Modeling flocks and prices: jumping particles with an attractive interaction (Q2451105)

Consider \(n\) particles on the real line. The position in \({\mathbb R}\) of the particles at time \(t\) is denoted by \(x_1(t )\), \(x_2(t ), \dots{} , x_n(t)\), respectively. The mean position of the particles is \(m_n(t) = \frac 1 n \sum_{i=1}^n x_i (t)\). Next, let \(w:{\mathbb R}\to {\mathbb R}_+\) be a positive and monotone decreasing function. We now describe the dynamics which is a continuous-time Markov jump process. The \(i\)th particle, positioned at \(x_i (t)\) at time \(t\), jumps with rate \(w(x_i (t) - m_n(t))\). That is, conditioned on the configuration of the particles, jumps happen independently after an exponentially distributed time, with the parameter of the \(i\)th particle being \(w(x_i (t) - m_n(t))\). When a particle jumps, the length of the jump is a random positive number \(Z\) from a specific distribution. This is independent of time and position of the particle and of all other particles as well. Assume that \({\mathbb E}Z = 1\), and \(Z\) has a finite third moment. The authors focus on the empirical measure of the \(n\) particle process at time \(t\): \(\mu_n(t) =\frac 1 n \sum_{i=1}^n \delta_{x_i (t)}\). The main result of this paper says that, under suitable conditions, as \(n\to\infty\), \(\mu_n\) converges in the Skorokhod space \(D([0, \infty), {\mathcal P}_1(\mathbb R))\) to some \(\mu\), where \({\mathcal P}_1(\mathbb R)\) is the space of probabilities having finite first moment, and \(\mu\) is the unique solution to the mean field equation \[ \langle f, \mu (t)\rangle - \langle f, \mu (0)\rangle -\int_0^t \Big[{\mathbb E}\big(f(x+Z)-f(x)\big) w\big(x-\mu(s)(\mathbb R)\big)\Big] \mu(s) \,ds \] for every \(t\geq 0\) and a class of test functions \(f\). Here, \(\langle f, \mu \rangle=\int f \,d\mu\), as usual.