On Analytically Tractable Multidimensional Diffusions via Doob h-Transforms, with Resetting and Applications to Wiener and Ornstein--Uhlenbeck Processes
We investigate a class of drift transformations of multidimensional diffusion processes generated through Doob $h$-transforms. These transformations provide a systematic approach for constructing analytically tractable stochastic models with prescribed probabilistic and statistical properties. We derive sufficient conditions under which the transformed diffusion admits an explicit transition density expressed through a product form involving a strictly positive harmonic function. Particular choices of this function lead to mixture representations of the transition density and to bimodality. We further analyze the effects of the transformation on stochastic ordering, diffusions in potential landscapes, and Poissonian resetting dynamics. In particular, we show that the product-form relation is preserved under resetting, enabling explicit characterization of the corresponding stationary distributions. Two multidimensional examples based on Wiener and Ornstein--Uhlenbeck processes illustrate the theory, providing closed-form expressions for transition densities, weight functions, and effective potentials. The two-dimensional setting is explored in detail, including symmetry effects and absorbing boundaries.