Then, for any $t\in(0,T)$ we have

[TABLE]

where $\gamma$ is a constant in (4.7) and $K_{t}$ is defined as

[TABLE]

Proof.

Note that $f_{t}^{n,\varepsilon}$ satisfies relation (4.6) and $f_{t}^{n,\varepsilon}$ is compactly supported in $x$ and $v$, since $f^{in,\varepsilon}$ is compactly supported in the phase space and $\varphi_{t}^{n,\varepsilon}$ is a $\mathcal{C}^{m}$-diffeomorphisim. Thus, we may follow the arguments in Lemma 3.3 to derive the conservation estimates.

For the dissipation estimate of $M_{2}^{n,\varepsilon}$, we use a similar argument to Lemma 3.3 to have

[TABLE]

where we used Young’s inequality on the second inequality. In a differential form, we have

[TABLE]

Then, it follows from (4.8) and comparison theorem (in Lemma A.2) that

[TABLE]

where the process $X_{t}$ satisfiesf

[TABLE]

It follows from Lemma A.1 that $X_{t}$ can be represented as

[TABLE]

This implies

[TABLE]

Now, we set

[TABLE]

Then, it satisfies

[TABLE]

We use Lemma 4.1 in the way from Remark 4.1 to get

[TABLE]

This yields the desired result. ∎

We also provide uniform estimates for the stochastic characteristic flows.

Proposition 4.2**.**

For each $n\in\mathbb{N}$ and $T\in(0,\infty)$, let $(X_{t}^{n,\varepsilon},V_{t}^{n,\varepsilon})$ be the stochastic characteristic flow for (4.2) with the initial data:

[TABLE]

Then for $t\in(0,T)$, we have

[TABLE]

Proof.

(i)  It follows from Itô’s lemma and (4.4) that

[TABLE]

where $dV_{t}^{n,\varepsilon}\cdot dV_{t}^{n,\varepsilon}$ denotes a handy notation for a quadratic variation of $V_{t}^{n,\varepsilon}$.

We use Proposition 4.1 and Lemmas A.1-A.2 to get

[TABLE]

(ii) For the estimate of spatial process, we use Cauchy-Schwarz inequality to get

[TABLE]

This yields the desired result.

∎

As a corollary of Proposition 4.2, we have estimates for the sizes of velocity and spatial supports: We set

[TABLE]

Corollary 4.1**.**

For each $n\in\mathbb{N}$ and $T\in(0,\infty]$, let $(X_{t}^{n,\varepsilon},V_{t}^{n,\varepsilon})$ be the stochastic characteristic flow for (4.2) with the initial data:

[TABLE]

Then for $t\in(0,T)$, we have

[TABLE]

where $\mathcal{X}^{\infty}(t)$ and $\mathcal{V}^{\infty}(t)$ are given by the following relations:

[TABLE]

Proof.

It follows from Proposition 4.2 that

[TABLE]

This yields the first estimate for velocity support. On the other hand, we also use Proposition 4.2 to get

[TABLE]

∎

Remark 4.2**.**

Note that $f_{t}^{n,\varepsilon}$ has compact supports in $x$ and $v$ for every sample path which are bounded uniformly in $n$ and $\varepsilon$.

Now, we are ready to state the results on the uniform bound for the sequence $\{f_{t}^{n,\varepsilon}\}$.

Proposition 4.3**.**

For every $n$, $m\in\mathbb{N}$ and $t\in(0,T)$, there exists a nonnegative process $\mathcal{A}^{m}_{t}$ which has continuous sample paths and is independent of $n$ and $\varepsilon$ such that

[TABLE]

Proof.

Since the proof is quite lengthy, we postpone it to Appendix C. ∎

Remark 4.3**.**

It is easy to see that for fixed $t$ and $\omega$, $\mathcal{A}_{t}^{m}$ is monotonically increasing with respect to $m$.

Next, we prove that the sample paths of approximate solutions become a Cauchy sequence in a suitable functional space.

