Some remarks and examples concerning the transience and recurrence of random diffusions
Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment
Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site , the probability of jumping to the right is ()∈[½, 1), until the first time the process jumps to the left from site , from which time onward the probability of jumping to the right is ½. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments {()}∈. In deterministic environments, we also study the speed of the...
Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift
We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases – a transient to mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain non-negative constant when the diffusion is sufficiently near...
The lifetimes of conditioned diffusion processes
The compact support property for measure-valued processes
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