Flux and First-Passage Time Distributions in One-Dimensional Integrated Stochastic Processes with Arbitrary Temporal Correlation and Drift

Abstract

The arrival of tracers at boundaries with defined distances from the origin of their motion in stochastically fluctuating advection processes is investigated. The advection model is a stationary one-dimensional integrated stochastic process with an arbitrary a priori known correlation and with possible mean drift. The current (direction-sensitive), the total flux (direction-insensitive) of tracers through a non-absorbing boundary, and the first-passage times of the tracers at an absorbing boundary are derived depending on the correlation function of the carrying flow velocity. While the general derivations are universal with respect to the distribution function of the advection’s increments, the current and the total flux are explicitly derived for a Gaussian distribution. The first-passage time is derived implicitly through an integral that is solved numerically in the present study. No approximations or restrictions to special cases of the advection process are used. One application is one-dimensional Gaussian turbulence, where the one-dimensional random velocity carries tracer particles through space. Finally, subdiffusive or superdiffusive behavior can temporarily be reached by such a stochastic process with an adequately designed correlation function.

1. Introduction

Currents and arrival rates on non-absorbing boundaries, or first-arrival times on absorbing boundaries (also called first-arrival rates or first-passage times), in integrated stochastic processes are relevant to practical applications such as diffusion reactions, turbulence, animal movements, epidemic dynamics, and financial markets. Flow meters often use the time of flight of natural tracers or artificial markers between two detection locations separated by a given distance to determine the flow rate.

Currents, counting tracer arrivals at a boundary moving upwards with a positive sign and those in the opposite direction with a negative sign, and total arrival rates, counting all tracer arrivals at the boundary regardless of their direction of motion, basically depend on the distribution of tracer-particle locations over time and their actual velocities at that time. If both parts of that are known, then these quantities can be obtained by integrating over all possible arrival velocities with their respective probability densities, either from explicit solutions or through one-dimensional numerical integration.

Unfortunately, the prediction of first-arrival statistics is much more complicated. If tracers cross the virtually defined boundary for the first time, then a certain probability remains that the same tracers will return to the boundary later—a second, third, or even more times. A possible solution to match a virtual boundary for prediction with the absorbing character is to take the total arrival rate without absorption, including the first and all higher-order arrivals, and reduce it by the portions that result from tracers returning after having reached the boundary once before. This leads to an integral formulation of the relation between first arrivals and all arrivals together, including returns, see [1], which can be arbitrarily complex, e.g., if the velocity has a temporal correlation.

There exist a few methods for solving such problems, e.g., involving the Laplace transform in cases where the integral formulation can be interpreted as a deconvolution, solved by dividing the respective Laplace transforms [2]. A second method is that of reflected negative images of the tracers, which can be used when tracers that have reached the boundary are introduced and superimposed with a negative amplitude to imitate their removal from the ensemble of tracers [3]. This requires symmetry in the distribution of their motion and independence from history. Therefore, this method is limited to special cases in which the processes comply with these conditions. Otherwise, approximations must be made, and the deviations between the process under investigation and the predictions for the approximated process need quantification; see [4].

In the present paper, stationary one-dimensional integrated stochastic processes are investigated with arbitrary correlations and possible mean drift. The tracers start at a fixed position $x_0$ at a fixed time $t_0$, and travel with the time-dependent velocity $u\left(t\right)$.

First, general formulations of currents, total arrival rates, and first arrivals are derived for the general case of an arbitrary probability density function $h\left(u\right)$ of velocities u. Then, for the case of Gaussian-distributed velocities with the mean drift velocity ${\mu }_u=\langle u\rangle$ and the variance ${\sigma }_u^2=\langle {(u-{\mu }_u)}^2\rangle$, where $\langle ·\rangle$ denotes the expectation, the current and the total arrival rate of the tracers at non-absorbing boundaries are given as explicit solutions of the respective integrals involving the arbitrary but a priori known correlation function $\rho \left(\tau \right)$

$$\rho \left(\tau \right)={\displaystyle \frac{〈u\left(t\right)u(t+\tau )〉}{{\sigma }_u^2}}.$$

(1)

The process is assumed to be stationary; thus, $h\left(u\right)$ and $\rho \left(\tau \right)$ do not change with time. While $x_0$ and $t_0$ are fixed, $u_0$ is not given; it is randomly chosen from the distribution $h\left(u\right)$.

Unfortunately, the probability density of the first-passage times for such processes with correlation does not comply with the boundary conditions of the effective methods described above. Therefore, the method of numerical integration is used here, including for arbitrary correlation functions.

For random walks, the initial conditions are usually chosen as $t_0=0$ and $x_0=0$, and the tracers typically prefer the positive direction of motion (${\mu }_u>0$) if a mean drift is encountered. However, for first-passage problems, the initial position at the starting time $t_0=0$ is commonly chosen as positive, $x_0>0$, and the absorbing boundary is placed at $x=0$. However, the calculations below use only relative quantities like distances traveled in a certain time interval, such that they are independent of the specific arrangements. In the present study, the example scenarios simulated comply with the usual arrangement for first-passage problems. The derivations in the appendix then consistently yield equations for relative displacements within time intervals. The only difference that requires consideration is whether the tracer is bound to the positive or negative side of the boundary. Coming from the positive side, the tracer passes the virtual boundary for the first time in the negative direction, that is, with a negative velocity; the second time with a positive velocity, and so on. Conversely, coming from the negative side, the first and all odd counts of boundary crossings occur with positive velocity, and all even boundary crossings occur with negative velocity. The drift velocity can also be chosen arbitrarily. However, to ensure that all tracers released will hit the boundary sooner or later, the drift velocity must be negative if the tracer is released above the boundary. For positive drift, tracers may move away from the boundary and never reach it. While the calculations below stay valid, the interpretation of the first-passage distribution as a probability density function becomes problematic because some tracers are lost and do not yield any specific first-passage time. The situation can be resolved by interpreting all lost tracers as having an infinite first-passage time. Such cases can be constructed and treated with the framework below, and the equations will still lead to meaningful estimates. However, this is beyond the scope of the present investigation.