Proposition 4.4**.**

For every $n$ and $t\in(0,T)$, there exists a nonnegative process $\tilde{\mathcal{D}}_{t}$ which has continuous sample paths and is independent of $n$ and $\varepsilon$ such that

[TABLE]

Proof.

Since the proof is almost the same as that of Theorem 3.1, we only point out some differences. In the proof of Theorem 3.1, we just replace $\mathcal{R}(t)$, $\mathcal{P}(t)$, $\max(\|f^{in}\|_{L^{\infty}},\|\tilde{f}^{in}\|_{L^{\infty}})$ and $\max(\|f_{t}\|_{W^{1,\infty}},\|\tilde{f}_{t}\|_{W^{1,\infty}})$ by $\mathcal{X}^{\infty}(t)$, $\mathcal{V}^{\infty}(t)$, $\|f^{in}\|$ and $\|f^{in}\|_{W^{1,\infty}}\mathcal{A}_{t}^{1}$, respectively. Then it becomes our desired estimate and hence, we can actually get

[TABLE]

where

[TABLE]

This gives the desired result.

∎

For each $t$ and $\omega\in\Omega$, we define

[TABLE]

Corollary 4.2**.**

The functional $\Delta_{n}^{\varepsilon}(t)$ satisfies

[TABLE]

where $\mathcal{K}=\mathcal{K}(\omega)$ is a nonnegative random variable.

Proof.

It follows from Proposition 4.4 that

[TABLE]

Since $\tilde{\mathcal{D}}_{t}$ is a nonnegative process with continuous sample paths, there exists a nonnegative random variable $D=D(\omega)$ such that

[TABLE]

Thus, we can use the Grönwall-type lemma in Lemma 4.1 to deduce

[TABLE]

where $\mathcal{K}=\mathcal{K}(\omega)$ depends on $D(\omega)$. ∎

Remark 4.4**.**

Corollary 4.2 implies that for every $\omega$,

[TABLE]

Since $f_{t}^{n,\varepsilon}$ is $\mathcal{F}_{t}$-adapted (where $\mathcal{F}_{t}$ is a filtration generated by the Wiener process) and $f_{t}^{\varepsilon}$ is a pointwise limit of $f^{n,\varepsilon}$ over $\Omega$, we have $f$ is $\mathcal{F}_{t}$-adapted. Moreover, we have a uniform boundedness of $f_{t}^{n,\varepsilon}$ in $L^{\infty}([0,T];W^{m,p}(\mathbb{R}^{2d}))$ for any $p\in[1,\infty)$. By the property of reflexive Banach space, there exists a subsequence $\{f^{n_{k},\varepsilon}(\omega)\}\subseteq\{f^{n,\varepsilon}(\omega)\}$ which is weakly convergent to $\tilde{f}_{t}(\omega)$ in $L^{\infty}([0,T];W^{m,p}(\mathbb{R}^{2d}))$ for each $\omega\in\Omega$ and every $p\in[1,\infty)$. Since we have already a strong convergence in the lower order, we can conclude that $f_{t}^{\varepsilon}(\omega)=\tilde{f}_{t}(\omega)$. However, we can not proceed further, since it is not clear whether $f_{t}^{\varepsilon}$ satisfies the equation (1.3) at this moment. This is due to the noise term in the right-hand side of (4.2). It is not certain whether the Stratonovich integral of $f_{t}^{\varepsilon}$ can be defined or not. In addition, even if the noise term can be well-defined, it is also not clear whether the Stratonovich integral of $f^{n,\varepsilon}$ converges to that of $f_{t}^{\varepsilon}$ or not.

4.3. Proof of Theorem 2.3

In this subsection, we prove a global existence of a solution to system (4.2) by showing that the limit of the sequence $\{f_{t}^{n,\varepsilon}\}$ exists as $n\to\infty$ for each $\varepsilon$, and that this limit is indeed a strong solution to (3.6) corresponding to the regularized initial datum $f^{in,\varepsilon}$.

