Drift transformations of multidimensional diffusion processes: a framework with applications to Wiener and Ornstein--Uhlenbeck processes

We investigate a class of drift-based transformations of multidimensional diffusion processes. These mappings yield a flexible mechanism for generating analytically tractable diffusion models with controlled probabilistic properties. Specifically, we derive sufficient conditions under which the transformed process admits a transition density with an explicit product-form representation in the multidimensional setting. The transformation is defined through a weight function $w$, obtained as the solution of a suitable partial differential equation. Moreover, specific choices of $w$ yield mixture representations of the transformed transition density, leading naturally to bimodal behavior. We establish closed-form relations between the original and transformed diffusions and investigate properties including stochastic ordering, Poissonian resetting, and diffusions in potential fields. The analysis of the usual stochastic order reveals how the transformation modifies the probabilistic behavior of the processes. Furthermore, the product-form relation is shown to persist under Poisson-paced resets, allowing for explicit stationary distributions in certain cases. We present two case studies involving transformations of the Wiener and Ornstein–Uhlenbeck processes, for which explicit expressions for the weight function, potential function, and transition densities are derived. Particular attention is devoted to the two-dimensional setting, where symmetry properties and absorbing boundaries are explored, providing further insight into the structure and behavior of the transformed diffusions.