Self-intersection local time for Gaussian \({\mathcal S}'(\mathbb{R} ^ d)\)-processes: Existence, path continuity and examples
The problem which the authors consider is to calculate the following formal expression of self-intersection local time of a generalized process \(X_t\), \(\int^t_0 \int^t_0 \langle X_s \otimes X_r, \delta (x-y) \varphi (y) \rangle ds dr\). The authors consider the above as a limit of smeared local time \(L^f_\varepsilon\) defined by \[ \langle L^f_\varepsilon (t), \varphi \rangle= \int^t_0 \int^t_0 \langle :X_s \otimes X_r :, \Phi^f_{\varepsilon, \varphi} \rangle ds dr, \] where \(\Phi^f_{\varepsilon, \varphi} (x,y)= \varphi (x) (\varepsilon^{-d} f_\varepsilon ({{x-y} \over \varepsilon}))\), the test function \(\varphi\) smeared by the modifying function \(f\), and \(:\cdot:\) is the Wick product. They obtain some existence conditions and path continuity conditions for this limit. They also apply this limit to several examples of generalized Gaussian processes valued in \({\mathcal S}' ({\mathcal R}^d)\) and show dimensional gap properties of them.
scientific article; zbMATH DE number 859314
scientific article