uniform measure attractors of McKean-Vlasov stochastic reaction-diffusion equations on unbounded thin domain

In this paper, our objective is to investigate the limiting behaviour of uniform measure attractors for the following non-autonomous, almost periodic stochastic reaction-diffusion equation driven by nonlinear noise defined on $\mathcal{O}_{\varepsilon}$.

with initial data

where $\lambda>0$ denotes a positive constant, $\mathcal{L}_{\hat{u}^{\varepsilon}(t)}$ represents the probability distribution of $\hat{u}^{\varepsilon}(t)$, $f$ is a nonlinear function possessing an arbitrary growth rate, and $g$ signifies an almost-periodic external term in $t$. Furthermore, $\nu_{\varepsilon}$ denotes the unit outward normal vector to the boundary $\partial\mathcal{O}_{\varepsilon}$, $\kappa\in L^{2}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$, $\sigma_{k}$ is defined on $\widetilde{{\mathcal{O}}}$, $\varpi_{k}$ is a nonlinear diffusion term, and $(W_{k})_{k\in\mathbb{N}}$ is a sequence of independent two-sided real-valued standard Wiener processes on a complete filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in\mathbb{R}},P)$ which satisfies the usual condition.

The unbounded thin domain, denoted as $\mathcal{O}_{\varepsilon}$, is rigorously defined as

where $\rho\in C^{2}\left(\mathbb{R}^{n},(0,+\infty)\right)$, and $0<\varepsilon\leq 1$. Consequently, there exist constants $\rho_{1}$ and $\rho_{2}$ such that

Let $\mathcal{O}=\mathbb{R}^{n}\times(0,1)$ and $\widetilde{\mathcal{O}}=\mathbb{R}^{n}\times\left[0,\rho_{2}\right]$, where $\widetilde{\mathcal{O}}=\mathbb{R}^{n}\times\left[0,\rho_{2}\right]$ contains $\mathcal{O}_{\varepsilon}$ for $0<\varepsilon\leq 1$.

It is noteworthy that the unbounded thin domain $\mathcal{O}_{\varepsilon}$ undergoes a collapse, converging to the space $\mathbb{R}^{n}$ as $\varepsilon\rightarrow 0$. We will investigate the existence and uniqueness and the limit of uniform measure attractors of (1.1)-(1.2) as $\varepsilon\rightarrow 0$.

For $\varepsilon=0$, the limit equation of (1.1) reduces to the follow system defined on $\mathbb{R}^{n}$

with initial condition

where $y^{*}=(y_{1},\cdots,y_{n})\in\mathbb{R}^{n}$.

The McKean-Vlasov stochastic differential equations (MVSDEs) were initially examined in the seminal works of M. McKean [33] and V. Vlasov [45]. These equations frequently emerge from the realm of interacting particle systems, as evidenced in the extensive literature[3, 12, 13]. It is the distinctive quality of these differential equations that they depend not only on the states of the solutions, but also on the distributions of the solutions. Consequently, the Markov operators pertinent to MVSDEs cease to constitute semigroups are no longer semigroups (as illustrated in [47]), thereby rendering the methodologies devised for addressing stochastic equations independent of distributions inapplicable to MVSDEs in a straightforward manner.

At present, a considerable number of academic publications are dedicated to the examination of solutions to MVSDEs. For instance, the existence of solutions to MVSDEs has been examined in previous research [2, 14, 21]. Additionally, the existence and ergodicity of invariant measures for finite-dimensional MVSDEs have been investigated in other studies [4, 22, 48]. In a recent publication, Shi et al. [42] innovatively utilized the theory of pullback measure attractors to demonstrate the existence of invariant measures and periodic measures for infinite-dimensional MVSDEs.

For non-autonomous random dynamical systems, there are typically two kinds of pathwise pullback random attractor which have drawn much attention in last years: cocycle attractors introduced in [46] and uniform random attractors introduced in [10]. For further details, please refer to the following sources:[11, 19, 20, 24, 25].

It bears noting that the notion of a measure attractor for autonomous dynamical systems in measure spaces was initially posited by Schmalfu$\ss$ in [38]. For further details concerning the existence of measure attractors for autonomous stochastic equations, the reader is directed to the references [34, 35, 39]. A recent investigation by Li and Wang [31] explored the limiting dynamical behaviour of pullback measure attractors in a system of semilinear parabolic stochastic equations with deterministic non-autonomous forcing in bounded thin domains. Notably, this investigation did not assume any particular constraints on the external forces involved, including translation-bounded. Subsequently, Li et al. initially explored uniform measure attractors in the context of stochastic Navier-Stokes equations, as outlined in their work cited in [28]. In this paragraph, however, it is worth noting that in all of the articles mentioned above, the stochastic equations do not depend on the distributions of the solutions.