In order to cope with the problems discussed in Remark 4.4, we employ a stopping time argument. First, we define a sequence of stopping times $\{\tau_{M}\}_{M\in\mathbb{N}}$ as follows:

[TABLE]

where $k_{*}:=\max\{k,4\}$ and $\mathscr{D}_{t}$ is a nonnegative process with continuous sample paths which will be specified later. Now, we verify the existence of regularized solutions step by step.

$\bullet$ (Step A: The limit $n\to\infty$): First, we obtain the limit function $f_{t\wedge\tau_{M}}^{\varepsilon}$ which is a classical solution to equation (1.3) with the regularized initial data, based on the estimates in the previous subsection.

$\diamond$ (Step A-1: Extracting a limit function): We can find out that for each $n\in\mathbb{N}$,

[TABLE]

Thus, we can use the same argument as in Corolllary 4.2 to yield that as $n\to\infty$, there exists a limit function $f_{t\wedge\tau_{M}}^{\varepsilon}$ such that, up to a subsequence,

[TABLE]

$\diamond$ (Step A-2: Verification of relation (2.7)): Now, we need to show that $f_{t\wedge\tau_{M}}^{\varepsilon}$ satisfies (4.2) in the sense of Definition 2.1. Since $f_{t\wedge\tau_{M}}^{n,\varepsilon}$ satisfies (4.6) and conditions of Lemma 3.2, it satisfies the following relation:

[TABLE]

where $\Sigma:=\mathbb{R}^{2d}$ and $dz=dvdx$.

Next, our job is to pass $n\to\infty$ in the integral relation (LABEL:C-9) to derive an integral relation (2.7) for $f^{\varepsilon}_{t\wedge\tau_{M}}$. For this, note that the $x$- and $v$-supports of $f^{n,\varepsilon}_{t\wedge\tau_{M}}$ and $f^{\varepsilon}_{t\wedge\tau_{M}}$ are uniformly bounded by $|\mathcal{X}_{t\wedge\tau_{M}}^{\infty}|$ and $|\mathcal{V}_{t\wedge\tau_{M}}^{\infty}|$ (see Corollary 4.1). Moreover, we can find out that $|\mathcal{X}_{t\wedge\tau_{M}}^{\infty}|$ and $|\mathcal{V}_{t\wedge\tau_{M}}^{\infty}|$ are bounded by $\mathcal{A}_{t\wedge\tau_{M}}^{k_{*}}$, and hence by $M$. We combine the strong convergence on the lower order with these facts to yield

[TABLE]

uniformly in $\omega$, as $n$ goes to infinity.

Now it remains to check with the stochastic integral term in (LABEL:C-9). For this term, one has

[TABLE]

This $L^{2}$-convergence over $\Omega$ implies that there exists a subsequence $\{f^{n_{l},\varepsilon}_{t\wedge\tau_{M}}\}$ such that

[TABLE]

for a.s. $\omega$, as $l$ goes to infinity. Thus, we can conclude that for a.s. $\omega\in\Omega$, $f_{s\wedge\tau_{M}}^{\varepsilon}$ satisfies

[TABLE]

for every $\psi\in\mathcal{D}(\mathbb{R}^{2d})$. One also has $f_{t\wedge\tau_{M}}^{\varepsilon}$ is a $\mathcal{F}_{t}$-semimartingale. Here, we use Lemma 3.1 to obtain that $f_{t\wedge\tau_{M}}^{\varepsilon}$ satisfies (1.3) in the sense of distribution.

$\bullet$ (Step B: The limit $\varepsilon\to 0$): Here, we address the convergence of solutions to the regularized system (LABEL:C-0). Since $k^{*}\geq 4$, one uses the Sobolev embedding theorem to get $f_{t\wedge\tau_{M}}^{\varepsilon}\in L^{\infty}(\Omega;\mathcal{C}([0,T];\mathcal{C}^{3,\delta}(\mathbb{R}^{2d})))$. Thus, it follows from Lemma 3.2 and Remark 3.1 that $f_{t\wedge\tau_{M}}^{\varepsilon}$ becomes a classical solution to (LABEL:C-0) corresponding to the regularized initial datum $f^{in,\varepsilon}$ .

