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Generalized self-intersection local time for a superprocess over a stochastic flow. (English) Zbl 1278.60126

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, \[ \sigma _{i j } ( x , y ) = \sum _{k = 1 } ^{m } c _{i k } ( x ) c _{j k } ( y ), \]
\[ a _{i j } ( x , y ) = \delta _{i j } b _{i } ( x ) b _{j }( y ) + \sigma _{i j } ( x , y ), \]
\[ (L \varphi ) ( x ) = ( 1 / 2 ) \sum _{i , j = 1 }^{d } a _{i j }( x , x ) ( \partial _{i j } ^{2 } \varphi ) ( x ), \] supposed uniformly elliptic, \[ ( \Lambda \varphi ) ( x , y ) = \sum _{i , j = 1 } ^{d }\sigma _{i j } ( x , y )( \partial _{i } \varphi ) ( x ) ( \partial _{j } \varphi ) ( x ), \] 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), \[ \mu _{t}(\varphi)-\mu _{0 } ( \varphi )-\int _{0} ^{t } \mu _{s } ( L \varphi ) d s \] 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 G. Skoulakis and 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.
Consider \(\alpha = ( \alpha _{0 } ,\dotsc , \alpha _{| \alpha | } )\), \(\alpha _{i } \in [ 0 , 1 ]\), 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 \[ B _{0 } ^{\alpha , n }= x _{\alpha _{0}}^{n \;}\text{ for}\;| \alpha | = 0,\;\;B _{t } ^{\alpha , n } = B _{t } ^{\alpha - 1 , n} \] (for \(0 \leq i \leq | \alpha | / n\)), \(W _{0 } ^{n } = 0\) and the rest is independent, the solution \(Y ^{\alpha , n }\) of \[ d Y _{t } = b ( Y _{t } ) d B _{t } ^{\alpha , n } + c ( Y _{t } ) d W _{t } ^{n } \] on \([ 0 , ( k + 1 ) n ^{- 1 } ]\), for \( \alpha \in [ k / n , ( k - 1 ) / n ),\) and \[ Y _{0 } ^{\alpha , n } = x _{\alpha _{0 }} ^{n }, \tau ^{\alpha , n } = 0 \] 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 \(\mu _{t } ^{( n ) } ( U ) = n^{- 1 } \operatorname{card} \{ \alpha ; t \in [ | \alpha | / n , ( | \alpha | + 1 ) / n ) , X _{t } ^{\alpha , n } \in U \} \).
The 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, \[ \int _{0 } ^{T } d t _{3 } \int _{0 } ^{t_3} d t _{2 } \int _{0} ^{t_2 } d t _{1 } \operatorname{E} ( ( \mu _{t _{1 }} \mu _{t _{2 }} \mu _{t _{3 }} ) ( \psi ) ) \leq C \| \phi \| _{1 } ^{2 } \] and \[ \int _{0 } ^{t_3} d t _{2 } \int _{0 } ^{t_2 } d t _{1 } \operatorname{E} ( ( \mu _{t _{1 }} \mu _{t _{2 }} \mu _{t _{3 }} ^{2 } ) ( \varphi ) ) \leq \| \phi \| _{1 } ^{2 }. \] 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 \[ \gamma _{\varepsilon } ^{\lambda } ( u , T ) = \int _{0} ^{T } d t \int _{0 } ^{t} d s ( ( \lambda -L ) ( \mu _{s } \mu _{t } ) ( G _{\varepsilon } ^{\lambda , u } ) ), \]
\[ {\mathcal L} _{\varepsilon } ^{\lambda } ( u , T ) = \gamma _{\varepsilon } ^{\lambda } ( u , T )- \int _{0 } ^{T } d t ( \mu _{t } \mu _{t } ) ( G _{\varepsilon } ^{\lambda , u } ). \] For \(d \leq 3\), if \(Z\) is the martingale measure for \[ ( L _{2 } \Psi ) ( x , y ) = (1/2)\sum _{i , j = 1 } ^{d } a _{i j } ( y ) \partial _{2 , i } \partial _{2 , j } \Psi ( x , y ), \]
\[ \lim _{\varepsilon \downarrow 0 } {\mathcal L} _{\varepsilon } ^{\lambda } ( u , T ) \] exists in \(L^{2 }\), uniformly in \(u\), being equal to \[ \lambda \int _{0 } ^{T } d t \int _{0 } ^{t } d s ( \mu _{s } \mu _{t } ) ( G ^{\lambda , u } ) - \int _{0 } ^{T } d t ( \mu _{t } \mu _{T } ) ( G ^{\lambda , u } ) + \int _{0 } ^{T } \int Z ( d t , d y ) \int _{0 } ^{t } \mu _{s } ( G ^{\lambda , u } ( \cdot-y )), \] the self intersection local time for the superprocess. The proofs are given in appendices.