The dynamics of deterministic partial differential equations (PDEs) on thin domains were initially investigated by Hale and Raugel [15, 16, 17]. Subsequently, a considerable number of models have extended these initial findings, as demonstrated by the following references [1, 23, 36, 37]. Recently, several results have been published concerning stochastic dynamical systems on thin domains. For example, see the references [27, 29, 32] for studies of random attractors on bounded thin domains and the references [40, 41] for studies of random attractors on unbounded thin domains.

In order to illustrate the principal conclusions of this paper and for the sake of convenience, we shall use equations (3.21), which are the equivalent of equations (1.1)-(1.2). we introduce the notation $\mathcal{P}\left(L^{2}\left(\mathcal{O}\right)\right)$ to represent the space of probability measures on $\left(L^{2}\left(\mathcal{O}\right),\mathcal{B}\left(L^{2}\left(\mathcal{O}\right)\right)\right.$, where $\mathcal{B}\left(L^{2}\left(\mathcal{O}\right)\right)$ denotes the Borel $\sigma$-algebra associated with $L^{2}\left(\mathcal{O}\right)$. Notably, the weak topology of $\mathcal{P}\left(L^{2}\left(\mathcal{O}\right)\right)$ is metrizable, and the corresponding metric is denoted by $d_{\mathcal{P}\left(L^{2}\left(\mathcal{O}\right)\right)}$. Set

Then $\left(\mathcal{P}_{4}\left(L^{2}\left(\mathcal{O}\right)\right),d_{\mathcal{P}\left(L^{2}\left(\mathcal{O}\right)\right)}\right)$ is a metric space. Given $r>0$, denote by

Given $\tau\leq t$ and $\mu\in\mathcal{P}_{4}\left(L^{2}\left(\mathcal{O}\right)\right)$, let $P_{*}^{g,\varepsilon}(t,\tau)\mu$ be the law of the solution of (3.21) with initial law $\mu$ at initial time $\tau$. If $\phi:L^{2}\left(\mathcal{O}\right)\rightarrow\mathbb{R}$ is a bounded Borel function, then we write

where $u^{\varepsilon}\left(t,\tau,\xi^{\varepsilon}\right)$ is the solution of (3.21) with initial value $\xi^{\varepsilon}$ at initial time $\tau$. Note that for the McKean-Vlasov stochastic equations like (3.21), $P_{*}^{g,\varepsilon}(t,\tau)$ is not the dual of $P^{g,\varepsilon}(\tau,t)$ (see [47]) in the sense that

where $\mu\in\mathcal{P}_{4}\left(L^{2}\left(\mathcal{O}\right)\right)$ and $\phi:L^{2}\left(\mathcal{O}\right)\rightarrow\mathbb{R}$ is a bounded Borel function.

To prove the existence of uniform measure attractors for (3.21), we first need to show $\left\{P^{g,\varepsilon}_{*}(t,\tau)\right\}_{\tau\leq t}$ is a jointly continuous process, i.e., it is continuous in $\mathcal{P}_{4}(L^{2}(\mathcal{O}))\times\mathcal{H}(g_{0})$. In general, if a stochastic equation does not depend on distributions of the solutions, then the continuity of $\left\{P^{g,\varepsilon}_{*}(t,\tau)\right\}_{\tau\leq t}$ on $\left(\mathcal{P}_{4}\left(L^{2}\left(\mathcal{O}\right)\right),d_{\mathcal{P}\left(L^{2}\left(\mathcal{O}\right)\right)}\right)$ follows from the Feller property of $\left\{P^{g,\varepsilon}(t,\tau)\right\}_{\tau\leq t}$ and the duality relation between $P^{g,\varepsilon}_{*}(t,\tau)$ and $P^{g,\varepsilon}(t,\tau)$. However, this method does not apply to the McKean-Vlasov stochastic reaction-diffusion equation (1.1)-(1.2), because $P^{g,\varepsilon}_{*}(t,\tau)$ is no longer the dual of $P^{g,\varepsilon}(t,\tau)$ as demonstrated by (1.6). To surmount this impediment, we leverage the regularity properties inherent in $\mathcal{P}_{4}(L^{2}(\mathcal{O}))$ and invoke the Vitali theorem to establish the continuity of $\mathcal{P}^{g,\varepsilon}_{*}(t,\tau)$ within the restricted domain$(B_{\mathcal{P}_{4}(L^{2}(\mathcal{O}))},d_{\mathcal{P}(L^{2}(\mathcal{O}))})\times\mathcal{H}(g_{0})$ rather than endeavoring to prove it across the entire space $(\mathcal{P}_{4}(L^{2}(\mathcal{O})),d_{\mathcal{P}(L^{2}(\mathcal{O}))})\times\mathcal{H}(g_{0})$.