$\diamond$ (Step B-1: Extracting a limit function):  Note that the strong convergence in Step A implies that the $x$-support and the $v$-support of $f_{t\wedge\tau_{M}}^{\varepsilon}$ are bounded by $\mathcal{X}^{\infty}$ and $\mathcal{V}^{\infty}$, respectively, uniformly in $\varepsilon$. Thus, we can follow the stability estimate in Theorem 3.1 to get

[TABLE]

where $\mathscr{D}_{t}$ can be obtained if $\mathcal{R}(t)$, $\mathcal{P}(t)$, $\max(\|f^{in}\|_{L^{\infty}},\|\tilde{f}^{in}\|_{L^{\infty}})$ and $\max(\|f_{t}\|_{W^{1,\infty}},\|\tilde{f}_{t}\|_{W^{1,\infty}})$ in the formulation of $\mathcal{D}_{t}$ from Theorem 3.1 are substituted by $\mathcal{X}^{\infty}(t)$, $\mathcal{V}^{\infty}(t)$, $\|f^{in}\|$ and $\|f^{in}\|_{W^{1,\infty}}\mathcal{A}_{t}^{1}$, respectively.

Since $f^{in,\varepsilon}$ converges uniformly to $f^{in}$, it follows from the stability estimate (LABEL:D-1) that there exists $f_{t\wedge\tau_{M}}$ such that

[TABLE]

Moreover, it follows from the weak convergence and ($\mathcal{F}^{\varepsilon}1$) that

[TABLE]

Hence, we can follow the arguments in Step A to yield that $f_{t\wedge\tau_{M}}$ satisfies relation (3.1) and hence (2.7). Moreover, $f_{t\wedge\tau_{M}}$ is compactly supported in $x$ and $v$.

$\diamond$ (Step B-2: Regularity of a strong solution): Now, we prove that $f_{t\wedge\tau_{M}}$ has the desired regularity. Since $f_{t\wedge\tau_{M}}^{\varepsilon}$ is a classical solution to (1.3) with initial data $f^{in,\varepsilon}$, it can be uniquely written as

[TABLE]

(for detail, we refer to Appendix B). Since we also obtain the uniform convergence of the characteristics $\varphi_{t\wedge\tau_{M}}^{\varepsilon}$ as $\varepsilon\to 0$ from (LABEL:D-1), the solution $f_{t\wedge\tau_{M}}$ satisfies the following relation:

[TABLE]

and the limit $\varphi_{t\wedge\tau_{M}}(x,v)=(X_{t\wedge\tau_{M}}(x,v),V_{t\wedge\tau_{M}})$ is a solution to the following SDE:

[TABLE]

Since the kernel $F_{a}[f_{t}]$ is smooth, $\varphi_{t\wedge\tau_{M}}(x,v)$ can be shown to be a $\mathcal{C}^{m}$-diffeomorphism for any $m\in\mathbb{N}$, and so is its inverse $\psi_{t\wedge\tau_{M}}(x,v):=(\varphi_{t}(x,v))^{-1}$. Thus, if we write

[TABLE]

it directly follows from the regularity of $f^{in}$ and $\psi_{t}$ that $f_{t\wedge\tau_{M}}$ has the desired regularity.

$\bullet$ (Step C: Properties of a strong solution):  We recall several properties of regularized solutions. First, it is obvious from (4.12) that

[TABLE]

Since $f_{t\wedge\tau_{M}}^{\varepsilon}$ is a classical solution to (LABEL:C-0) corresponding to the regularized initial datum $f^{in,\varepsilon}$, Lemma 3.3 gives

[TABLE]

and the strong convergence together with compact supports gives

[TABLE]

Moreover, it is obvious that

[TABLE]

Thus, we choose a sufficiently large $M$ for each $\omega\in\Omega$ such that $f_{t\wedge\tau_{M}}(\omega)$ satisfies the relation (2.7) on $[0,T]$.

