A framework for drift transformations of multidimensional diffusion processes with applications to Wiener and Ornstein--Uhlenbeck dynamics

We investigate a class of drift-based transformations between multidimensional diffusion processes.

The approach allows to construct a new process whose transition probability density function (p.d.f.)\ can be expressed in a product form involving the p.d.f.\ of the original diffusion.

The framework is formulated in terms of stochastic differential equations providing general conditions under which the transformed p.d.f.\ remains analytically tractable in the multidimensional setting.

The transformation is defined through a weight function $w$, derived as the solution of a suitable partial differential equation.

Also, specific forms of $w$ yield certain mixture representations of the transformed p.d.f., which also leads to identify a bimodal feature.

We establish closed-form relations between the original and transformed diffusions focusing on features such as stochastic ordering, Poissonian resetting, and diffusions in potential fields.

The analysis of the usual stochastic order highlights how the transformation can substantially alter the probabilistic behavior of the process.

Furthermore, the product-form relation is shown to persist under Poisson-paced resets, allowing for explicit stationary distributions in certain cases.

Two fundamental case studies are presented, based on transformations of the Wiener and Ornstein-Uhlenbeck processes, for which explicit expressions of the weight function, potential function, and transition densities are derived.

Special attention is given 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.