The first goal of this paper is to prove the existence and uniqueness of uniform measure attractors for the almost periodic external term in McKean-Vlasov stochastic equation (1.1)-(1.2) which is dependent on the laws of solutions. To this end, the estimates of the solutions must be uniform with respect to all translations of the external term involved in the system.

The secondary objective of this paper is to investigate the limiting behavior of the uniform measure attractors associated with the system of (1.1)-(1.2) as $\varepsilon\rightarrow 0$. Specifically, we aim to understand how these attractors, originating from $(n+1)$-dimensional unbounded thin domains, undergo a collapse into the space $\mathbb{R}^{n}$.

The following sections of the paper are organised as follows: In Section $2$, we recall the theory of uniform measure attractors for processes defined on the space of probability measures. In Section $3$, we reformulate the problem and the transformation from the varying thin domain to the fixed domain, denoted by $\mathcal{O}$. Section $4$ is devoted to the uniform estimates and the tail estimates of the solutions. In Section $5$, we show the existence and uniqueness of almost periodic measures of (1.1)-(1.2). In the last section, we prove the upper semi-continuity of uniform measure attractors of (1.1)-(1.2) as $\varepsilon\rightarrow 0$.

This section reviews the theory of uniform measure attractors for processes in the space of probability measures. Since this space is metrizable, the processes can be seen as in a metric space.

In what follows, we will denote the separable Banach space with norm $\|\cdot\|_{X}$ by $X$. Let us define the space of bounded continuous functions on $X$ as $C_{b}(X)$ equipped with the norm

Let $L_{b}(X)$ denote the space of bounded Lipschitz functions on $X$ which consists of all functions $\varphi\in C_{b}(X)$ such that

The space $L_{b}(X)$ is equipped with the norm

Denote by $\mathcal{P}(X)$ be the set of probability measures on $\left(X,\mathcal{B}(X)\right)$, where $\mathcal{B}(X)$ is the Borel $\sigma$-algebra of $X$. Given $\varphi\in C_{b}(X)$ and $\mu\in\mathcal{P}(X)$, we write

Define a metric on $\mathcal{P}(X)$ by

Then $(\mathcal{P}(X),d_{\mathcal{P}(X)})$ is a polish space. Moreover, a sequence $\{\mu_{n}\}^{\infty}_{n=1}\subset\mathcal{P}(X)$ convergence to $\mu$ in $(\mathcal{P}(X),d_{\mathcal{P}(X)})$ if and only if $\{\mu_{n}\}^{\infty}_{n=1}$ convergence to $\mu$ weakly.

Given $p\geq 1$, denote by $\left(\mathcal{P}_{p}(X),\mathbb{W}_{p}\right)$ as defined by

and

where $\Pi(\mu,\nu)$ is the set of all coupling of $\mu$ and $\nu$. The metric $\mathbb{W}_{p}$ is called the Wasserstein distance.

Given $r>0$, denote by

A subset $S\subset\mathcal{P}_{p}(X)$ is bounded if there is $r>0$ such that $S\subset B_{\mathcal{P}_{p}(X)}(r)$. If $S$ is bounded in $\mathcal{P}_{p}(X)$, then we set

Note that $(\mathcal{P}_{p}(X),\mathbb{W}_{p})$ is a polish space, but $(\mathcal{P}_{p}(X),d_{\mathcal{P}(X)})$ is not complete. Since for every $r>0$, $B_{\mathcal{P}_{p}(X)}(r)$ is a closed subset of $\mathcal{P}(X)$ with respect to the metric $d_{\mathcal{P}(X)}$, we know that the space $(B_{\mathcal{P}_{p}(X)}(r),d_{\mathcal{P}(X)})$ is complete for every $r>0$.

