Well-posedness of a coupled system of Skorohod-like stochastic differential equations^††thanks: Received date, and accepted date (The correct dates will be entered by the editor).

We consider a two dimensional domain, which we refer to as $\Lambda$. As a building geometry, parts of the domain are filled with obstacles. Their collection is denoted by $G=\bigcup_{k=1}^{N_{\text{obs}}}G_{k}$, for all $k\in\{1,\ldots,N_{\text{obs}}\in\mathbb{N}\}$. A fire $F$ is introduced somewhere in this domain and is treated in this context as an obstacle for the motion of the crowd. Moreover, the domain has the exit denoted by $E$. Our domain represents the environment where the crowd of pedestrians is located. The crowd tries to find the fastest way to the exit, avoiding the obstacles and the fire. Let $D:=\Lambda\backslash(G\cup E\cup F)\subset\mathbb{R}^{2}$ with the boundary $\partial D$ such that $\partial\Lambda\cap\partial G_{k}=\emptyset$, $\partial\Lambda\cap\partial G_{k}=\emptyset$ and $F\cap G_{k}=\emptyset$, see in Section 4.2 for our assumption of the regularity of the involve set. We also denote $S=(0,T)$ for some $T\in\mathbb{R}_{+}$. We refer to $\bar{D}\times S$ as $D_{T}$, note that $\bar{D}$ denotes the closure of $D$. Furthermore, $N_{A}$ is the total number of active agents, $N_{P}$ is the total number of passive particles with $N:=N_{A}+N_{P}$ and $N_{A},N_{P},N\in\mathbb{N}$.

For $i\in\{1,\ldots,N_{A}\}$ and $t\in S$, let $\textbf{x}_{a_{i}}$ denote the position of the pedestrian $i$ belonging to the active population at time $t$. We assume that the dynamics of active pedestrians is governed by

where $\textbf{x}_{a_{i0}}$ represents the initial configuration of active pedestrians inside $\bar{D}$. In (3.1), $\nabla\phi$ is the minimal motion path of the distance between particle positions $\textbf{x}_{a_{i}}$ and the exit location $E$ (it solves the Eikonal-like equation). The function $\phi(\cdot)$ encodes the familiarity with the geometry; see also [42] for a related setting. We refer to as the motion planning map. In this context, $p(\textbf{x},t)$ is the local discomfort (a realization of the social pressure) so that

for $\{\textbf{x}_{c_{j}}\}:=\{\textbf{x}_{a_{i}}\}\cup\{\textbf{x}_{b_{k}}\}$ for $i\in\{1,\dots,N_{A}\},k\in\{1,\dots,N_{P}\},j\in\{1,\ldots,N_{A}+N_{P}\}$ and $\textbf{x}_{b_{k}}$ represents the position of the pedestrian $k$ belonging at time $t$ to the passive population. In (3.2), $\delta$ is the Dirac (point) measure, $\mu(t)$ is a finite measure and $B(\textbf{x},\tilde{\delta})$ is a ball center x with small enough radius $\tilde{\delta}$ such that $\tilde{\delta}>0$. Hence, the discomfort $p(\textbf{x},t)$ represents a finite measure on the set $D\cap B(\textbf{x},\tilde{\delta})$. In addition, we assume the following structural relation between the smoke extinction and the walking speed (see in [24] and [35]) as a function $\Upsilon:\mathbb{R}_{+}^{2}\longrightarrow\mathbb{R}^{2}$ such that

where $\zeta,\eta$ are given real positive numbers. The dependence of the model coefficients on the local smoke density is marked via a smooth relationship with respect to an a priori given function $s(\textbf{x}(t))$ describing the distribution of smoke inside the geometry at position $x$ and time $t$.

For $k\in\{1,\ldots,N_{P}\}$ and $t\in S$, let $\textbf{x}_{p_{k}}$ denote the position of the pedestrian $k$ belonging at time $t$ to the passive population. The dynamics of the passive pedestrians is described here as a system of stochastic differential equations as follows:

where $\textbf{x}_{p_{k0}}$ represents the initial configuration of passive pedestrians inside $\bar{D}$ and $\epsilon>0$. In (3.4), $\omega$ is a Morse-like potential function ⁱⁱiIn general, the interactions between particles are defined in the senses of a pairwise potential, i.e. repulsive at short ranges and attractive at longer ranges. Therefore, one of the typical choices is the exponentially decaying Morse potential, which is known to reproduce certain types of collective motion observed in nature, particularly aligned flocks and rotating mills (see e.g. in [3] and [6]). (see e.g. Ref. [6] for a setting where a similar potential has been used). We take $\omega:\mathbb{R}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ to be

