Locomotion characteristics are often recorded within bounded spaces, a constraint which introduces geometry-specific biases and potentially complicates the inference of behavioural features from empirical observations. We describe how statistical properties of an uncorrelated random walk, namely the steady-state stopping location probability density and the empirical step probability density, are affected by enclosure in a bounded space. The random walk here is considered as a null model for an organism moving intermittently in such a space, that is, the points represent stopping locations and the step is the displacement between them. Closed-form expressions are derived for motion in one dimension and simple two-dimensional geometries, in addition to an implicit expression for arbitrary (convex) geometries. For the particular choice of no-go boundary conditions, we demonstrate that the empirical step distribution is related to the intrinsic step distribution, i.e. the one we would observe in unbounded space, via a multiplicative transformation dependent solely on the boundary geometry. This conclusion allows in practice for the compensation of boundary effects and the reconstruction of the intrinsic step distribution from empirical observations.

运动特征通常记录在有界空间内，这是一种引入几何特定偏差的约束，并可能使从经验观察中推断出行为特征变得复杂。我们描述了一个不相关的随机游走的统计特性，即稳态停止位置概率密度和经验步骤概率密度，是如何受到有界空间中的包围的影响的。这里的随机游走被认为是一个有机体在这样一个空间中间歇性运动的零模型，即点代表停止位置，步长是它们之间的位移。除了任意（凸）几何的隐式表达式外，还为一维运动和简单的二维几何导出了封闭形式的表达式。对于禁止边界条件的特定选择，我们证明了经验阶跃分布与固有阶跃分布有关，即我们将在无界空间中观察到的分布，通过仅依赖于边界几何的乘法变换。该结论允许在实践中补偿边界效应和根据经验观察重建固有阶跃分布。