Martingale problem for superprocesses with non-classical branching functional

Abstract

The martingale problem for superprocesses with parameters (𝛏, Ф, 𝑘) is studied where 𝑘(𝒹𝑠) may not be absolutely continuous with respect to the Lebesgue measure. This requires a generalization of the concept of martingale problem: we show that for any process X which partially solves the martingale problem, an extended form of the liftings defined in [8] exists; these liftings are part of the statement of the full martingale problem, which is hence not defined for processes X who fail to solve the partial martingale problem. The existence of a solution to the martingale problem follows essentially from Itô’s formula. The proof of uniqueness requires that we find a sequence of (𝛏, Ф, 𝑘𝑛) -superprocesses “approximating” the (𝛏, Ф, 𝑘)-superprocess, where 𝑘𝑛(𝒹𝑠) has the form λ𝑛 (𝑠,𝛏𝑠)𝒹𝑠. Using an argument in [9], applied to the (𝛏, Ф, 𝑘𝑛)-superprocesses, we prove, passing to the limit, that the full martingale problem has a unique solution. This result is applied to construct superprocesses with interactions via a Dawson–Girsanov transformation.