Recall that the Hausdorff semi-metric between subsets of $\mathcal{P}_{p}(X)$ is given by

We assume that $g_{0}(t)$ is an almost periodic function in $t\in\mathbb{R}$ with values in $X$. Denote by $C_{b}(\mathbb{R},X)$ the space of bounded continuous functions on $\mathbb{R}$ with the norm $\|g\|_{C_{b}(\mathbb{R},X)}=\sup\limits_{t\in\mathbb{R}}\|g(t)\|_{X}$ for $g\in C_{b}(\mathbb{R},X)$. Since an almost periodic function is bounded and uniformly continuous on $\mathbb{R}$ (see, e.g., [26]), it follows that ${g}_{0}\in C_{b}(\mathbb{R},X)$. Further, by Bochners criterion in [26], whenever ${g}_{0}:\mathbb{R}\rightarrow X$ is almost periodic, the set of all translations $\left\{g_{0}(\cdot+h):h\in\mathbb{R}\right\}$ is precompact in $C_{b}(\mathbb{R},X)$. Let $\mathcal{H}\left({g}_{0}\right)$ be the closure of this set in $C_{b}(\mathbb{R},X)$. Then, for any $g\in\mathcal{H}\left({g}_{0}\right),g$ is almost periodic and $\mathcal{H}(g)=\mathcal{H}\left({g}_{0}\right)$. For each $h\in\mathbb{R}$, denote by $T(h)$ the translation on $\mathcal{H}\left({g}_{0}\right)$ with $T(h)g=g(\cdot+h)$ for all $g\in\mathcal{H}({g}_{0})$. It is evident that $\{T(h)\}_{h\in\mathbb{R}}$ is a continuous translation group on $\mathcal{H}({g}_{0})$ that leaves $\mathcal{H}({g}_{0})$ invariant:

It is assumed that the following translation identity holds for the processes $\left\{U^{g}(t,\tau)\right\}_{g\in\mathcal{H}\left(g_{0}\right)}$ and the translation group $\{T(h)\}_{h\in\mathbb{R}}$ :

The kernel of the process $U^{g}(t,\tau)$ is the collection $\mathcal{K}_{g}$ of all its bounded complete solutions. The kernel section of the process $U^{g}(t,\tau)$ at time $s\in\mathbb{R}$ is the set

If the family of processes $\left\{U^{g}(t,\tau)\right\}_{g\in\mathcal{H}\left(g_{0}\right)}$ has a uniform measure attractor, then it must be unique. To prove the existence of such a uniform measure attractor, it is convenient to transfer the family of processes to a semigroup of nonlinear operators, and then use the semigroup theory to investigate the uniform measure attractor of the processes. As in [5], we define a nonlinear semigroup $\{S(t)\}_{t\geq 0}$ acting on the extended phase space $\mathcal{P}_{p}(X)\times\mathcal{H}\left(g_{0}\right)$ by the following formula, for every $t\geq 0,\mu\in\mathcal{P}_{p}(X)$ and $g\in\mathcal{H}\left(g_{0}\right)$,

By the translation identity and Definition 2.1 of the process, it is clear that $\{S(t)\}_{t\geq 0}$ satisfies the semigroup identities: for any $t\geq s\geq 0$,

We know from [5] that if $\{S(t)\}_{t\geq 0}$ has a global attractor in the extended phase space $\mathcal{P}_{p}(X)\times\mathcal{H}\left(g_{0})\right.$ then the family of the processes $\left\{U^{g}(t,\tau)\right\}_{g\in\mathcal{H}\left(g_{0}\right)}$ possesses a uniform measure attractor in the phase space $\mathcal{P}_{p}(X)$, which is actually the projection onto $\mathcal{P}_{p}(X)$ of the global attractor of $\{S(t)\}_{t\geq 0}$.

In consequence of the uniform attractors theory set forth in [5], we have the following theorem for the family of processes $\left\{U^{g}(t,\tau)\right\}_{g\in\mathcal{H}\left(g_{0}\right)}$. We also refer the reader to [6, 18, 43, 44] for the attractors theory of semigroups.

In accordance with the aforementioned notation from reference [28], the following criterion is established for the existence and uniqueness of uniform measure attractors.

In this section, we consider the following equation

with initial data

Throughout this paper, we use $\delta_{0}$ for the Dirac probability measure at $0$, and we assume $f:\widetilde{\mathcal{O}}\times\mathbb{R}\times\mathcal{P}_{2}(L^{2}(\mathcal{O}))\rightarrow\mathbb{R}$ is continuous and differentiable with respect to the first and second arguments, which further satisfies the conditions:

$\mathbf{(A1)}$. for all $x\in\widetilde{\mathcal{O}}$, $u,u_{1},u_{2}\in\mathbb{R}$ and $\mu,\mu_{1},\mu_{2}\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$,

where $p\geq 2$, $\alpha_{1},\alpha_{2}>0$, $\psi_{1}\in L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ and $\phi_{i}\in L^{\infty}(\mathbb{R})\cap L^{1}(\mathbb{R}^{n})$ for $i=1,2,3,4,5$.

