This research identifies a harmonic function for positive random walks in time-varying environments, suggesting new insights into stochastic processes.
Consider a random walk in a time-inhomogeneous random environment. When the environment is stationary and ergodic, we identify a quenched harmonic function for almost every realization of the environment. This function allows us to define a random walk in a random environment conditioned to stay positive, using Doob’s h -transform.
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Hong et al. (2026) studied this question.
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