Intersection local times of all orders for Brownian and stable density processes -- construction, renormalisation and limit laws
Let \(X^ 1\), \(X^ 2,\ldots\) be a sequence of independent symmetric stable processes with index \(\alpha\in(0,2]\) started at 0 (the case \(\alpha=2\) corresponds to Brownian motion). Consider a Poisson point process \(\Pi^ \lambda=\{x^ 1,x^ 2,\ldots\}\) with intensity \(\lambda dx\;(\lambda>0)\) and \(\sigma^ 1,\sigma^ 2\ldots\) a sequence of independent Rademacher variables. Introduce the process \(\eta^ \lambda\) valued in the Schwartz space \({\mathcal S}'\) of temperate distributions given by \[ \eta^ \lambda_ t(\varphi)=\lambda^{-1/2}\sum^ \infty_{i=1}\sigma^ i\varphi(x^ i+X^ i_ t)\qquad(\varphi\in{\mathcal S},\;t\geq 0). \] The density process \(\eta\) is the weak limit of \(\eta^ \lambda\) as \(\lambda\to\infty\), and satisfies a stochastic partial differential equation of the type studied by \textit{J. B. Walsh} [École d'été de probabilités de Saint-Flour XIV-1984, Lect. Notes Math. 1180, 265-437 (1986; Zbl 0608.60060)]. The paper under review is concerned with the \(k\)th order intersection local times (ILT) \(\gamma_ k\) of \(\eta\) \((k=2,3,\ldots)\). For simplicity, we will only discuss the case \(k=2\), but the paper has also results for general \(k\). Formally, \(\gamma_ 2\) is defined by \[ \gamma_ 2(t,\varphi)=\int^ t_ 0du\int^ t_ 0dv\int_{\mathbb{R}^ d}\int_{\mathbb{R}^ d} (\eta_ u\times\eta_ v)\delta(x-y)\varphi(x)dxdy, \tag{*} \] where \(\delta\) stands for the Dirac delta function and \(\varphi\) is a function in the Schwartz space \({\mathcal S}\). In order to make a rigorous definition, one introduces first the approximative ILT, \(\gamma_{2,\varepsilon}\), by replacing \(\delta\) in the right-hand side of \((*)\) by \(f_ \varepsilon\), where \(f_ \varepsilon(x)=\varepsilon^{-d}f(x/\varepsilon)\) and \(f\) is a smooth symmetric function with compact support and \(\int f(x)dx=1\). The main concern of the paper is to study the convergence of \(\gamma_{2,\varepsilon}\) as \(\varepsilon\downarrow 0\). When \(d<2\alpha\), \(\gamma_{2,\varepsilon}(t,\varphi)\) converges in \({\mathcal L}^ 2\), the limit is denoted by \(\gamma_ 2(t,\varphi)\) and is shown to satisfy an evolution equation. This feature is lost when \(d\geq 2\alpha\), nonetheless the authors prove limit theorems for the approximative ILT after suitable renormalisation. The limits are then identified in terms of the Gaussian white noise.
scientific article; zbMATH DE number 428673