A CTRW-driven subdiffusive fractional Brownian bridge in the reconstruction of missing experimental data

Single-particle tracking experiments frequently suffer from missing observations that bias their analysis. To address this challenge, we introduce the fractional Brownian bridge, a novel stochastic process for reconstructing incomplete subdiffusive trajectories, whose subdiffusion mechanism arises from a continuous-time random walk with power-law waiting times. We use this process for reconstructing incomplete single-particle tracking trajectories. Extending the classical Brownian bridge, the model incorporates memory effects via a fractional Fokker-Planck framework, enabling gap-filling that preserves the power-law waiting times characteristic of subdiffusion. We derive the probability density function and mean-square displacement of the process, establish its theoretical properties, and develop an efficient simulation algorithm for practical implementation. Numerical experiments demonstrate that the proposed method accurately reconstructs missing trajectory segments and maintains key statistical features of subdiffusive motion, even when large portions of data are absent. To address the problem of possible violations of the boundary condition at the end point of time interval, we modify the fractional Brownian bridge simulation algorithm. The modified algorithm fully reconstructs the probabilistic structure of the original trajectories under the assumed continuous-time random walk (CTRW)-driven subdiffusive model. This framework provides a robust tool for the reliable gap filling and analysis of experimental single-particle tracking data. The proposed reconstruction procedure is a model-conditional imputation scheme. It is designed for trajectories whose dynamics are assumed, on the basis of independent evidence, to be governed by a CTRW mechanism with power-law waiting times.