2. General Cases for Arbitrary Distribution and Correlation

2.1. The Current of Tracers Through a Boundary

Assuming $f(\xi ,t)$ is the spatial probability density that a tracer has traveled the distance $\xi$ within time t, the current through a boundary at distance x (relative to $x_0$) at time t is (method Q₁):

$$q(x,t)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\underset{x}{\overset{\infty }{\int }}f(\xi ,t)\mathrm{d}\xi \right].$$

(2)

The integral is the probability that the tracer is above the boundary (between $x_0+x$ and ∞) at time t, regardless of whether it was released above or below the boundary (the distance x can be positive or negative). The temporal derivative of the probability given by the integral, then, is the current through the boundary per tracer released. Note that the tracers may cross the boundary multiple times. The upwards flux from below the boundary is counted as positive, whereas the opposite flux is counted as negative, similar to a current of charges through the boundary, where equal positive and negative fluxes cancel each other out. This method of deriving the current through a non-absorbing boundary is very general with respect to the investigated process and the distributions of velocities. The task is to find an appropriate $f(\xi ,t)$. For stationary integrated processes with increments u having a Gaussian distribution $h\left(u\right)$ and correlation $\rho \left(\tau \right)$, explicit expressions are given in Section 3 for all quantities occurring in the derivations, including $f(x,t)$, which is the same function as $f(\xi ,t)$ with its argument replaced; see 3 in Section 3.

Since the different methods for deriving the various currents, fluxes, and arrival rates will be compared quantitatively in the Results section, the methods will be indicated by a numerical index for the specific quantity, e.g., the method above will be denoted as Q₁, and the following alternative method as Q₂.

Alternatively (method Q₂), the current through the boundary is obtained from the probability density that a tracer is positioned at the boundary at distance x from the origin $x_0$ at time t. However, the current additionally depends on the velocity of the tracer u at its arrival at the boundary. If a certain spatial number density of tracers is in infinitesimal proximity to the boundary, then this spatial number density moves through the boundary with its arrival velocity, and the current is therefore proportional to that velocity. Consequently, the relation between the spatial number density f and the current q depends on the arrival velocity u, namely,

$$q(x,t,u)=uf(x,t,u),$$

(3)

where $q(x,t,u)$ is the current–velocity probability density and $f(x,t,u)$ is the space–velocity probability density of having traveled the distance x within time t and having reached the boundary (at position $x_0+x$) with the velocity u. Taking the arrival velocity as the independent variable, $f(x,t,u)$ can be obtained as

$$f(x,t,u)=f(x,t|u\left)h\right(u)$$

(4)

where $f(x,t|u)$ is the spatial probability density of having traveled the distance x within time t, conditioned on the arrival velocity u at the boundary at time t, and $h\left(u\right)$ is the probability density of all possible arrival velocities u. Then, the current through the boundary at time t is given by the integral

$$q(x,t)=\underset{-\infty }{\overset{\infty }{\int }}q(x,t,u)\mathrm{d}u=\underset{-\infty }{\overset{\infty }{\int }}uf(x,t|u)h\left(u\right)\mathrm{d}u$$

(5)

over all these possible arrival velocities, u. Note that $h\left(u\right)$ can be any distribution. However, for Gaussian distributed velocities, explicit expressions are given in Section 3 for all probability densities occurring in the derivations, which make use of an arbitrary but a priori known correlation function $\rho \left(\tau \right)$. For $h\left(u\right)$, see 1 in Section 3; for $f(x,t|u)$, see 4. The explicit solution of the integral in Equation (5) is given in 7.

2.2. The Arrival Rate (Absolute Flux) of Tracers at a Boundary

Similarly, the arrival rate, or absolute flux, of tracers at a non-absorbing boundary depends on the absolute magnitude of the arrival velocity and on the spatial probability density of having reached the distance x after time t, as

$$j(x,t,u)=\left|u\right|f(x,t,u)=\left|u\right|f(x,t|u\left)h\right(u),$$

(6)

where both directions of tracer arrivals at the boundary are counted as positive. Then, the total arrival rate at the non-absorbing boundary at time t (method J₁) is given by the integral

$$j(x,t)=\underset{-\infty }{\overset{\infty }{\int }}j(x,t,u)\mathrm{d}u=\underset{-\infty }{\overset{\infty }{\int }}\left|u\right|f(x,t|u)h\left(u\right)\mathrm{d}u$$

(7)

over all possible arrival velocities u from its probability density function $h\left(u\right)$.

Limiting the integration to only positive velocities

$$j^{+}(x,t)=\underset{0}{\overset{\infty }{\int }}j(x,t,u)\mathrm{d}u=\underset{0}{\overset{\infty }{\int }}\left|u\right|f(x,t|u)h\left(u\right)\mathrm{d}u$$

(8)

yields the flux of tracers passing the boundary in the positive x direction, that is, those with positive velocities u. Accordingly,

$$j^{-}(x,t)=\underset{-\infty }{\overset{0}{\int }}j(x,t,u)\mathrm{d}u=\underset{-\infty }{\overset{0}{\int }}\left|u\right|f(x,t|u)h\left(u\right)\mathrm{d}u$$

(9)

yields the flux of tracers passing the boundary in the negative x direction, that is, with negative velocities u. For the arrangement where the tracers are released above the boundary, $j^{-}(x,t)$ is the sum of all arrivals of tracers passing the boundary away from their origin with negative velocity, which occurs for the first, third, and all further arrivals of odd order. The remaining arrivals with even orders all have positive velocities and are included in $j^{+}(x,t)$. If the third and all other higher odd arrivals can be neglected, $j^{-}(x,t)$ may be used as an approximation for first-passage times (method P₁). This is the case only for strong mean drift velocities or strong temporal correlations of the varying advection velocity. This approximation is limited in its applicability and is shown in the Results section below for comparison only. Again, the expressions above are valid for various probability densities $h\left(u\right)$ of velocities u. For the Gaussian distribution, Section 3 gives explicit expressions for $h\left(u\right)$ in 1, $f(x,t|u)$ in 4, and for $j(x,t)$, $j^{+}(x,t)$ and $j^{-}(x,t)$ in 8.

2.3. First-Passage Probability Density

2.3.1. Rough Approximation

For the arrangement of releasing the tracers above the boundary, $q(x,t)$ counts the first and all higher odd orders of arrivals as negative and all even orders of arrivals as positive; $-q(x,t)$ inverts the counting directions; $j(x,t)$ counts the first and all higher orders together; and $j^{-}(x,t)$ counts the first and all higher odd orders of arrivals. Therefore, all of these quantities deviate from the distribution of first arrivals because of the included contributions from higher-order arrivals. In Figure 1a, the quantities $-q(x,t)$, $j(x,t)$, and $j^{-}(x,t)$ are shown in comparison with the first-passage distribution obtained from a numerical simulation of a stochastic process with an artificial correlation function (see Section 4.4), chosen such that repeated returns of tracers to the boundary location are possible. In the diagram, intervals can be identified with changing contributions of different orders of arrivals, which can be identified in the drawn example traces in Figure 1b.

