Nonlinear stochastic motion presents significant challenges for Bayesian particle tracking. To address this challenge, we propose a lifting framework that constructs a higher-dimensional linear stochastic representation of nonlinear state-space models. The resulting surrogates enable the use of computationally efficient linear filtering techniques while retaining a direct connection to the underlying nonlinear dynamics. The paper derives the necessary conditions for such transformations using Ito's lemma and variational calculus, and illustrates the method on a bistable cubic motion model, radial Brownian process model, and a logistic model with multiplicative noise. Simulations confirm that the transformed linear systems, when projected back, accurately reconstruct the nonlinear dynamics and, in distinct regimes of stiffness and singularity, yield tracking accuracy competitive with conventional filters, while avoiding their structural instabilities.

翻译：非线性随机运动为贝叶斯粒子跟踪带来显著挑战。为应对这一挑战，我们提出一种提升框架，通过构建更高维度的线性随机表示来表征非线性状态空间模型。得到的代理模型既能在计算上采用高效的线性滤波技术，又能保留与潜在非线性动力学的直接关联。本文利用伊藤引理和变分法推导了此类变换的必备条件，并在双稳态三次运动模型、径向布朗过程模型和含乘性噪声的逻辑斯蒂模型上验证了该方法。模拟证实，当投影回原空间时，变换后的线性系统能够准确重构非线性动力学，且在刚性和奇异性不同区域中，其跟踪精度可媲美传统滤波器，同时规避了后者的结构不稳定性。