Self-collisions of superprocesses: renormalization and limit theorems
Let \(z\) be a Brownian motion in \(\mathbb R^{d}\), denote by \(Z\) the superprocess over \(z\) taking values in the space of finite measures on \(\mathbb R^{d}\). A formal expression \(\int ^{T}_{0} \langle \delta (x-y),Z_{s}( dx) Z_{s}( dy)\rangle ds\), which may be intuitively interpreted as a ``measure of self-collisions'' of \(Z\), is given a rigorous meaning by a renormalization procedure, i.e. by approximating the Dirac ``delta function'' by a suitable sequence of smooth functions and subtracting a singular term before passing to the limit; cf. the results on self-intersections of the Brownian motion in \(\mathbb R^{2}\), see e.g. \textit{J.-F. Le~Gall} [in: Sémin. de probabilités XIX. Lect. Notes Math. 1123, 314-331 (1985; Zbl 0563.60072)]. More precisely: Let \(f\geq 0\) be a continuous symmetric function whose support is contained in the unit ball of \(\mathbb R^{d}\) and \(\int f dx =1\), set \(f_\varepsilon (x) = \varepsilon ^{-d}f(\varepsilon ^{-1}x)\). Denoting by \(p_{r}(y)\) the transition density for the \(d\)-dimensional Brownian motion and by \(|Z_{s}|\) the mass of the measure \(Z_{s}\), define \[ \begin{aligned} &\varphi _\varepsilon (t) = \int ^{t}_{0}\left ( \iint p_{r}(x)f_\varepsilon (x-y)p_{r}(y) dx dy\right)dr, \\ &\gamma _\varepsilon (T) = \int ^{T}_{0}\left (\iint f_\varepsilon (x-y) Z_{s}( dx) Z_{s}( dy) \right)ds - 2\int ^{T}_{0} \varphi _\varepsilon (T-s)|Z_{s}|ds. \end{aligned} \] It is shown that \(\gamma _\varepsilon (T)\) converges in \(L^{2}\) as \(\varepsilon \downarrow 0\) if \(d=2\) or 3. If \(d=4\), then \(\gamma _\varepsilon (T)/\ln \varepsilon ^{-1}\) converges weakly. Moreover, analogous results are obtained in the case when \(z\) is a Brownian motion in \(\mathbb R^{d}\) killed at an independent exponential time, or \(z\) is a diffusion in \(\mathbb R^{d}\) with a uniformly elliptic generator with smooth coefficients, and for symmetric stable processes \(z\).
scientific article; zbMATH DE number 1554291
scientific article