• Below a certain time (interval I), no arrivals occur because the tracers need a certain time to reach the boundary.
• Then (interval II) tracers arrive at the boundary for the first time. Because of the correlation of the velocity, which corresponds to a certain inertia of the tracers, no immediate return is expected, and higher-order arrivals do not occur. Up to this time of flight, there is no difference between the various quantities shown; they all follow the distribution of first arrivals from the simulated process.
• Then (interval III) tracers return to the boundary with positive velocities. Here, the current $-q(x,t)$ and the arrival rate $j(x,t)$ start to deviate because of their opposite sign when counting arrivals with positive velocities. If only negative velocities are considered with $j^{-}(x,t)$, the second arrivals are not counted, and this flux still follows the distribution of first arrivals from the simulated process.
• Then (interval IV), the third arrivals occur. Unfortunately, they contribute to $j^{-}(x,t)$ because of their negative velocities, resulting in a deviation of this flux from the first-arrival distribution.
• Later (interval V), the fifth and all higher odd orders of arrivals occur. These also contribute to $j^{-}(x,t)$ and cause a further deviation of this flux from the first-arrival distribution.

With long-lasting correlations or a strong mean velocity, the probability of returning tracers may be low enough, or delayed long enough, to allow the prediction of the flux $j^{-}(x,t)$ to be used as an approximation of the first-arrival distribution for small times of flight (method P₁). However, if third- and higher-order odd arrivals must be considered, this prediction requires further corrections.

The following numerical solution primarily makes use of the arrival rate, allowing only negative velocities, counting first arrivals together with all higher-order odd arrivals. Then, the fraction of this arrival rate is calculated from the returns of tracers that have been at the boundary previously. After removing these fractions of higher-order arrivals, only the first arrivals remain in the predicted rate.

2.3.2. Numerical Solution

To distinguish between first and higher-order odd arrivals within the actual arrival rate, the possibility of tracer returns must be considered, which strongly depends on the velocity over time. For that, the actual arrival rate–velocity density $j^{-}(x,t,u)$ at the boundary at distance x from the origin of the motion is split into first arrivals $p(x,t,u)$ at time t with arrival velocity u at the boundary, and those in which the tracer has returned to the boundary at time t with velocity u from a previous first arrival at the boundary at time $t^{\prime }<t$ and with first-arrival velocity $u^{\prime }$, according to an appropriate first-arrival rate–velocity density $p(x^{\prime },t^{\prime },u^{\prime })$. For clarity in notation, velocity and position at a particular time t are denoted without the prime symbol, as u and x, whereas both are denoted with the prime symbol at a different time $t^{\prime }$, namely $u^{\prime }$ and $x^{\prime }$. In the given case, where the tracer returns to the same distance x from the origin, $x^{\prime }$ is identical to x. Figure 2 shows the composition of $j^{-}(x,t,u)$ from first arrivals and higher-order arrivals with a previous first arrival at the boundary.

By integrating over all possible times $t^{\prime }$ and all possible velocities $u^{\prime }$ at the first arrival before t, all portions of $j^{-}(x,t,u)$ originating from tracers that have been at the boundary previously are identified. For that, the correlation of the velocities must be considered, because $u^{\prime }$ and u are correlated as a function of the time lapse between $t^{\prime }$ and t, and the location of the tracer at the two time instances strongly depends on the velocities and their correlation. With $j^{-}(x,t,u|x^{\prime },t^{\prime },u^{\prime })$ being the arrival rate–velocity density at the boundary at distance x from the origin at time t with arrival velocity u, conditioned on having a previous observation of the tracer at distance $x^{\prime }$ from the origin at time $t^{\prime }<t$ with arrival velocity $u^{\prime }$, the actual rate of tracer returns with negative velocities is given by

$$\underset{0}{\overset{t-}{\int }}\underset{-\infty }{\overset{0}{\int }}{j}^{-}(x,t,u|x^{\prime },t^{\prime },u^{\prime })p(x^{\prime },t^{\prime },u^{\prime })\mathrm{d}{u}^{\prime }\mathrm{d}{t}^{\prime }.$$

(10)

Together with the first arrivals at time t and velocity u, this yields the actual total arrival rate–velocity density, consisting of the first arrivals together with all odd-order arrivals for a specific arrival velocity u.

$$\begin{array}{c}\hfill j^{-}(x,t,u)=p(x,t,u)+\underset{0}{\overset{t-}{\int }}\underset{-\infty }{\overset{0}{\int }}{j}^{-}(x,t,u|x^{\prime },t^{\prime },u^{\prime })p(x^{\prime },t^{\prime },u^{\prime })\mathrm{d}{u}^{\prime }\mathrm{d}{t}^{\prime }\end{array}$$

(11)

For confirmation of the correctness of the units—$p(x^{\prime },t^{\prime },u^{\prime })$ is a first-arrival rate–velocity density with units of $1{\mathrm{m}}^{-1}$. Integrated over all velocities and over a time t, this becomes unitless, because the first-arrival rate–velocity density p integrated over all possible arrival velocities and over all possible first-arrival times (before and beyond t) must ultimately be unity. The remaining quantities, namely $j^{-}(x,t,u)$, $p(x,t,u)$, and $j^{-}(x,t,u|x^{\prime },t^{\prime },u^{\prime })$, all are arrival rate–velocity densities with the unit $1{\mathrm{m}}^{-1}$.

The actual first-passage rate–velocity density can just be obtained from $j^{-}(x,t,u)$ by subtracting the returning tracers as

$$p(x,t,u)=j^{-}(x,t,u)-\underset{0}{\overset{t-}{\int }}\underset{-\infty }{\overset{0}{\int }}{j}^{-}(x,t,u|x^{\prime },t^{\prime },u^{\prime })p(x^{\prime },t^{\prime },u^{\prime })\mathrm{d}{u}^{\prime }\mathrm{d}{t}^{\prime },$$

(12)

where, obviously, p is implicitly reused in the integral. Note that the distances x and $x^{\prime }$ are arguments of the function p; however, p is a joint probability density for time and velocity only. If x and $x^{\prime }$ are chosen to be identical, as for the return to the boundary, then $p(x,t,u)$ and $p(x^{\prime },t^{\prime },u^{\prime })$ are the same function evaluated at different arguments—t and u at the actual arrival, and $t^{\prime }$ and $u^{\prime }$ at the previous (first) arrival, respectively. In this way, it is possible to derive the actual first-arrival rate–velocity density $p(x,t,u)$ numerically by integration or summation over all values of p obtained up to times $t^{\prime }$ shortly before t. The integration or summation for numerical integration over $t^{\prime }$ must stop before t, because $j^{-}(x,t,u|x,t,u)$ becomes infinite and describes the actual tracers passing the boundary instead of the tracers that are returned.

