Generalized self-intersection local time for a superprocess over a stochastic flow
Let \(b:{\mathbb R}^{d } \rightarrow {\mathbb R}^{d }\), \(c :{\mathbb R}^{d } \rightarrow \mathbb{R}^{d \times m}\) (the real \(d\times m\)-matrices) be maps satisfying a Lipschitz condition bounded together with their first and second derivatives, NEWLINE\[NEWLINE\sigma _{i j } ( x , y ) = \sum _{k = 1 } ^{m } c _{i k } ( x ) c _{j k } ( y ),NEWLINE\]NEWLINE NEWLINE\[NEWLINEa _{i j } ( x , y ) = \delta _{i j } b _{i } ( x ) b _{j }( y ) + \sigma _{i j } ( x , y ),NEWLINE\]NEWLINE NEWLINE\[NEWLINE(L \varphi ) ( x ) = ( 1 / 2 ) \sum _{i , j = 1 }^{d } a _{i j }( x , x ) ( \partial _{i j } ^{2 } \varphi ) ( x ),NEWLINE\]NEWLINE supposed uniformly elliptic, NEWLINE\[NEWLINE( \Lambda \varphi ) ( x , y ) = \sum _{i , j = 1 } ^{d }\sigma _{i j } ( x , y )( \partial _{i } \varphi ) ( x ) ( \partial _{j } \varphi ) ( x ),NEWLINE\]NEWLINE and let \(\mu _{t } \in M ({\mathbb R}^{d } )\) (finite measures for \(t \in [ 0, \infty )\); a superprocess over a stochastic flow defined by \(d Y _{t } = c ( Y _{t } ) d W _{t }\), \(W\) a Brownian motion) be the unique continuous solution of a martingale problem: for \(\varphi \in C _{K } ^{2 } ({\mathbb R}^{d })\) (K stands for compactly supported), NEWLINE\[NEWLINE\mu _{t}(\varphi)-\mu _{0 } ( \varphi )-\int _{0} ^{t } \mu _{s } ( L \varphi ) d sNEWLINE\]NEWLINE is a continuous martingale, having quadratic variation \(\int _{0} ^{t } ( \mu _{s } ( \varphi ^{2 } ) + \mu _{s } ^{2 } ( \Lambda \varphi ) ) d s\) (\(\mu _{s } ^{2 }\) means \(\mu _{s } \otimes \mu _{s }\), \(\mu _{s } ( f ) = \int f d \mu _{s }\)). This \(\mu\) was constructed by \textit{G. Skoulakis} and \textit{R. J. Adler} [Ann. Appl. Probab. 11, No. 2, 488--543 (2001; Zbl 1018.60052)] as the limit of \(\mu ^{( n ) }\), a fact essential in the proofs in this paper. NEWLINENEWLINEConsider \(\alpha = ( \alpha _{0 } ,\dotsc , \alpha _{| \alpha | } )\), \(\alpha _{i } \in [ 0 , 1 ]\), and let \(\alpha -1 = ( \alpha _{0 } ,\dotsc , \alpha _{| \alpha | - 1 } )\), \(\alpha | _{i } = (\alpha _{0 } ,\dotsc , \alpha _{i } )\). Set \(\mu _{0 } ^{( n ) } = \sum _{i = 0 } ^{M ( n ) } \varepsilon _{x _{i }^{n} } \rightarrow \mu _{0 }\), take families \(( B _{t } ^{\alpha , n } ) _{0 \leq t \leq ( | \alpha | + 1 ) / n }\), \(( W _{t } ^{n } ) _{t \in [ 0 , \infty ) }\), \(N ^{\alpha , n }\), are families of Brownian motions on \({\mathbb R}^{d }\), \({\mathbb R}^{m }\) and \(( \varepsilon _{2 } + \varepsilon _{0 } ) / 2\) random variables, respectively; we have NEWLINE\[NEWLINEB _{0 } ^{\alpha , n }= x _{\alpha _{0}}^{n \;}\text{ for}\;| \alpha | = 0,\;\;B _{t } ^{\alpha , n } = B _{t } ^{\alpha - 1 , n}NEWLINE\]NEWLINE (for \(0 \leq i \leq | \alpha | / n\)), \(W _{0 } ^{n } = 0\) and the rest is independent, the solution \(Y ^{\alpha , n }\) of NEWLINE\[NEWLINEd Y _{t } = b ( Y _{t } ) d B _{t } ^{\alpha , n } + c ( Y _{t } ) d W _{t } ^{n }NEWLINE\]NEWLINE on \([ 0 , ( k + 1 ) n ^{- 1 } ]\), for \( \alpha \in [ k / n , ( k - 1 ) / n ),\) and NEWLINE\[NEWLINEY _{0 } ^{\alpha , n } = x _{\alpha _{0 }} ^{n }, \tau ^{\alpha , n } = 0NEWLINE\]NEWLINE for \(\alpha _{0 } > k / n\), \(\min ( ( i + 1 ) / n ; 0 \leq i \leq | \alpha | , N ^{^{\alpha } i ^{, n }} = 0 )\) for \(\alpha _{0 } \leq M ( n )\) and \(( \cdot ) \neq \emptyset\), and \(( 1 + | \alpha | ) / n\) otherwise. Then \(X _{t } ^{\alpha , n } = Y _{t } ^{\alpha , n }\) for \(t < \tau ^{\alpha , n }\), \(\Delta\) otherwise and \(\mu _{t } ^{( n ) } ( U ) = n^{- 1 } \operatorname{card} \{ \alpha ; t \in [ | \alpha | / n , ( | \alpha | + 1 ) / n ) , X _{t } ^{\alpha , n } \in U \} \).NEWLINENEWLINE NEWLINEThe authors express \(\operatorname{E} ( ( \mu _{t _{1 }} \mu _{t _{2 }} \mu _{t _{3 }} \mu _{t _{4 }} ) ( \psi ) )\) as a sum of 14 terms (sums of integrals). They deduce, when \(\mu _{0 }\) has compact support and a bounded density, that, for \(\psi ( x ) = \phi ( x _{1 }-x _{3 } )\phi ( x _{2 }-x _{3 } )\), \(\varphi ( y ) =\phi ( y _{1 }-y _{3 } )\phi ( y _{2 }-y _{4 } )\), \(\phi\) rapidly decreasing, NEWLINE\[NEWLINE\int _{0 } ^{T } d t _{3 } \int _{0 } ^{t_3} d t _{2 } \int _{0} ^{t_2 } d t _{1 } \operatorname{E} ( ( \mu _{t _{1 }} \mu _{t _{2 }} \mu _{t _{3 }} ) ( \psi ) ) \leq C \| \phi \| _{1 } ^{2 }NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\int _{0 } ^{t_3} d t _{2 } \int _{0 } ^{t_2 } d t _{1 } \operatorname{E} ( ( \mu _{t _{1 }} \mu _{t _{2 }} \mu _{t _{3 }} ^{2 } ) ( \varphi ) ) \leq \| \phi \| _{1 } ^{2 }.NEWLINE\]NEWLINE For every rapidly decreasing \(\phi\), \(\mu _{t } (\phi )\) is a.s. a continuous semimartingale. The resolvent of \(L\), written as \(G ^{\lambda , u } ( x )\), appears as a tempered distribution, it is approximated by some \(G _{\varepsilon } ^{\lambda , u } \in C _{K } ^{\infty } \; \;\), \(\varepsilon \downarrow 0\); the authors define NEWLINE\[NEWLINE\gamma _{\varepsilon } ^{\lambda } ( u , T ) = \int _{0} ^{T } d t \int _{0 } ^{t} d s ( ( \lambda -L ) ( \mu _{s } \mu _{t } ) ( G _{\varepsilon } ^{\lambda , u } ) ),NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal L} _{\varepsilon } ^{\lambda } ( u , T ) = \gamma _{\varepsilon } ^{\lambda } ( u , T )- \int _{0 } ^{T } d t ( \mu _{t } \mu _{t } ) ( G _{\varepsilon } ^{\lambda , u } ).NEWLINE\]NEWLINE For \(d \leq 3\), if \(Z\) is the martingale measure for NEWLINE\[NEWLINE( L _{2 } \Psi ) ( x , y ) = (1/2)\sum _{i , j = 1 } ^{d } a _{i j } ( y ) \partial _{2 , i } \partial _{2 , j } \Psi ( x , y ),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim _{\varepsilon \downarrow 0 } {\mathcal L} _{\varepsilon } ^{\lambda } ( u , T )NEWLINE\]NEWLINE exists in \(L^{2 }\), uniformly in \(u\), being equal to NEWLINE\[NEWLINE\lambda \int _{0 } ^{T } d t \int _{0 } ^{t } d s ( \mu _{s } \mu _{t } ) ( G ^{\lambda , u } ) - \int _{0 } ^{T } d t ( \mu _{t } \mu _{T } ) ( G ^{\lambda , u } ) + \int _{0 } ^{T } \int Z ( d t , d y ) \int _{0 } ^{t } \mu _{s } ( G ^{\lambda , u } ( \cdot-y )),NEWLINE\]NEWLINE the self intersection local time for the superprocess. The proofs are given in appendices.
scientific article; zbMATH DE number 6067449
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