while $C_{A}>0,C_{R}>0$ are the attractive and repulsive strengths and $\ell_{A}>0,\ell_{R}>0$ are the respective length scales for attraction and repulsion. Moreover, the coefficient $\beta$ is a regularized version of the Heaviside step function. As in Subsection 3.2, the dependence of the model coefficients on the smoke is marked via a smooth relationship with respect to an a priori given function $s(x,t)$ describing the distribution of smoke inside the geometry at position $x$ and time $t$. Note that the passive pedestrians do not posses any knowledge on the geometry of the walking space. In particular, the location of the exit is unknown; see [13] for a somewhat related context.

We recall the classical Ascoli-Arzelà Theorem:
A family of functions $U\subset C(\bar{S};\mathbb{R}^{d})$ is relatively compact (with respect to the uniform topology) if

1. i.

   for every $t\in\bar{S}$, the set $\{f(t);f\in U\}$ is bounded.

2. ii.

   for every $\varepsilon>0$ and $t,s\in\bar{S}$, there is $\bar{\delta}>0$ such that

   $\displaystyle|f(t)-f(s)|\leq\varepsilon,$

   (4.6)

   whenever $|t-s|\leq\bar{\delta}$ for all $f\subset U$.

For a function $f:\bar{S}\to\mathbb{R}^{d}$, we introduce the definition of Hölder seminorms as

for $\alpha\in(0,1)$ and the supremum norm as

We refer to [1] and [20] for more details on the used function spaces.

Using Arzelà-Ascoli Theorem based on the facts:

1. i’.

   there is $M_{1}>0$ such that $\|f\|_{L^{\infty}(S;\mathbb{R}^{d})}\leq M_{1}$ for all $f\in U$,

2. ii’.

   for some $\alpha\in(0,1)$, there is an $M_{2}>0$ such that $[f]_{C^{\alpha}(\bar{S};\mathbb{R}^{d})}\leq M_{2}$ for all $f\in U$,

we infer that the set

is relatively compact in $C(\bar{S};\mathbb{R}^{d})$.

For $\alpha\in(0,1)$, $T>0$ and $p>1$, the space $W^{\alpha,p}(S;\mathbb{R}^{d})$ is defined as the set of all $f\in L^{p}(S;\mathbb{R}^{d})$ such that

This space is endowed with the norm

Moreover, we have the following embedding

and $[f]_{C^{\gamma}(S;\mathbb{R}^{d})}\leq C_{\gamma,\alpha,p}\|f\|_{W^{\alpha,p}(S;\mathbb{R}^{d})}$. Relying on the Ascoli-Arzelà Theorem, we have the following situation:

1. ii”.

   for some $\alpha\in(0,1)$ and $p>1$ with $\alpha p>1$, there is $M_{2}>0$ such that $[f]_{W^{\alpha,p}(S;\mathbb{R}^{d})}\leq M_{2}$ for all $f\in U$.

If i’ and ii” hold, then the set

is relatively compact in $C(\bar{S};\mathbb{R}^{d})$, if $\alpha p>1$ (see e.g. [19], [11]).

To be successful with our mathematical analysis, we rely on the following assumptions:

• ($\text{A}_{1}$)

  The functions $b:D_{T}\times D_{T}\longrightarrow\mathbb{R}^{2}\times\mathbb{R}^{2},$ and $\sigma:D_{T}\times D_{T}\longrightarrow\mathbb{R}^{2\times 2}\times\mathbb{R}^{2\times 2}$ satisfy $|\sigma(x,t)|\leq L$, $|b(x,t)|\leq L$ for all $x\in\bar{D}\times\bar{D}$ and $t\in S$. Here $\sigma$ and $b$ incorporate the right-hand sides of the SDEs (3.1) and (3.4) in their respective dimensionless form indicated in Appendix B.

• ($\text{A}_{2}$)

  $p_{\text{max}}=N|\bar{D}|$, where $|\bar{D}|$ denotes the area of $\bar{D}$.

• ($\text{A}_{3}$)

  $\Upsilon,\omega,\beta\in C^{1}(\mathbb{R}^{2})$.

• ($\text{A}_{4}$)

  $s\in C^{1}(\bar{S};\mathbb{R}^{2})$.

• ($\text{A}_{5}$)

  $\partial D$ is $C^{2,\alpha}$ with $\alpha\in(0,1)$.

