Self-intersection local time of order \(k\) for Gaussian processes in \({\mathcal S}^ \prime (\mathbb R^ {d})\).
Let \((X_ {t})_ {t\in [0,1]}\) be a centered continuous Gaussian process in \({\mathcal S}^ \prime (\mathbb R^ {d})\) (the space of tempered distributions on \(\mathbb R^ {d}\)). Its self-intersection local time of order \(k\) is, non-rigorously, defined by \(\int _ {[0,t]^ {k}}\langle X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}},\varphi (x_ 1)\delta (x_ 2 -x_ 1)\cdots \delta ( x_ {k}-x_ 1)\rangle \, ds_ 1\cdots ds_ {k}\), where \(\delta \) is the Dirac distribution, \(\varphi \in {\mathcal S} (\mathbb R^ {d})\), and \(\otimes \) denotes the tensor product in \({\mathcal S}^ \prime (\mathbb R^ {d})\). More precisely: define \(\:\!X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}\:\) by the relation \[ \langle :X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}:,\;\varphi _ 1\otimes \cdots \otimes \varphi _ {k}\rangle =\! :\langle X_ {s_ 1},\varphi _ 1\rangle \cdots \langle X_ {s_ {k}}, \varphi _ {k}\rangle\!:, \] the right-hand side being the Wick product of the jointly Gaussian variables \(\langle X_ {s_ {i}},\varphi _ {i}\rangle \). Let \({\mathcal F} \subset {\mathcal S}(\mathbb R^ {d})\) be the subset of all symmetric nonnegative functions \(f\) vanishing outside the unit ball of \(\mathbb R ^ {d}\) and satisfying \(\int _ {\mathbb R^ {d}} f\, dx=1\). For \(f\in {\mathcal F}\), \(\varepsilon >0\) and \(\varphi \in {\mathcal S}(\mathbb R^ {d})\) set \[ \Phi _ {f,\varepsilon ,\varphi }(x_ 1,\ldots ,x_ {k})= {1\over \varepsilon ^ {d(k-1)}} \varphi (x_ 1)f\Bigl ({x_ {2}-x_ {1}\over \varepsilon }\Bigr )\cdots f\Bigl ({x_ {k}-x_ {1}\over \varepsilon }\Bigr ), \] and define an \({\mathcal S}^ \prime (\mathbb R^ {d})\)-valued process \(L_ {f,\varepsilon }(t)\) by \[ \langle L_ {f,\varepsilon }(t), \varphi \rangle = \int _ {[0,t]^ {k}} \langle :\!X_ {s_ 1}\otimes \cdots \otimes X_ {s_ {k}}\;1:,\;\Phi _ {f, \varepsilon ,\varphi }\rangle \, ds_ {1}\cdots ds_ {k}, \quad \varphi \in {\mathcal S}(\mathbb R^ {d}). \] An \({\mathcal S}^ \prime (\mathbb R^ {d})\)-valued process \(L(t)\) is called the self-intersection local time of order \(k\) of the process \(X\) provided \(\langle L_ {f,\varepsilon }(t),\varphi \rangle \to \langle L(t),\varphi \rangle \) in \(L^ 2\) as \(\varepsilon \downarrow 0\) for all \(\varphi \in {\mathcal S}(\mathbb R^ {d})\), \(t\in [0,1]\) and \(f\in {\mathcal F}\). Self-intersection local times of order 2 have been recently thoroughly investigated by \textit{T. Bojdecki} and \textit{L. G. Gorostiza}, see e.g. [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2, 569-615 (1999; Zbl 1043.60508)]. In the case \(k>2\), however, serious additional difficulties arise. In the paper under review, existence and continuity of self-intersection local times of order \(k\) are studied for several classes of Gaussian processes, including (time-inhomogeneous) Wiener processes \(W_ {t}\) in \({\mathcal S}^ \prime (\mathbb R^ {d})\) and Ornstein-Uhlenbeck processes, defined by the equation \( dX_ {t} = \Delta ^ \prime _ {\alpha }X_ {t}\, dt + dW_ {t}\) with a Gaussian initial condition, where \(\Delta _ \alpha = -(-\Delta )^ { \alpha /2}\), \(0<\alpha \leq 2\).
scientific article; zbMATH DE number 2138903