We extend the class of $(ξ,ψ,K)$-superprocesses known so far by applying a simple transformation induced by a \lq\lq weight function\rq\rq\ for the one-particle motion. These transformed superprocesses may exist under weak conditions on the branching parameters, and their state space automatically extends to a certain space of possibly infinite Radon measures. It turns out that a number of superprocesses which were so far not included in the general theory fall into this class. For instance, we are able to extend the hyperbolic branching catalyst of Fleischmann and Mueller to the case of $β$-branching. In the second part of this paper, we discuss regularity properties of our processes. Under the assumption that the one-particle motion is a Hunt process, we show that our superprocesses possess right versions having \cadlag\ paths with respect to a natural topology on the state space. The proof uses an approximation with branching particle systems on Skorohod space.
Regularity of the Value Function in Discounted Infinite-Time Mean Field Games
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions
Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size
School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Department of Mathematical Sciences, University of Durham（杜伦大学数学科学学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Department of Mathematics and Risk Management Institute, National University of Singapore（新加坡国立大学数学系与风险管理研究所）
