-measures for branching exit Markov systems and their applications to differential equations
-measures for branching exit Markov systems and their applications to differential equations
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DOI：
10.1007/s00440-003-0333-8
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发表时间：
2004-03
影响因子：
2
通讯作者：
E. Dynkin;S. E. Kuznetsov
中科院分区：
文献类型：
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作者：
E. Dynkin;S. E. Kuznetsov
用(L,ψ)-超扩散研究了一类半线性方程slu =ψ(u)，其中ψ是一个椭圆微分算子，ψ是一个正函数。在一个特殊情况Δu=u2a中，Le Gall和他的学派成功地应用了他引入的强大的概率工具——布朗蛇，得到了该方程解的深刻结果。其中一些结果（但不是全部）被Dynkin和Kuznetsov通过应用超过程扩展到一般方程。连续路径空间的测量在布朗蛇理论及其应用中起着重要的作用。我们的目标是引入与超过程（以及一般分支出口马尔可夫系统）相关的类似度量。它们是在测度空间上定义的我们称它们为测度。使用-测度可以将布朗蛇和超过程的一些优点结合起来，作为研究半线性偏微分方程的工具。
Semilinear equationsLu=ψ(u) whereLis an elliptic differential operator and ψ is a positive function can be investigated by using (L,ψ)-superdiffusions. In a special case Δu=u2a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measuresxon the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them -measures. Using -measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.