[10] Continuous random walks are formulated in a similar fashion to equation (2) where the summation is replaced with integration and the probabilities are no longer conditional to intervals in time and age. By continuous, we mean that the PDFs describing particle motion are continuous and not discrete functions. Continuous-time random walks (CTRWs) typically use a continuous PDF for time transitions but a discrete PDF for spatial jumps; here we will allow continuous PDFs for space, time, and age. The original *Montroll and Weiss* [1965] random walk model will be augmented with an age dimension which is defined to be a positive “internal variable” (convective) dimension over which the mass density of water will be distributed. Most of this derivation closely follows the work of *Montroll and Weiss* [1965], *Scher and Lax* [1973], *Shlesinger* [1974], *Klafter and Silbey* [1980], *Klafter et al.* [1987], *Metzler et al.* [1998], and *Metzler and Klafter* [2000], but we retraverse these derivations since we have distributed the problem space over an additional dimension. It is important to recognize that many articles have been written on CTRW in the last 20 years, and many novel applications have been found that we do not mention because the essential foundation for CTRW is laid down by these earlier works. For compactness, we define a new vector for the convective dimensions so our problem space, , becomes . The probability for a tagged particle to just arrive at is,

where the primes denote initial locations, is the initial condition in , is the initial condition in , is the joint probability distribution function (PDF) of the displacements in **x** and **w**, and the limits on the integrals span the full range of initial locations; hence, (6) represents integration over five dimensions. We are interested in the likelihood of finding a particle at , which is given by the product of the probability of arrival at time-age , , and the probability that the particle has not since vacated the site, integrated over all possible time-age arrivals :

where

where is assumed independent of spatial position and is the marginal PDF of both time and age increments. To find a general solution for the expression in (6), we follow *Scher and Lax* [1973] who Laplace transform (LT) equation (7) and solve for . Unlike previous random walk derivations, however, our LTs of (7) are executed with respect to time and age. We define , where and are the Laplace parameters of time and age, respectively, and the (double) LT of (7) is,

where overbars denote an LT in time and tildes denote an LT in age. Solve (9) for , substitute that result into the LT of (6), and simplify to find,

[12] Equation (11) is the FT-LT Green's function for the probability propagator of a tagged particle in space, time, and age. The leading term on the right-hand side of (11) accounts for the initial conditions and waiting times and the second term accounts for the movement of our tagged particle. The Laplace parameters in the denominator are the result of taking the 2-D LT of a constant. For a given set of transition time and waiting time distributions, and respectively, equation (11) is the exact solution of the system. The vector notation for time and age has put this result in an almost identical form to that of previous authors [e.g., *Montroll and Weiss* 1965; *Scher and Lax*, 1973; *Dentz and Berkowitz*, 2003] but with the distinct feature that is distributed across another dimension and there is a second Laplace parameter in (11). Although we derived it using age, we could have derived (11) for a generalized exposure time [e.g., *Ginn*, 1999] without any changes. This is a result of the fact that we have not yet specified the form of the two joint PDFs required to solve (11), which we do in the next sections (3.2 and 3.3).