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Pinsky","submitted_at":"2005-04-19T15:55:36Z","abstract_excerpt":"The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form \\[ \\begin{aligned}& u_t=Lu+\\beta u-\\alpha u^p \\text{in} R^d\\times (0,\\infty), p\\in(1,2]; &u(x,0)=f(x) \\text{in} R^d; &u(x,t)\\ge0 \\text{in} R^d\\times[0,\\infty). \\end{aligned} \\] In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. 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