Aging continuous time random walks

We investigate biased and nonbiased aging continuous time random walks (ACTRW), using fractal renewal theory. For example, a biased ACTRW process describes a Montroll-Weiss CTRW process which starts at time -t(a) and then at time t=0 a bias is added to the random walk (i.e., an external field is switched on). Statistical behaviors of the displacement of the random walker r=r(t)-r(0) in the time interval (0,t) are obtained, after aging the random walk in the time interval (-t(a),0). In ACTRW formalism, the Green function P(r,t(a),t) depends on the age of the random walk t(a) and the forward time t. We derive a generalized Montroll-Weiss equation, which yields an exact expression for the Fourier double Laplace transform of the ACTRW Green function. Asymptotic long times t(a) and t behaviors of the Green function are shown to be related to the arc-sine distribution and Levy stable laws. In the limit of t>t(a), we recover the standard nonequilibrium CTRW behaviors, while the important regimes t