Random walks in a time-inhomogeneous random environment conditioned to stay positive

Key Points

• This research aims to explore random walks conditioned to remain positive within varying random environments.
• Analyzed random walks under time-inhomogeneous random environments
• Identified a quenched harmonic function for the environment
• Utilized Doob’s h-transform to define positive conditioned random walks
• Established the existence of a harmonic function for almost every environment realization
• Showed that positive conditioning alters the behavior of random walks in non-constant environments