It is worth mentioning that assumptions ($\text{A}_{1}$) and ($\text{A}_{2}$) correspond to the modeling of the situation, while ($\text{A}_{3}$)-($\text{A}_{5}$) are of technical nature, corresponding to the type of solution we are searching for; clarifications in this direction are given in the next Section.

Take $x\in\partial D$ arbitrarily fixed. We define the set $\mathcal{N}_{x}$ of inward normal unit vectors at $x\in\partial D$ by

where $B(z,r)=\{y\in\mathbb{R}^{2}:|y-z|<r\},z\in\mathbb{R}^{2},r>0$. Mind that, in general, it can happen that $\mathcal{N}_{x}=\emptyset$. In this case, the uniform exterior sphere condition is not satisfied (see, for instance, the examples in [8], Fig. 5 and in [9], page $4$).
We complement our list of assumptions ($\text{A}_{1}$)–($\text{A}_{5}$) with three specific conditions on the geometry of the domain $D$:

• ($\text{A}_{6}$)

  (Uniform exterior sphere condition). There exists a constant $r_{0}>0$ such that

  $$\mathcal{N}_{x}=\mathcal{N}_{z,r_{0}}\neq\emptyset\text{ for any }z\in\partial D.$$

• ($\text{A}_{7}$)

  There exits constants $\delta>0$ and $\delta^{\prime}\in[1,\infty)$ with the following property: for any $x\in\partial D$ there exists a unit vector $\textbf{l}_{x}$ such that

  $$\langle\textbf{l}_{x},\textbf{n}\rangle\geq 1/\delta^{\prime}\text{ for any }\textbf{n}\in\bigcup_{y\in B(x,\delta)\cap\partial D}\mathcal{N}_{y},$$

  where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^{2}$.

• ($\text{A}_{8}$)

  There exist $\delta^{\prime\prime}>0$ and $\nu>0$ such that for each $x_{0}\in\partial D$ we can find a function $f\in C^{2}(\mathbb{R}^{2})$ satisfying

  $\displaystyle\langle y-x,\textbf{n}\rangle+\frac{1}{\nu}\langle\nabla f(x),\textbf{n}\rangle|y-x|^{2}\geq 0,$

  (5.12)

  for any $x\in B(x_{0},\delta^{\prime\prime})\cap\partial D,y\in B(x_{0},\delta^{\prime\prime})\cap\partial\bar{D}$ and $\textbf{n}\in\mathcal{N}_{x}$.

The following relation is called the Skorohod equation: Find $(\xi,\phi)\in C(\bar{S},\mathbb{R}^{2})$ such that

where $w\in C(\bar{S},\mathbb{R}^{2})$ is given so that $w(0)\in\bar{D}$. The solution of (5.13) is a pair $(\xi,\phi)$, which satisfies the following two conditions:

• (a)

  $\xi\in C(\bar{S},\bar{D})$;

• (b)

  $\phi\in C(\bar{S})$ with bounded variation on each finite time interval satisfying $\phi(0)=0$ and

  	$\displaystyle\phi(t)$	$\displaystyle=\int_{0}^{t}\textbf{n}(y)d|\phi|_{y},$
  	$\displaystyle|\phi|_{t}$	$\displaystyle=\int_{0}^{t}\mathbbm{1}_{\partial D}(\xi(y))d|\phi|_{y},$		(5.14)

  where

  	$\displaystyle\textbf{n}(y)$	$\displaystyle\in\mathcal{N}_{\xi(y)}\text{ if }\xi(y)\in\partial D,$		(5.15)
  	$\displaystyle|\phi|_{t}$	$\displaystyle=\text{ total variation of }\phi\text{ on }[0,t]$
  		$\displaystyle=\sup_{\mathcal{T}\in\mathcal{G}([0,t])}\sum_{k=1}^{n_{\mathcal{T}}}|\phi(t_{k})-\phi(t_{k}-1)|.$		(5.16)

  In (5.15), we denote by $\mathcal{G}([0,t])$ the family of all partitions of the interval $[0,t]$ and take a partition $\mathcal{T}=\{0=t_{0}<t_{1}<\ldots<t_{n}=t\}\in\mathcal{G}([0,t])$. The supremum in (5.15) is taken over all partitions of type $0=t_{0}<t_{1}<\ldots<t_{n}=t$.

Conditions (a) and (b) guarantee that $\xi$ is a reflecting process on $\bar{D}$.

It is easily seen from the definition that

and

We define a multidimensional Skorohod’s map $\Gamma:C(\bar{S})\longrightarrow C(\bar{S})$ such that

Hence, the pair $(\xi_{d},\phi)$ is the exact solution of the one-dimensional Skorohod problem $\xi_{d}$. Therefore, it holds

The multidimensional Skorohod’s map $\Gamma$ satisfies the Lipschitz condition in a space of continuous functions.

