Open-world egocentric activity recognition poses a fundamental challenge due to its unconstrained nature, requiring models to infer unseen activities from an expansive, partially observed search space. We introduce ProbRes, a Probabilistic Residual search framework based on jump-diffusion that efficiently navigates this space by balancing prior-guided exploration with likelihood-driven exploitation. Our approach integrates structured commonsense priors to construct a semantically coherent search space, adaptively refines predictions using Vision-Language Models (VLMs) and employs a stochastic search mechanism to locate high-likelihood activity labels while minimizing exhaustive enumeration efficiently. We systematically evaluate ProbRes across multiple openness levels (L0--L3), demonstrating its adaptability to increasing search space complexity. In addition to achieving state-of-the-art performance on benchmark datasets (GTEA Gaze, GTEA Gaze+, EPIC-Kitchens, and Charades-Ego), we establish a clear taxonomy for open-world recognition, delineating the challenges and methodological advancements necessary for egocentric activity understanding.