Basic study on immigration super processes and catalytic super diffusions

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We constructed a mathematical model (say a stochastic process) that is able to describe the status changing its branching mechanism according to whether the circumstance is good or not dependent on the location, and investigated the uniqueness and regularity of the process. On this account, based on the process we proposed a random model for immune response against the cancer cells, by taking cytotoxicity via effectors into consideration. As a result of analysis, we found that the local extinction revealed by the model is of extreme importance on an applicational basis, because the property just corresponds to the situation that effects provided by the effectors expel cancer cells from the tissue.