The net charge process for interacting, signed diffusions (Q918047)

One of the motivations of the present paper is to construct systems of interacting particles in which the ``fluctuation theory'' is non- Gaussian. Let \(\sigma^ 1,...,\sigma^ N\) be \(N\geq 1\) independent random signs such that \[ P\{\sigma^ i=+1\}=P\{\sigma^ i=- 1\}=1/2,\quad 1\leq i\leq N, \] and let \(W^ 1,...,W^ N\) be independent standard, real-valued Brownian motions which are also independent of the \(\sigma^ i\). Let \(\beta >0\) and let b(\(\cdot,\cdot,\cdot)\) be a ``nice'' function on \(R^ 3\). Let \(X^ 1,...,X^ N\) be interacting diffusions given by the solution of the N stochastic differential equations \[ dX^ i_ t=dW^ i_ t+\sigma^ i\beta N^{- 3/2}\sum_{i\neq k\neq j\neq i}\sigma^ j\sigma^ kb(X^ i_ t,X^ j_ t,X^ k_ t)dt,\quad 1\leq i\leq N. \] In order to study the temporal development of this system, the author concentrates on the following measure-valued stochastic processes: \[ \mu^+_ N(t,A)=\sum^{N}_{i=1}1_{\{\sigma^ i=+1\}}1_ A(X^ i_ t),\quad \mu^-_ N(t,A)=\sum^{N}_{i=1}1_{\{\sigma^ i=-1\}}1_ A(X^ i_ t), \] \[ \mu_ N(t,A)=\sum^{N}_{i=1}1_ A(X^ i_ t),\quad \mu_ N^{\sigma}(t,A)=\sum^{N}_{i=1}\sigma^ i1_ A(X^ i_ t) \] (e.g., \(\mu_ N^{\sigma}(t,A)\) is a signed measure measuring the ``net charge'' in A). These four processes are studied via their corresponding distribution-valued versions. Thus each of these four processes can be thought of as an \({\mathcal S}'\)-valued process, \({\mathcal S}'\) denoting the space of tempered distributions. It turns out that each of the pairs \(<\mu_ N,\mu_ N^{\sigma}>\) and \(<\mu^+_ N,\mu^-_ N>\) is an \({\mathcal S}'\times {\mathcal S}'\)-valued Markov process. One of the main results is that (under certain conditions), as \(N\to \infty\), the pair \(<N^{-1}\mu_ N\), \(N^{-1/2}\mu_ N^{\sigma}>\) converges, in the sense of weak convergence of measures on C([0,1], \({\mathcal S}'\times {\mathcal S}')\), to a pair \(<\mu_{\infty},\mu^{\sigma}_{\infty}>\), the latter being the solution of a certain martingale problem as an \({\mathcal S}'\times {\mathcal S}'\)-valued stochastic process.