Using the ideal of Dynkin’s branching particle systems for reference,We constructa widespread class of two-type superprocesses by taking limit to the renormalized two-typebranching particle systems,and study the properties of the additive functional of the two-typesuperprocesses.

In this paper, we investigate the Balayage Problem of a new form for X which is finer than the Balayage Problem in classical case and prove a representation Theorem of additive functionals for X: If A= {A_t,t≥0} is a nature additive functional of X,then, under some conditions, there exists a non-negative function g_A such that A_t=∫_0~gA(X_)ds andthe exact form of g_A is given.

With Doob-Meyer decomposition theorem, we prove that an excessive function should correspond the unique predictable additive functional in the first part; Using the results in the first part, we construct the local time of Markov chains at S_∞, and give the sufficient and necessary condition of continuity of local time.

The extremum of function in mathematical physics;