We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure. For this scheme, we prove general method stability. The work is motivated by Partial Differential Equation (PDE) models of flocculation for which the shape of the post-fragmentation conditional probability measure greatly impacts the solution dynamics. To illustrate our methodology, we apply the theory to a particular PDE model that arises in the study of population dynamics for flocculating bacterial aggregates in suspension, and provide numerical evidence for the utility of the approach.

中文翻译：

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一类条件概率测度相关演化方程的反问题

我们研究了在规模结构人口建模中出现的依赖于度量的进化方程中识别条件概率度量的逆问题。我们将逆问题表述为概率测度估计的最小二乘问题。使用 Prohorov 度量框架，我们证明了最小二乘估计的存在性和一致性，并概述了近似条件概率度量的离散化方案。对于这个方案，我们证明了一般方法的稳定性。这项工作的动机是偏微分方程 (PDE) 絮凝模型，​​其中破碎后条件概率度量的形状极大地影响了溶液动力学。为了说明我们的方法，