Note the inherent dependence of the first-arrival rates on both velocities, u and $u^{\prime }$. Therefore, arrival rate–velocity densities have been kept so far without integrating over one or the other velocity. Accordingly, the numerical integration or summation requires the stepwise calculation of the two-dimensional joint probability density along the time axis, resolved for various arrival velocities. The integration over the arrival velocity u is performed last to obtain the final first-passage time density as a function of time in Equation (16).

Including the fact that all arrival rates $j^{-}$ depend on the appropriate spatial number density f and the actual velocity magnitude $\left|u\right|$, this further leads to

$$p(x,t,u)=\left|u\right|f(x,t,u)-\underset{0}{\overset{t-}{\int }}\underset{-\infty }{\overset{0}{\int }}\left|u\right|f(x,t,u|x^{\prime },t^{\prime },u^{\prime })p(x^{\prime },t^{\prime },u^{\prime })\mathrm{d}{u}^{\prime }\mathrm{d}{t}^{\prime }$$

(13)

The spatial number density f conditioned on the previous arrival needs to be reformulated, considering the dependence on u and $u^{\prime }$ as

$$f(x,t,u|x^{\prime },t^{\prime },u^{\prime })=f(x,t|u,x^{\prime },t^{\prime },u^{\prime })h(u,t-t^{\prime }|u^{\prime }),$$

(14)

where $f(x,t,u|x^{\prime },t^{\prime },u^{\prime })$ is the joint spatial number–velocity density at distance x from the origin at time t with arrival velocity u, conditioned on the previous arrival at the boundary at time $t^{\prime }$ with arrival velocity $u^{\prime }$, as used before, whereas $f(x,t|u,x^{\prime },t^{\prime },u^{\prime })$ is the spatial density at distance x from the origin at time t, conditioned on the previous arrival at the boundary at time $t^{\prime }$ with arrival velocity $u^{\prime }$ and on the actual velocity u. The probability density of the actual velocity u at time t is $h(u,t-t^{\prime }|u^{\prime })$, conditioned on having previously had velocity $u^{\prime }$ at time $t^{\prime }$. Since the process is assumed to be stationary and time-invariant, $h(u,t-t^{\prime }|u^{\prime })$ depends not on a specific $t^{\prime }$ but on the time span $\tau =t-t^{\prime }$ between the two arrival instances at $t^{\prime }$ and t only. Therefore, the formal condition of $h(u,t|u^{\prime },t^{\prime })$ on $t^{\prime }$ has been dropped. Since the process is assumed to be linear, $h(u,t-t^{\prime }|u^{\prime })$ is closely related to the correlation function of the velocity. For Gaussian distributed velocities, Section 3 gives an explicit expression for $h(u,t-t^{\prime }|u^{\prime })$ in 2.

On the other hand, the spatial number density $f(x,t|u,x^{\prime },t^{\prime },u^{\prime })$ conditioned on the previous arrival at $t^{\prime }$ with $u^{\prime }$ and the actual velocity u, can be obtained from the joint probability density $f(x,t,x^{\prime },t^{\prime }|u,u^{\prime })$ of having reached distance $x^{\prime }$ at $t^{\prime }$ and x at t, conditioned on velocity $u^{\prime }$ at $t^{\prime }$ and velocity u at t, together with the spatial number density of having reached $x^{\prime }$ at $t^{\prime }$ under the condition of velocity $u^{\prime }$ at $t^{\prime }$ and velocity u at t, independent of x.

$$f(x,t|u,x^{\prime },t^{\prime },u^{\prime })={\displaystyle \frac{f(x,t,x^{\prime },t^{\prime }|u,u^{\prime })}{f(x^{\prime },t^{\prime },t|u^{\prime },u)}}$$

(15)

Note that in the specifications for the Gaussian distribution in Section 3, for the denominator, the quantities with and without prime symbols are exchanged as $f(x,t,t^{\prime }|u,u^{\prime })$ in 5, however, with exactly the same function. The numerator is given in 6.

Finally, by integrating over all possible arrival velocities u from its distribution $h\left(u\right)$, one obtains the first-arrival rate (method P₂):

$$p(x,t)=\underset{-\infty }{\overset{0}{\int }}p(x,t,u)\mathrm{d}u.$$

(16)

Because of the negative drift velocity, all tracers will eventually arrive at the boundary. Therefore, the predicted rate of first arrivals per tracer is identical to the probability density of the flight times until the first arrivals.

3. Specifications for Gaussian Distribution of Increments

For the numerical solution of the integrals above, probability densities of the increments (velocities) and of the distances traveled within time t are needed. To derive the portions of trajectories that return to the boundary, in addition to the unconditioned probability densities, the probability densities of observing velocity u at time t conditioned on one observation of a specific velocity $u^{\prime }$ at time $t^{\prime }$, and those conditioned on two observations, $u^{\prime }$ at $t^{\prime }$ and $u^{″}$ at time $t^{″}$, are required. If a Gaussian distribution of increments is assumed (velocity observed by a fluid element along its path), the following ingredients of the above formulas can be specified:

• $h\left(u\right)$ is the (unconditioned) probability density of a tracer with velocity u. Then, $h\left(u\right)\mathrm{d}u$ is the probability that a fluid element observes a velocity within the range $u:u+\mathrm{d}u$. Assuming a Gaussian distribution of velocities, this becomes

  $$h\left(u\right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_u^2}}}{\mathrm{e}}^{-{\displaystyle \frac{{(u-{\mu }_u)}^2}{2{\sigma }_u^2}}}$$

  (17)

  with a given mean drift velocity ${\mu }_u$ and velocity variance ${\sigma }_u^2$.
• $h(u,\tau |u^{\prime })$ is the probability density of a tracer having velocity u at a time $\tau$ after a previous observation, conditioned on having had velocity $u^{\prime }$ at that observation. Then, $h(u,\tau |u^{\prime })\mathrm{d}u$ is the probability that a fluid element has a velocity within the range $u:u+\mathrm{d}u$ under the condition of having had velocity $u^{\prime }$ at the previous observation $\tau$ ago. Since the process is assumed to be stationary, velocity statistics are time-reversible; hence, $h(u,\tau |u^{\prime })=h(u,-\tau |u^{\prime })$. Assuming a Gaussian distribution of velocities, this is

  $$h(u,\tau |u^{\prime })={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{u|u^{\prime }}^2\left(\tau \right)}}}{\mathrm{e}}^{-{\displaystyle \frac{{\left[u-{\mu }_{u|u^{\prime }}\left(\tau \right)\right]}^2}{2{\sigma }_{u|u^{\prime }}^2\left(\tau \right)}}}$$

