8/3/2019 Guo Junyi- The oscillation of the occupation time process of super-Brownian motion on Sierpinski gasket

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V O I . 43 NO . 12 SCIENCE IN CHINA (Ser ies A ) December 2000

The oscillation of the occupation time process ofsuper-Brownian motion on Sierpinski gasketGUO Junyi ( $ 2 f A)Department of Mathematics, Nankai University, Tianjin 300071, China (email: jyguo@public. pt . j .cn)Received October 22 , 1999Abstract The occupation time process of super-Brownian motion on the Sierpinski gasket is stud-ied. It is shown that this process does not possess stable property in the long run, but oscillates peri-odically in some sense. Other convergence properties are also studied.

Keywords: occupation time process, superprocess, catalytic point.Let G denote the Sierpinski gasket which is a fractal subset of 2, . The Brownian motion

B ( t ) on G was constructed by Barlow and ~erk ins ' ' ]n 1988. Later on, different kind of diffu-sion were constructed in different kinds of fractal structure^[^-^' . Up to now, the properties ofthis kinds of diffusions are comparatively clear. The corresponding superprocess and the superpro-cess with spatial motion in Euclidian space[4.51behave differently. As will be proved in this pa-per, the occupation time process of this super-Brownian motion oscillates regularly in the long runwhich differs from the known fact that, in Euclidian spaces, the occupation time process of super-Brownian motion has a tendency to stabilize in the critical case.

Let X, be the super-Brownian motion with spatial motion B ( t ) and branching mechanism2'' where 0 < /3 < 1 . Then the Laplace functionals for X, satisfy

EPexpl- ( X , , Y ) ~ expi- ( , u , u ( t ) ) / , ( 1 )where u ( t ) is the solution to the equation

In the above equation a special measure ,u on G is used as the initial measure of X ( t ) and actu-ally serves as the reference measure of the transition density function p ( t , ,y ) of B ( t ) (ref.[ l I ) . Briefly speaking, it is a Hausdorff measure normalized to be 1 on the Sierpinski gasketwith unit side. For the properties of ,u and p ( t , ,y ) one can refer to ref. [ 1 ] . We will also

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