多目标遭遇传播子：Dirichlet-to-Neumann谱形式方法

中文摘要 AI 辅助

我们开发了一种基于遭遇的受限扩散形式化方法，用于具有多个目标的有限域，分别解析每个目标上累积的边界局部时间。与单目标问题不同，目标投影算子通常不与控制性的Dirichlet-to-Neumann (DtN)算子交换，因此其特征基不能对角化局部时间依赖性。我们通过将DtN算子分解为目标受限块来克服这一困难。多维拉普拉斯反演随后产生一个收敛的切换展开，其连续项描述了目标之间的交替扩散转移。我们还推导了一个等价的算子值更新方程。对于两个目标，对角化目标受限块提供了基于两个DtN块谱和靶间耦合矩阵的显式谱表示。当耦合保持谱模式时，切换级数可以精确地用修正贝塞尔函数重新求和。一般来说，由于分离目标的非对角DtN块是平滑的，可以求助于低秩有限维近似。我们研究了小型、良好分离目标的有效双模约化，并确定了一个重复靶间转移被逐渐抑制的机制。讨论了概率解释和对扩散控制反应的影响。

英文摘要

We develop an encounter-based formulation of restricted diffusion in a bounded domain with multiple targets, resolving separately the boundary local time accumulated on each target. Unlike the single-target problem, the target projection operators do not generally commute with the governing Dirichlet-to-Neumann (DtN) operator, so that its eigenbasis does not diagonalize the local-time dependence. We overcome this difficulty by decomposing the DtN operator into target-restricted blocks. Multi-dimensional Laplace inversion then yields a convergent switching expansion, whose successive terms describe alternating diffusive transfers between the targets. We also derive an equivalent operator-valued renewal equation. For two targets, diagonalizing the target-restricted blocks provides an explicit spectral representation in terms of two DtN block spectra and an inter-target coupling matrix. When the coupling preserves spectral modes, the switching series can be resummed exactly in terms of modified Bessel functions. In general, as the off-diagonal DtN block is smoothing for separated targets, one can resort to low-rank finite-dimensional approximations. We examine an effective two-mode reduction for small, well-separated targets and identify a regime in which repeated inter-target transfers are progressively suppressed. Probabilistic interpretations and implications for diffusion-controlled reactions are discussed.

发表机构

• Laboratoire de Physique de la Matière Condensée, CNRS – École Polytechnique, Institut Polytechnique de Paris（凝聚态物质物理实验室，法国国家科学研究中心-巴黎综合理工学院，巴黎理工学院）

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