In this paper, we study the nonlinear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation of this equation, which is also related to the supercooled Stefan problem, as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.

中文翻译：

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McKean-Vlasov 方程的半解析解，通过击中边界得到反馈

在本文中，我们研究了与粒子系统相关的非线性扩散方程，其中常见的漂移取决于粒子在边界处的吸收率。我们提供了这个方程的解释，它也与过冷的 Stefan 问题有关，作为一个在大型相互关联的银行系统中具有违约传染的结构性信用风险模型。使用热势的方法，我们推导出了过渡密度和吸收损失的 Volterra 积分方程的耦合系统。对于小的交互参数，给出了扩展的近似值。我们还提出了一种数值求解算法并进行了计算测试。