In this paper we consider a stochastic process that may experience random
reset events which bring suddenly the system to the starting value and analyze
the relevant statistical magnitudes. We focus our attention on monotonous
continuous-time random walks with a constant drift: the process increases
between the reset events, either by the effect of the random jumps, or by the
action of the deterministic drift. As a result of all these combined factors
interesting properties emerge, like the existence|for any drift strength|of a
stationary transition probability density function, or the faculty of the model
to reproduce power-law-like behavior. General formulas for two extreme
statistics, the survival probability and the mean exit time, are also derived.
To corroborate in an independent way the results of the paper, Monte Carlo
methods were used. These numerical estimations are in full agreement with the
analytical predictions.Comment: 15 pages, 4 figures, revtex4-1; considerable revision, 4 appendices
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