Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.

翻译：许多物理和生物科学中的随机过程可建模为含乘性噪声的布朗动力学。然而，当扩散项依赖于构型时，这些过程的数值积分器可能丧失精度甚至无法收敛。一种解决方法是构造到恒定扩散过程的变换，并对变换后的过程进行采样。本文阐释如何单独或组合使用基于坐标和基于时间重缩放的变换，将一类广义变扩散布朗运动过程映射为恒定扩散过程。这些变换是可逆的，因此能够恢复原始动力学。我们首先通过一维实例论证该方法，进而考虑多元扩散过程。通过数值模拟展示变换的优势，证明积分器与变换的恰当结合如何提升计算效率及对不变分布的收敛阶数。特别地，本文推导的变换适用于一类在多体各向异性斯托克斯-爱因斯坦扩散问题中的应用，此类问题在生物物理建模中具有重要应用价值。