  (18)

  with

  $${\mu }_{u|u^{\prime }}\left(\tau \right)={\mu }_u+(u^{\prime }-{\mu }_u)\rho \left(\tau \right)$$

  (19)

  and

  $${\sigma }_{u|u^{\prime }}^2\left(\tau \right)={\sigma }_u^2\left[1-{\rho }^2\left(\tau \right)\right]$$

  (20)
• $f(x,t)$ is the (unconditioned) probability density that a tracer has traveled the distance x within time t. Then, $f(x,t)\mathrm{d}x$ is the probability that a fluid element is within the range $x:x+\mathrm{d}x$ at a given time t. Assuming a Gaussian distribution of velocities, this is

  $$f(x,t)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_x^2\left(t\right)}}}{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_x\left(t\right)]}^2}{2{\sigma }_x^2\left(t\right)}}}$$

  (21)

  with

  $${\mu }_x\left(t\right)={\mu }_{u}t$$

  (22)

  and following [5]

  $${\sigma }_x^2\left(t\right)=2{\sigma }_u^2\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\tau }{\int }}\rho \left(\xi \right)\mathrm{d}\xi \mathrm{d}\tau$$

  (23)

  which is, following [6]

  $${\sigma }_x^2\left(t\right)=2{\sigma }_u^2{R}_2\left(t\right)$$

  (24)

  with

  $$R_2\left(t\right)=\underset{0}{\overset{t}{\int }}(t-\tau )\rho \left(\tau \right)\mathrm{d}\tau$$

  (25)

  where, again, ${\mu }_u$ is the given mean drift velocity and ${\sigma }_u^2$ is the given variance of the velocity.
• $f(x,t|u)$ is the probability density that a tracer has traveled the distance x within time t, conditioned on the arrival velocity u at the boundary at distance x to the origin of motion at time t. Then, $f(x,t|u)\mathrm{d}x$ is the probability that a fluid element is within the range $x:x+\mathrm{d}x$ at a given time t under the condition of having the arrival velocity u at the boundary at distance x at time t. Assuming a Gaussian distribution of velocities, this is given by (derivation in Appendix A.2)

  $$f(x,t|u)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{x|u}^2\left(t\right)}}}{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{x|u}\left(t\right)]}^2}{2{\sigma }_{x|u}^2\left(t\right)}}}$$

  (26)

  with

  $${\mu }_{x|u}\left(t\right)={\mu }_{u}t+(u-{\mu }_u)R_1\left(t\right)$$

  (27)

  and

  $${\sigma }_{x|u}^2\left(t\right)={\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]$$

  (28)

  with

  $$R_1\left(t\right)=\underset{0}{\overset{t}{\int }}\rho \left(\tau \right)\mathrm{d}\tau$$

  (29)

  and $R_2\left(t\right)$, as above, where, again, ${\mu }_u$ is the given mean drift velocity and ${\sigma }_u^2$ is the given variance of the velocity.
  Both integrals $R_1$ and $R_2$ are characteristic of the diffusive behavior of the process. Therefore, these two integrals are shown for the various test processes in the following test section, together with the corresponding correlation functions.
• $f(x,t,t^{\prime }|u,u^{\prime })$ is the probability density that a tracer has traveled distance x within time t conditioned on having velocity u at the boundary at distance x at time t and velocity $u^{\prime }$ at a previous (or later, as in the case of the derivations of the numerical solution of first arrivals above) observation at time $t^{\prime }$. Then, $f(x,t,t^{\prime }|u,u^{\prime })\mathrm{d}x$ is the probability that a fluid element is within the range $x:x+\mathrm{d}x$ at a given time t under the condition of having the arrival velocity u and having velocity $u^{\prime }$ at time $t^{\prime }$. Assuming a Gaussian distribution of velocities, this is given by (derivation in Appendix A.4)

  $$f(x,t,t^{\prime }|u,u^{\prime })={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{x|u,u^{\prime }}^2\left(t\right)}}}{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{x|u,u^{\prime }}\left(t\right)]}^2}{2{\sigma }_{x|u,u^{\prime }}^2\left(t\right)}}}$$

  (30)

  with

  $$\begin{array}{cc}\hfill & {\displaystyle {\mu }_{x|u,u^{\prime }}\left(t\right)={\mu }_{u}t+\frac{R_1\left(t\right)-\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}(u-{\mu }_u)}\\ \hfill & {\displaystyle +\frac{R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)-R_1\left(t\right)\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}(u^{\prime }-{\mu }_u)}\end{array}$$

  (31)

  and

  $$\begin{array}{cc}\hfill & {\displaystyle {\sigma }_{x|u,u^{\prime }}^2\left(t\right)={\sigma }_u^2\left\{2R_2\left(t\right)-\frac{{\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]}^2+R_1^2\left(t\right)}{1-{\rho }^2(t-t^{\prime })}\right.}\\ \hfill & {\displaystyle \left.+2\frac{\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]R_1\left(t\right)\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}\right\}.}\end{array}$$

  (32)
  Note that both ${\mu }_{x|u,u^{\prime }}\left(t\right)$ and ${\mu }_{x|u,u^{\prime }}\left(t\right)$ also depend on $t^{\prime }$. The functional dependence is dropped in the notations because of the simplification of the notation in the following derivations. Particular specifications are given in each of the following cases.
• $f(x,t,x^{\prime },t^{\prime }|u,u^{\prime })$ is the joint probability density that a tracer has traveled the distance x within time t and distance $x^{\prime }$ at time $t^{\prime }$, conditioned on the arrival velocity u at distance x at time t and arrival velocity $u^{\prime }$ at distance $x^{\prime }$ at time $t^{\prime }$. Then, $f(x,t,x^{\prime },t^{\prime }|u,u^{\prime }){\left(\mathrm{d}x\right)}^2$ is the probability that a fluid element is within the range $x:x+\mathrm{d}x$ at a given time t and within the range $x^{\prime }:x^{\prime }+\mathrm{d}x$ at time $t^{\prime }$ under the condition of having the arrival velocity u at time t and arrival velocity $u^{\prime }$ at time $t^{\prime }$. Assuming a Gaussian distribution of velocities, this is given by (derivation in Appendix A.5)

  $$\begin{array}{cc}& {\displaystyle f(x,t,x^{\prime },t^{\prime }|u,u^{\prime })=}\\ & {\displaystyle \frac{1}{2\pi \sqrt{{\sigma }_{x|u,u^{\prime }}^2\left(t\right){\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)\left[1-r_{x{x}^{\prime }|u,u^{\prime }}^2(t,t^{\prime })\right]}}{\mathrm{e}}^{-\frac{z(t,t^{\prime })}{2\left[1-r_{x{x}^{\prime }|u,u^{\prime }}^2(t,t^{\prime })\right]}}}\end{array}$$