For the expectation estimates of the solution, we use Fatou’s lemma to get, for any $p\in(1,\infty)$,

[TABLE]

where we used the fact $X_{t}=\exp(aW_{t}-a^{2}t/2)$ is a martingale, Hölder’s inequality and Lebesgue’s dominated convergence theorem. Then we take the limit $p\to 1$ on both sides to obtain the desired result. For the dissipation of the second velocity moment, we use a similar argument to get the desired estimate.

5. Conclusion

In this paper, we presented a global existence of strong solutions and their asymptotic emergent dynamics for the stochastic kinetic Cucker-Smale equation perturbed by a multiplicative white noise. For a global well-posedness, we first derive a sequence of classical solutions to the stochastic kinetic C-S equation with regularized initial data. Then, using the properties of classical solutions, we obtained the existence of a strong solution corresponding to the original initial data and asymptotic emergent stochastic dynamics of strong solutions. Of course, there are lots of interesting issues to be addressed in a future work, e.g., a global existence of weak solutions, emergent dynamics under other types of random perturbations and zero noise limit, etc. These topics will be discussed in future works.

Appendix A Elementary lemmas

In this appendix, we provide two useful lemmas used in previous sections. First, we begin with estimate on a variant of geometric brownian motion.

Lemma A.1**.**

Let $\{X_{t}\}_{t\geq 0}$ be a solution satisfying the following Cauchy problem:

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where $\{a_{t}\}_{t\geq 0}$ and $\{b_{t}\}_{t\geq 0}$ are stochastic processes with continuous sample paths, and $c$ is a constant. Then one has

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Proof.

The proof is exactly given in Example 19.7 from [36]. So, we refer to [36] for its proof.

∎

Lemma A.2**.**

(Comparision principle)* Suppose that two stochastic proceeses $\{X_{t}\}_{t\geq 0}$ and $\{Y_{t}\}_{t\geq 0}$ satisfy*

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where $\{a_{t}\}_{t\geq 0}$ is a stochastic process with continuous sample paths. Then, we have

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Proof.

Let $\{Y_{t}^{\delta}\}_{t\geq 0}$, ($\delta>0$) be a stochastic process satisfying

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and we set

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Then, we have

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We use Itô’s lemma to get

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Again, we integrate the above relation to get

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This yields

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It follows from the representation formula in Lemma A.1 that

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This yields the desired result:

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∎

Appendix B A proof of Theorem 3.1

First, we define $\eta$ and $\tilde{\eta}$ as follows:

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We use the generalized Itô’s formula from Theorem 3.3.2 in [26] to obtain that $\tilde{\eta}_{t}$ satisfies the relation (1.3). Since the classical solutions can become measure-valued solutions and the uniqueness of measure-valued solutions is guaranteed in Theorem 2.1, we have

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Moreover, since $\eta$ is a geometric Brownian motion, a unique classical solution $f$ corresponding to the initial datum $f^{in}$ can be represented by

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Now, we consider another classical solution $\tilde{f}$ corresponding to the initial datum $\tilde{f}^{in}$ and the associated stochastic flow $\tilde{\varphi}_{t}(x,v)$. Moreover, we set

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Then, we claim

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where $\mathcal{B}_{t}^{i}$ and $\mathcal{C}_{t}^{i}$ ($i=1,2$) are nonnegative processes which have continuous sample paths.

(i)   First, we derive the $L^{\infty}$-estimates for classical solutions:

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$\bullet$ (Estimate of $\mathcal{I}_{21}$): By direct estimate, one has

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where we used the mean-value theorem, and we have

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Thus, we get

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$\bullet$ (Estimate of $\mathcal{I}_{22}$): By direct estimate, one has

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Hence, we take the supremum over $(x,v)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$, and use Young’s inequality and the Cauchy-Schwarz inequality to get

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Setting

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we obtain the desired result (i) of (LABEL:T3-1.1).