It follows from (3.3) and (3) that for all $u\in\mathbb{R},x\in\widetilde{\mathcal{O}}$ and $\mu\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$,

for some $\alpha_{3}>0$ and $\phi_{6}\in L^{\infty}(\mathbb{R})\cap L^{1}(\mathbb{R}^{n})$.

Here, for the diffusion term $\{\sigma_{k}\}_{k=1}^{\infty}$ and $\{\varpi_{k}\}^{\infty}_{k=1}$, we assume the following conditions.

$\mathbf{(A2)}$ The function $\sigma=\{\sigma_{k}\}_{k=1}^{\infty}$ satisfies for all $x=(x^{*},x_{n+1})\in\widetilde{\mathcal{O}}$

and

where $\sigma_{1}=\{\sigma_{1,k}\}^{\infty}_{k=1},\sigma_{2}=\{\sigma_{2,k}\}^{\infty}_{k=1}\in L^{2}(\mathbb{R}^{n},l^{2})$ with $\\ \sum\limits^{\infty}_{k=1}\left(\|\sigma_{1,k}\|^{2}_{L^{2}(\mathbb{R}^{n})}+\|\sigma_{2,k}\|^{2}_{L^{2}(\mathbb{R}^{n})}\right)<+\infty$.

$\mathbf{(A3)}$ For each $k\in\mathbb{N}$, $\varpi_{k}:\mathbb{R}\times\mathcal{P}_{2}(L^{2}(\mathcal{O}))\rightarrow\mathbb{R}$ is continuous such that for all $u\in\mathbb{R}$ and $\mu\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$,

where $\beta=\{\beta_{k}\}^{\infty}_{k=1}$ and $\gamma=\{\gamma_{k}\}^{\infty}_{k=1}$ are nonnegative sequences with $\sum^{\infty}_{k=1}(\beta^{2}_{k}+\gamma^{2}_{k})<\infty$. Furthermore, we assume $\sigma_{k}(u,\mu)$ is differentiable in $u$ and Lipschitz continuous in both $u$ and $\mu$ uniformly for $t\in\mathbb{R}$ in the sense that for all $u_{1},u_{2}\in\mathbb{R}$ and $\mu_{1},\mu_{2}\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$,

where $L_{\varpi}=\{L_{\varpi,k}\}^{\infty}_{k=1}$ is a sequence of nonnegative numbers such that $\sum^{\infty}_{k=1}L^{2}_{\varpi,k}<\infty$.

It follows from (3.12) that for all $u\in\mathbb{R}$ and $\mu\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$,

Now, we transfer problem (3.1)-(3.2) into the fixed domain $\mathcal{O}$. To that end, define $T_{\varepsilon}(x^{*},x_{n+1})=(x^{*},\frac{x_{n+1}}{\varepsilon\rho(x^{*})})$ for $x=(x^{*},x_{n+1})\in\mathcal{O}_{\varepsilon}$. Let $y=(y^{*},y_{n+1})=T_{\varepsilon}(x^{*},x_{n+1})$. Then we have

and

where we denote by $u(y)=\hat{u}(x)$, $\Delta_{x}$ is the Laplace operator in $x\in\mathcal{O}_{\varepsilon}$, $\operatorname{div}_{y}$ is the divergence operator in $y\in\mathcal{O}$, and $P_{\varepsilon}$ is the operator given by

We often write $f(x,u,\mu)$ and $\sigma_{k}(x)$ as $f\left(x^{*},x_{n+1},u,\mu\right)$ and $\sigma_{k}\left(x^{*},x_{n+1}\right)$ for $x=\left(x^{*},x_{n+1}\right)$, respectively. For $y=\left(y^{*},y_{n+1}\right)\in\mathcal{O}$ and $t,s\in\mathbb{R}$, let us define $\sigma_{\varepsilon}=\{\sigma_{k,\varepsilon}\}^{\infty}_{k=1}$ and $f_{\varepsilon}$ as follows:

and

Denote by $l^{2}$ the space of square summable sequences of real numbers. For every $u^{\varepsilon}\in L^{2}(\mathcal{O})$, $\mu^{\varepsilon}\in\mathcal{P}_{2}(L^{2}(\mathcal{O}))$, we define a map $\varpi_{\varepsilon}(u^{\varepsilon},\mu^{\varepsilon}):l^{2}\rightarrow L^{2}(\mathcal{O})$ by

Similarly, For every $u^{0}\in L^{2}(\mathbb{R}^{n})$, $\mu^{0}\in\mathcal{P}_{2}(L^{2}(\mathbb{R}^{n}))$, we define $\varpi_{0}(u^{0},\mu^{0}):l^{2}\rightarrow L^{2}(\mathbb{R}^{n})$ by