  (33)

  with

  $$\begin{array}{cc}& {\displaystyle z(t,t^{\prime })=\frac{{\left[x-{\mu }_{x|u,u^{\prime }}\left(t\right)\right]}^2}{{\sigma }_{x|u,u^{\prime }}^2\left(t\right)}+\frac{{\left[x^{\prime }-{\mu }_{x^{\prime }|u,u^{\prime }}\left(t^{\prime }\right)\right]}^2}{{\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)}}\\ & {\displaystyle -2\frac{\left[x-{\mu }_{x|u,u^{\prime }}\left(t\right)\right]\left[x^{\prime }-{\mu }_{x^{\prime }|u,u^{\prime }}\left(t^{\prime }\right)\right]}{\sqrt{{\sigma }_{x|u,u^{\prime }}^2\left(t\right){\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)}}{r}_{x{x}^{\prime }|u,u^{\prime }}(t,t^{\prime })}\end{array}$$

  (34)

  and with

  $$\begin{array}{cc}\hfill & {\displaystyle {\mu }_{x|u,u^{\prime }}\left(t\right)={\mu }_{u}t+A(u^{\prime }-{\mu }_u)+B(u-{\mu }_u)}\end{array}$$

  (35)

  $$\begin{array}{cc}\hfill & {\displaystyle {\mu }_{x^{\prime }|u,u^{\prime }}\left(t^{\prime }\right)={\mu }_u{t}^{\prime }+A^{\prime }(u^{\prime }-{\mu }_u)+B^{\prime }(u-{\mu }_u)}\end{array}$$

  (36)

  with

  $$\begin{array}{ccc}\hfill A& =& {\displaystyle \frac{R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)-R_1\left(t\right)\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}}\hfill \end{array}$$

  (37)

  $$\begin{array}{ccc}\hfill B& =& {\displaystyle \frac{R_1\left(t\right)-\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}}\hfill \end{array}$$

  (38)

  $$\begin{array}{ccc}\hfill A^{\prime }& =& {\displaystyle \frac{R_1\left(t^{\prime }\right)-\left[R_1\left(t\right)-R_1(t-t^{\prime })\right]\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}}\hfill \end{array}$$

  (39)

  $$\begin{array}{ccc}\hfill B^{\prime }& =& {\displaystyle \frac{R_1\left(t\right)-R_1(t-t^{\prime })-R_1\left(t^{\prime }\right)\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}}\hfill \end{array}$$

  (40)

  and

  $$\begin{array}{cc}\hfill & {\displaystyle {\sigma }_{x|u,u^{\prime }}^2\left(t\right)={\sigma }_u^2\left\{2R_2\left(t\right)-\frac{{\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]}^2+R_1^2\left(t\right)}{1-{\rho }^2(t-t^{\prime })}\right.}\\ \hfill & {\displaystyle \left.+2\frac{\left[R_1(t-t^{\prime })+R_1\left(t^{\prime }\right)\right]R_1\left(t\right)\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}\right\}.}\end{array}$$

  (41)

  $$\begin{array}{cc}\hfill & {\displaystyle {\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)={\sigma }_u^2\left\{2R_2\left(t^{\prime }\right)-\frac{R_1^2\left(t^{\prime }\right)+{\left[R_1\left(t\right)-R_1(t-t^{\prime })\right]}^2}{1-{\rho }^2(t-t^{\prime })}\right.}\\ \hfill & {\displaystyle \left.+2\frac{R_1\left(t^{\prime }\right)\left[R_1\left(t\right)-R_1(t-t^{\prime })\right]\rho (t-t^{\prime })}{1-{\rho }^2(t-t^{\prime })}\right\}.}\end{array}$$

  (42)

  and

  $$\begin{array}{cc}& {\displaystyle r_{x{x}^{\prime }|u,u^{\prime }}(t,t^{\prime })=\frac{{\sigma }_{x|u,u^{\prime }}^2\left(t\right)+{\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)-{\sigma }_u^2\left[2R_2(t-t^{\prime })-C\right]}{2\sqrt{{\sigma }_{x|u,u^{\prime }}^2\left(t\right){\sigma }_{x^{\prime }|u,u^{\prime }}^2\left(t^{\prime }\right)}}}\end{array}$$

  (43)

  with

  $$\begin{array}{cc}& {\displaystyle C=\frac{2R_1^2(t-t^{\prime })}{1+\rho (t-t^{\prime })}}\end{array}$$

  (44)
• With $q(x,t|u)=uf(x,t\left|u\right)$ and $h\left(u\right)$ being Gaussian, one obtains

  $$q(x,t)=\underset{-\infty }{\overset{\infty }{\int }}q(x,t|u)h\left(u\right)\mathrm{d}u=\underset{-\infty }{\overset{\infty }{\int }}uf(x,t|u)h\left(u\right)\mathrm{d}u$$

  (45)
  Using $h\left(u\right)$ and $f(x,t|u)$ from Equations (17) and (26) above, including Equations (27) and (28), one obtains

  $$\begin{array}{ccc}\hfill q(x,t)& =& \underset{-\infty }{\overset{\infty }{\int }}u{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}}{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{u}t-(u-{\mu }_u)R_1\left(t\right)]}^2}{2{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}}\hfill \\ & & \times {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_u^2}}}{\mathrm{e}}^{-{\displaystyle \frac{{(u-{\mu }_u)}^2}{2{\sigma }_u^2}}}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\hfill \\ & & \times \underset{-\infty }{\overset{\infty }{\int }}u{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{u}t-(u-{\mu }_u)R_1\left(t\right)]}^2}{2{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}-{\displaystyle \frac{{(u-{\mu }_u)}^2}{2{\sigma }_u^2}}}\mathrm{d}u\hfill \end{array}$$

  (46)
  Using the substitutions

  $$\begin{array}{ccc}\hfill c_0& =& {\displaystyle \frac{\frac{1}{2}{(x-{\mu }_{u}t)}^2+(x-{\mu }_{u}t){\mu }_u{R}_1\left(t\right)+{\mu }_u^2{R}_2\left(t\right)}{{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}\hfill \end{array}$$

  (47)

  $$\begin{array}{ccc}\hfill c_1& =& {\displaystyle \frac{({\mu }_{u}t-x)R_1\left(t\right)-2{\mu }_u{R}_2\left(t\right)}{{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}\hfill \end{array}$$