(ii)  Now, we estimate $\|\varphi_{t}-\tilde{\varphi}_{t}\|_{L^{\infty}}$. It follows from (3.7) and Itô’s lemma that

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Here, we have

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We separately estimate the $\mathcal{I}_{23i}$’s as follows:

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Thus, by Lemma A.1 we get

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Moreover, it is easy to obtain that

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Thus, if we define $\mathcal{B}_{t}^{2}$ and $\mathcal{C}_{t}^{2}$ as

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then (ii) of (LABEL:T3-1.1) can be fulfilled with the above $\mathcal{B}_{t}^{2}$ and $\mathcal{C}_{t}^{2}$.

Therefore, we add $(i)$ in (LABEL:T3-1.1)1 to $(1+2\max(\|f_{t}\|_{W^{1,\infty}},\|\tilde{f}_{t}\|_{W^{1,\infty}}))$ times $(ii)$ in (LABEL:T3-1.1)2 and obtain

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where $\tilde{\mathcal{B}}_{t}$ is given by

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Then, letting y_{t}:=\int_{0}^{t}\Big{(}\|\varphi_{s}-\tilde{\varphi}_{s}\|_{L^{\infty}}^{2}+\|f_{s}-\tilde{f}_{s}\|_{L^{\infty}}^{2}\Big{)}ds, we have

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Then, by Grönwall’s lemma we get

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and this gives

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Hence, defining

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we arrive at the desired estimate.

Appendix C A proof of Proposition 4.3

Recall that $f_{t}^{n,\varepsilon}$ satisfies a differential form:

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i.e., it satisfies

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Next, we claim: there exists a nonnegative process $\mathcal{A}^{m}_{t}$ with continuous sample paths such that

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In the sequel, we provide $L^{\infty}$-estimate of $f_{t}$ and its derivatives to provide a proof of Proposition 4.3.

$\bullet$ (Zeroth-order estimate): It follows the formula (4.5) that

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This implies the zeroth-order estiamte:

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$\bullet$ (Higher-order estimates): Let $\alpha$ and $\beta$ be multi-indices satisfying

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Then, we apply $\partial_{x}^{\alpha}\partial_{v}^{\beta}$ to the relation (C.1) using Theorem 3.1.2 in [26]:

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where we used the relation:

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Note that the differentiation equality (C.3) is only true outside a $\mathbb{P}$-zero set in $\Omega$ which depends on $(x,v)$, according to Theorem 3.1.2 in [26]. However, we can use the argument in Lemma 3.2 to obtain that the equality also holds $\mathbb{P}\otimes dx\otimes dv$-a.s. Now, we rearrange the previous relation to obtain

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where the process $\mathcal{L}_{\alpha,\beta}$ is given by the following relation:

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Next, we define $\lambda$ and $\tilde{\lambda}$ as follows:

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By using generalized Itô’s formula from Theorem 3.3.2 in [26], $\tilde{\lambda}_{t}$ satisfies the relation (LABEL:App-B.2). Thus, by the uniqueness,

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and we use Itô’s formula on $\lambda_{t}$ to get

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For detailed explanation for the above realtion, we refer to the proof of Theorem 3.2 in [9].

Note that the following estimates hold:

• •

If $|\beta|=1$, one has

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• •

If $|\alpha|\geq 1$, one gets

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We set $C_{\alpha,\beta}(t)$ to be

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This yields

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Thus, we have

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Now, we take the supremum over all characteristic flow, sum (C.5) over all $1\leq|\alpha|+|\beta|\leq m$ and combine this with (C.2) to obtain

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where the process $\mathcal{M}_{t}^{m}$ is given by the following relation:

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Note that $\mathcal{M}_{t}^{m}$ is independent of $n$ and $\varepsilon$. We set

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Then, one gets

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where the process $\tilde{\mathcal{N}}_{s}^{m}$ is

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Thus, we can use Grönwall’s lemma to obtain

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where the process ${\mathcal{A}}_{t}^{m}$ is given by the following relation:

[TABLE]