We extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain \(\varOmega \) with non-local dynamic conditions on the boundary \(\partial \varOmega \). Due to the pioneering nature of the present research, we propose here the apparently simple case of \(\varOmega =(0, \infty )\) with boundary \(\{0\}\) of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

我们通过引入涉及非局部动态边界条件的一类新的边值问题来扩展[14]中获得的结果。我们专注于寻找边界上具有非局部动态条件的域\(\varOmega \)上的局部问题的解决方案\(\partial \varOmega \) 。由于本研究的开创性，我们在这里提出了明显简单的情况\(\varOmega =(0, \infty )\) ，其边界\(\{0\}\)为零勒贝格测度。我们的结果对于具有正（有限）Borel 测度的边界的一般情况具有指导意义。此外，我们认为，我们为动态边值问题和相关模型的概率描述带来了新的思路。