For the proof of this Theorem, we refer the reader to Theorem $4.1$ in [36].

To come closer to the model equations for active-passive pedestrian dynamics described in Section 3, we introduce the mappings

and consider the Skorohod-like system on the probability space $(\Omega,\mathcal{F},P)$

Note that (5.19) can be written component-wise as

with

where the inital value $X_{0}$ is assumed to be an $\mathcal{F}_{0}-$measurable random variable and $B(t)$ is a $2-$dimensional $\mathcal{F}_{t}-$Brownian motion with $B(0)=0$. Here, $\{\mathcal{F}_{t}\}$ is a filtration such that $\mathcal{F}_{0}$ contains all $P-$negligible sets and $\mathcal{F}_{t}=\cap_{\varepsilon>0}\mathcal{F}_{t+\varepsilon}$. The structure of (5.19) is provided in Section 6.2. Similarly to the deterministic case, we can now define the following concept of solutions to (5.19). More details of the structure of (5.19)-(5.20) are listed in Section 6.2.

Note that the Definition 1 ensures that $X_{t}$ entering (5.19) is a reflecting process on $\bar{D}$.

The main results of this paper are stated in Theorem 6.1, Theorem 6.2 and Theorem 6.3. In the frame of this paper, the focus lies on ensuring the well-posedness of Skorohod solutions to our crowd dynamics problem.

These statements are proven in the next two subsections.

For convenience, we rephrase the solution to the system (2.105) and (2.106) in terms of the vector $X_{t}^{n}$, $n\in\mathbb{N}$, such that

Furthermore, we set

with

where $\tilde{\sigma}_{IJ}:=(\tilde{\sigma}(X_{t}^{n},t))_{IJ}$ for $1\leq I,J\leq 2$ and the initial datum is

We denote by $\{\Phi_{t}^{n}\}$ the associated process of $\{X_{t}^{n}\}$ as in (5.19), viz.

We use the compactness method together with the continuity result of the deterministic case stated in Theorem 5.1 for proving the existence of solutions to (5.19)-(5.20). We follow the arguments by G. Da Prato and J. Zabczyk ($2014$) (cf. [14], Section $8.3$) and a result of F. Flandoli (1995) (cf. [19]) for martingale solutions. The starting point of this argument is based on considering a sequence $\{X_{t}^{n}\}$ of solutions of the following system of Skorohod-like stochastic differential equations

where $X_{0}^{n}\in\bar{D}$ is given, and

Moreover, by Theorem 5.1, we have a unique solution of (6.34). Hence, $X_{t}^{n}$ obtained for $0\leq t\leq k2^{-n}$ and for $k2^{-n}<t\leq(k+1)2^{-n}$ is uniquely determined as solution of the following Skohorod equation:

Let us call

Then $X_{t}^{n}(t)=(\Gamma Y_{t}^{n})(t)$, we also have

We define the family of laws

(6.40) is a family of probability distributions of $Y_{t}^{n}$. Let $\mathcal{P}^{n}$ be the laws of $Y_{t}^{n}$.

The compactness argument proceeds as follows. We begin with $Y_{t}^{n},n\in\mathbb{N}$, given cf. (6.39). The construction of $Y_{t}^{n}$ is investigated on a probability space $(\Omega,\mathcal{F},P)$ with a filtration $\{\mathcal{F}_{t}\}$ and a Brownian motion $B(t)$. Next, let $\mathcal{P}^{n}$ be the laws of $Y_{t}^{n}$ which is defined cf. (6.40). Then, by using Prokhorov’s Theorem (cf. [4], Theorem $5.1$), we can show that the sequence of laws $\{\mathcal{P}^{n}(Y_{t}^{n})\}$ is weakly convergent as $n\to\infty$ to $\mathcal{P}(Y_{t})$ in $C(\bar{S};\mathbb{R}^{2}\times\mathbb{R}^{2})$. Then, by using the “Skorohod Representation Theorem”(cf. [14], Theorem $2.4$), this weak convergence holds in a new probability space with a new stochastic process, for a new filtration. This leads to some arguments for weak convergence results of two stochastic processes in two different probability spaces together with the continuity result in Theorem 5.1 that we need to use to obtain the existence of our Skorohod-like system (5.19). Finally, we prove the uniqueness of solutions to our system.

Let us start with handling the tightness of the laws $\{\mathcal{P}^{n}\}$ through the following Lemma.