Markov properties of the fluctuation limit of a particle system
The paper investigates the weak limit of the fluctuation process of the empirical distribution of n independent Brownian particles in \(R^ d\), as \(n\to \infty\). The novelty point is that these particles need not ``exist'' from the very beginning but rather appear at random moments. More rigorously, the starting point of a particle is distributed according to a probability measure \(\mu\) in the space-time \(R^ d\times R_+.\) Let \(T_ 0=\inf \{t:\mu (R^ d\times [t,\infty))=0\}\). The fluctuation processes are proved to converge weakly in the Skorokhod spaces \(D([0,T_ 0], {\mathcal S}'(R^ d))\) and \(D([T_ 0,\infty), {\mathcal S}'(R^ d))\) to continuous, centered Gaussian Markov processes X and Y, respectively, and Langevin equations for X and Y are derived. The process X is said to be strictly Markov in [a,b] if for any s,t\(\in [a,b]\), \(s<t\), there exists a linear continuous mapping \(U_{st}:\;{\mathcal S}(R^ d)\to {\mathcal S}(R^ d)\) such that \[ Cov(<X_ r,\phi >,<X_ t,\psi >)=Cov(<X_ r,\phi >,<X_ s,U_{st}\psi >) \] for \(r\leq s\) and \(\phi\),\(\psi\in {\mathcal S}(R^ d)\). It is shown that the limit process X is strictly Markov in [a,b] if and only if \(\mu (R^ d\times (a,b])=0\).
scientific article; zbMATH DE number 4151522
scientific article