Synergy of Doob Transformation and Montroll Defect Theory for Random Walks in External Potentials

We present a systematic method for constructing stochastic processes by modifying simpler, analytically solvable random walks on discrete lattices. Our framework integrates the Doob $h$-transformation with the Montroll defect theory, overcoming the strict constraints associated with each method alone. By combining these two approaches, we map random walks in simple potentials onto processes involving more general external potentials and metastable states. Explicit analytical expressions relate the transformed process to the original one, facilitating direct investigation of exponential decay rates and additional dynamical modes. As an illustrative example, we demonstrate our method by analyzing a random walker in a linear potential modified to include a metastable state, revealing distinct exponential decay regimes relevant to escape dynamics.