Global martingale solutions to a stochastic superlinear cross-diffusion population system
Abstract
In this work we try to show a martingale solution exists to a stochastic cross-diffusion population system. The transition rate is superlinear. The diffusion matrix does not satisfy the local Lipschitz property. Once the diffusion matrix does not satisfy the local Lipschitz property, we can not apply the existence and uniqueness theorem to derive approximated solutions of this stochastic population system. We have to regularize the diffusion matrix in order to apply the existence and uniqueness theorem, and this is the key idea of this work. By applying the existence and uniqueness theorem, we derive a sequence of approximated solutions. We rely on the Itob formula to estimate approximated solutions. Then we derive the tightness of the approximated sequence in a topological space, with its limit a martingale solution of a stochastic cross-diffusion system. The diffusion matrix of this stochastic cross-diffusion system is a regularization of the original diffusion matrix. The limit of this sequence of regularized diffusion matrices is the diffusion matrix of the original stochastic population system. We show that the limit of this sequence of martingale solutions is also the martingale solution of the original stochastic population system. Nonnegative property for the martingale solution is proved via a standard Stampacchia-type argument.
Copyright (c) 2025 Xi Lin
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