  (48)

  $$\begin{array}{ccc}\hfill c_2& =& {\displaystyle \frac{R_2\left(t\right)}{{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}\hfill \end{array}$$

  (49)

  one obtains the following (method Q₃):

  $$\begin{array}{ccc}\hfill q(x,t)& =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\underset{-\infty }{\overset{\infty }{\int }}u{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\left[-{\displaystyle \frac{c_1\sqrt{\pi }}{2c_2\sqrt{c_2}}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\right]\hfill \\ & =& -{\displaystyle \frac{c_1}{4c_2\sqrt{\pi {\sigma }_u^2{R}_2\left(t\right)}}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\hfill \end{array}$$

  (50)
• With $j(x,t|u)=|u\left|f\right(x,t\left|u\right)$ and $h\left(u\right)$ being Gaussian, one obtains

  $$j(x,t)=\underset{-\infty }{\overset{\infty }{\int }}j(x,t|u)h\left(u\right)\mathrm{d}u=\underset{-\infty }{\overset{\infty }{\int }}\left|u\right|f(x,t|u)h\left(u\right)\mathrm{d}u$$

  (51)
  Using $h\left(u\right)$ and $f(x,t|u)$ from Equations (17) and (26) above, including Equations (27) and (28), one obtains

  $$\begin{array}{ccc}\hfill j(x,t)& =& \underset{-\infty }{\overset{\infty }{\int }}\left|u\right|{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}}{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{u}t-(u-{\mu }_u)R_1\left(t\right)]}^2}{2{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}}\hfill \\ & & \times {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_u^2}}}{\mathrm{e}}^{-{\displaystyle \frac{{(u-{\mu }_u)}^2}{2{\sigma }_u^2}}}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\hfill \\ & & \times \underset{-\infty }{\overset{\infty }{\int }}\left|u\right|{\mathrm{e}}^{-{\displaystyle \frac{{[x-{\mu }_{u}t-(u-{\mu }_u)R_1\left(t\right)]}^2}{2{\sigma }_u^2\left[2R_2\left(t\right)-R_1^2\left(t\right)\right]}}-{\displaystyle \frac{{(u-{\mu }_u)}^2}{2{\sigma }_u^2}}}\mathrm{d}u\hfill \end{array}$$

  (52)
  Using the substitutions $c_0$, $c_1$, and $c_2$ as above, one obtains the following (method J₂):

  $$\begin{array}{ccc}\hfill j(x,t)& =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\underset{-\infty }{\overset{\infty }{\int }}\left|u\right|{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\left[\underset{0}{\overset{\infty }{\int }}u{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\right.\hfill \\ & & \left.-\underset{-\infty }{\overset{0}{\int }}u{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\right]\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{c_2}}\right.\hfill \\ & & \left.+{\displaystyle \frac{c_1\sqrt{\pi }}{2c_2\sqrt{c_2}}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erf}\left({\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{2\sqrt{\pi {\sigma }_u^2{R}_2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{\sqrt{\pi c_2}}}+{\displaystyle \frac{c_1}{2c_2}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erf}\left({\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)\right]\hfill \end{array}$$

  (53)
  By limiting the integration range to positive velocities only, one obtains

  $$\begin{array}{ccc}\hfill j^{+}(x,t)& =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\underset{0}{\overset{\infty }{\int }}\left|u\right|{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{2c_2}}\right.\hfill \\ & & \left.-{\displaystyle \frac{c_1\sqrt{\pi }}{4c_2\sqrt{c_2}}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erfc}\left({\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{4\sqrt{\pi {\sigma }_u^2{R}_2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{\sqrt{\pi c_2}}}-{\displaystyle \frac{c_1}{2c_2}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erfc}\left({\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)\right]\hfill \end{array}$$

  (54)
  By limiting the integration range to negative velocities only, one obtains the following (method P₁):

  $$\begin{array}{ccc}\hfill j^{-}(x,t)& =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\underset{-\infty }{\overset{0}{\int }}\left|u\right|{\mathrm{e}}^{-c_0-c_{1}u-c_2{u}^2}\mathrm{d}u\hfill \\ & =& {\displaystyle \frac{1}{2\pi {\sigma }_u^2\sqrt{2R_2\left(t\right)-R_1^2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{2c_2}}\right.\hfill \\ & & +{\displaystyle \frac{c_1\sqrt{\pi }}{4c_2\sqrt{c_2}}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erfc}\left(-{\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)]\hfill \\ & =& {\displaystyle \frac{1}{4\sqrt{\pi {\sigma }_u^2{R}_2\left(t\right)}}}\left[{\displaystyle \frac{{\mathrm{e}}^{-c_0}}{\sqrt{\pi c_2}}}+{\displaystyle \frac{c_1}{2c_2}}{\mathrm{e}}^{{\displaystyle \frac{c_1^2}{4c_2}}-c_0}\mathrm{erfc}\left(-{\displaystyle \frac{c_1}{2\sqrt{c_2}}}\right)\right]\hfill \end{array}$$

  (55)

4. Test Cases

The following calculations (numerical integrations for the predictions and simulations) were carried out with a time step of $\Delta t=1$. The numerical integration for the first-passage prediction further requires integration over velocities, including the storage of the time- and velocity-dependent probability densities $p(x,t,u)$. For this, the velocity was resolved in steps of $0.02$ within plus/minus six standard deviations. The results shown in the diagrams were obtained using a notebook computer within a few hours. The programs used are provided on request to the corresponding author.

4.1. Telegrapher’s Process

The first test case is a random walk process with discrete time and continuous, random, and independent increments. This process is also known as the Telegrapher’s process. The time step is $\Delta t=1$, the increments are Gaussian distributed with a mean value of ${\mu }_u=0.4$ and a variance of ${\sigma }_u^2=0.36$. This yields a mean value of the position reached by summation of the steps within time t of ${\mu }_x={\mu }_{u}t$ and a variance of ${\sigma }_x^2={\sigma }_u^{2}t$. Figure 3a shows the correlation function of this process, which has a rectangular shape. It is one between $-0.5$ and $+0.5$, and zero outside this interval, since the velocity of motion is held constant over each time step $\Delta t$ and consecutive steps are uncorrelated. The two appropriate integrals $R_1\left(t\right)$ and $R_2\left(t\right)$ are shown in Figure 3b,c.

