For a biased aging continuous-time random walk with finite-mean, infinite-variance waiting times, the long-time position distribution is governed by a fractional advection-diffusion equation in space, while its far tail follows an infinite density that depends on the aging time.
It can be seen that the strong relation between the position and the number of renewals is x ∼ aN ∼ at/⟨τ ⟩ obtained using large deviations
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Diffusion equation and rare fluctuations of the biased aging continuous-time random walk model
For a biased aging continuous-time random walk with finite-mean, infinite-variance waiting times, the long-time position distribution is governed by a fractional advection-diffusion equation in space, while its far tail follows an infinite density that depends on the aging time.