MSC:

60J68 Superprocesses
60G57 Random measures
60J55 Local time and additive functionals
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1018.60052

References:

[1] Adler, R. J. and Lewin, M. (1991). An evolution equation for the intersection local times of superprocesses. In Stochastic Analysis : Proceeding of the Durahm Symposium on Stochastic Analysis . Cambridge Univ. Press, Cambridge. · Zbl 0768.60066 · doi:10.1017/CBO9780511662980.002
[2] Adler, R. J. and Lewin, M. (1992). Local time and Tanaka formulae for super Brownian motion and super stable processes. Stochastic Process. Appl. 41 45-67. · Zbl 0754.60086 · doi:10.1016/0304-4149(92)90146-H
[3] Billingsley, P. (1995). Probability and Measure . Wiley, New York. · Zbl 0822.60002
[4] Chung, K. L. (1974). A Course In Probability Theory . Academic Press, New York. · Zbl 0345.60003
[5] Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrsch. Verw. Gebiete 40 125-145. · Zbl 0343.60001 · doi:10.1007/BF00532877
[6] Dawson, D. A. (1993). Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XXI- 1991. Lecture Notes in Math. 1541 . Springer, Berlin. · Zbl 0799.60080
[7] Dawson, D. A., Li, Z. and Wang, H. (2001). Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 1-33. · Zbl 1008.60093 · doi:10.1214/EJP.v6-98
[8] Dynkin, E. B. (1965). Markov Processes , Vols I , II . Academic Press, New York. · Zbl 0132.37901
[9] Dynkin, E. B. (1988). Representation for functionals of superprocesses by multiple stochastic integrals, with application to self-intersection local times. Astérisque 157-158 147-171. · Zbl 0659.60105
[10] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes , Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[11] He, H. (2009). Discontinuous superprocesses with dependent spatial motion. Stochastic Process. Appl. 119 130-166. · Zbl 1169.60319 · doi:10.1016/j.spa.2008.02.002
[12] Hörmander, L. (1985). The Analysis of Linear Partial Differential Operators , Vol. III . Springer, Berlin. · Zbl 0601.35001
[13] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes . North-Holland, Amsterdam. · Zbl 0495.60005
[14] Karatzas, I. and Shreve, S. E. (2000). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0734.60060
[15] Lieb, E. H. and Loss, M. (2001). Analysis , 2nd ed. Graduate Studies in Mathematics 14 . Amer. Math. Soc., Providence, RI. · Zbl 0966.26002
[16] Mytnik, L. and Villa, J. (2007). Self-intersection local time of \((\alpha ,d,\beta)\)-superprocesses. Ann. Inst. H. Poincaré Probab. Stat. 43 481-507. · Zbl 1118.60041 · doi:10.1016/j.anihpb.2006.07.005
[17] Ren, Y., Song, R. and Wang, H. (2009). A class of stochastic partial differential equations for interacting superprocesses on a bounded domain. Osaka J. Math. 46 373-401. · Zbl 1170.60034
[18] Rosen, J. (1992). Renormalization and limit theorems for self-intersections of superprocesses. Ann. Probab. 20 1341-1368. · Zbl 0760.60024 · doi:10.1214/aop/1176989694
[19] Rudin, W. (1973). Functional Analysis . McGraw-Hill, New York. · Zbl 0253.46001
[20] Rudin, W. (1976). Principles of Mathematical Analysis . McGraw-Hill, New York. · Zbl 0346.26002
[21] Rudin, W. (1987). Real and Complex Analysis . McGraw-Hill, New York. · Zbl 0925.00005
[22] Skoulakis, G. and Adler, R. J. (2001). Superprocesses over a stochastic flow. Ann. Appl. Probab. 11 488-543. · Zbl 1018.60052 · doi:10.1214/aoap/1015345302
[23] Walsh, J. (1986). An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 265-437. Springer, Berlin. · Zbl 0608.60060
[24] Wang, H. (1998). A class of measure valued branching diffusions in a random medium. Stoch. Anal. Appl. 16 753-768. · Zbl 0913.60091 · doi:10.1080/07362999808809560
[25] Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 141-167. · Zbl 0159.46201
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