The simulated tracers are released at $x=20$ and the boundary assumed is located at $x=0$. Since the random walk, with a standard deviation of $\sqrt{{\sigma }_u^2}=0.6$ compared to a mean velocity of ${\mu }_u=0.4$, yields significant backflow, the currents $q(x,t)$ through the boundary and the total arrival rate of tracers at the boundary $j(x,t)$ differ in their absolute amplitudes (Figure 3d,e) due to different sign conventions for counting returning flows through the boundary toward the release origin of the tracers. The methods for predicting the current and the total arrival rate are Q₁, as given by Equation (2), Q₂, as given by Equation (5), and J₁, as given by Equation (7), which are numerically resolved integrals, as well as Q₃, as given by Equation (50), and J₂, as given by Equation (53), which are explicit solutions for the Gaussian distribution. All methods agree well with the results from a Monte Carlo simulation of the process, with ${10}^7$ numerical realizations of the process.

Due to the frequent steps in the positive direction, tracers may return to the boundary, and third- and higher-order odd crossings of the boundary become significant. Therefore, the approximation of the first-arrival rates by method P₁, as given in Equation (55), which counts fluxes with negative velocities only, is not appropriate. A deviation of this prediction method from the first arrivals obtained from the simulation can be seen in Figure 3f. Method P₂, as introduced in Equation (16), yields suitable results.

Assuming $\Delta t\to 0$, while the diffusion coefficient $D=\frac{1}{2}\Delta t{\sigma }_u^2$ remains finite leads to Wald’s distribution

$$p(x,t)={\displaystyle \frac{x}{\sqrt{2\pi \Delta t{\sigma }_u^2{t}^3}}}{\mathrm{e}}^{{\displaystyle \frac{-{(x-{\mu }_{u}t)}^2}{2\Delta t{\sigma }_u^{2}t}}}={\displaystyle \frac{x}{\sqrt{4\pi D{t}^3}}}{\mathrm{e}}^{{\displaystyle \frac{-{(x-{\mu }_{u}t)}^2}{4Dt}}}$$

(56)

for first arrivals (method P₃). Due to the finite time step for the simulated process, Wald’s distribution shows a small but noticeable systematic deviation from the values obtained from the simulated process in Figure 3f.

The maximum absolute deviations between the predictions and the simulations in Figure 3 are as follows: Q₁: $0.136\times {10}^{-3}$; Q₂: $0.136\times {10}^{-3}$; Q₃: $0.136\times {10}^{-3}$; J₁: $0.171\times {10}^{-3}$; J₂: $0.174\times {10}^{-3}$; P₁: $8.455\times {10}^{-3}$; P₂: $0.234\times {10}^{-3}$; P₃: $0.593\times {10}^{-3}$. The deviations between the various current predictions and those between the total arrival rates are negligible (<${10}^{-7}$ and <${10}^{-5}$, respectively). The deviation between P₂ and Wald’s distribution (P₃) is $0.748\times {10}^{-3}$, whereas the deviations between P₂ and the simulation are smaller than those between P₃ and the simulation.

4.2. Exponential Decay

The second test case is an integrated Ornstein–Uhlenbeck process with Gaussian distributed increments (velocities) with a mean value of ${\mu }_u=0.4$ and a variance of ${\sigma }_u^2=0.36$. The simulated tracers are released at $x=20$ and the boundary assumed is located at $x=0$. The correlation function for the Ornstein–Uhlenbeck process has an exponential decay (Figure 4a):

$$\begin{array}{ccc}\hfill \rho \left(\tau \right)& =& {\mathrm{e}}^{-\left|\tau \right|/T}\hfill \end{array}$$

(57)

and yields the appropriate integrated functions $R_1\left(t\right)$ and $R_2\left(t\right)$ (Figure 4b,c), given by

$$\begin{array}{ccc}\hfill R_1\left(t\right)& =& T\mathrm{sgn}\left(t\right)\left(1-{\mathrm{e}}^{-\left|t\right|/T}\right)\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill R_2\left(t\right)& =& T\left[\left|t\right|-T\left(1-{\mathrm{e}}^{-\left|t\right|/T}\right)\right]\hfill \end{array}$$

(59)

The integral timescale for the chosen process is $T=R_1(\infty )=10$, which can be read directly from Figure 4b.

The methods for predicting the current and the total arrival rate are Q₁, as given by Equation (2), Q₂, as given by Equation (5), and J₁, as given by Equation (7), which are numerically resolved integrals, as well as Q₃, as given by Equation (50) and J₂, as given by Equation (53), which are explicit solutions for the Gaussian distribution. All methods agree well with the results from a Monte Carlo simulation of the process with ${10}^7$ numerical realizations of the process.

Due to possible positive velocities and the resulting returns of tracers to the boundary, third- and higher-order odd crossings of the boundary become significant. Therefore, the approximation of the first-arrival rates by method P₁, as given by Equation (55), counting fluxes with negative velocities only, is appropriate for small travel times only, where higher-order arrivals are still rare. A deviation of this prediction method from the first arrivals obtained from the simulation can be seen in Figure 3f, at longer passage times on the descending slope of the curve. Method P₂ as introduced here in Equation (16), yields suitable results, which agree well with the simulation.

An explicit solution for the first-arrival rate is not provided here. Such solutions exist only for special cases, either without drift or with a cooperative relation between the mean drift, the velocity fluctuation, and the position of the boundary [7,8,9,10,11,12]. However, the general case has no explicit solution so far.

The maximum absolute deviations between the predictions and the simulations in Figure 4 are as follows: Q₁: $0.088\times {10}^{-3}$; Q₂: $0.088\times {10}^{-3}$; Q₃: $0.088\times {10}^{-3}$; J₁: $0.075\times {10}^{-3}$; J₂: $0.076\times {10}^{-3}$; P₁: $1.435\times {10}^{-3}$; P₂: $0.078\times {10}^{-3}$. The deviations between the various current predictions and those between the total arrival rates are negligible (<${10}^{-6}$).

4.3. Strong Correlation

The third test case is a process in which a tracer starts at time $t_0=0$ at position $x_0=20$ with a random velocity that remains constant during the travel of this particular tracer. Since this ballistic behavior yields infinitely lasting correlations, this test case is called a “strong” correlation.

The individual tracers move with a randomly chosen velocity that remains constant during the traveling time. Therefore, the traveling time to reach the boundary at distance x is inversely proportional to the tracer’s velocity. The velocities are Gaussian distributed with a mean value of ${\mu }_u=0.4$ and a variance of ${\sigma }_u^2=0.36$. This yields a reciprocal Gaussian distributed traveling time of the tracers to reach the boundary at $x=0$. Since the velocities stay unchanged, only negative velocities will let the tracer reach the boundary without any later backflow. Therefore, the magnitudes of all three given quantities—the current $q(x,t)$, the total arrival rate $j(x,t)$, and the first-arrival rate $p(x,t)$—are all identical,

$$-q(x,t)\equiv j(x,t)\equiv p(x,t)={\displaystyle \frac{x}{t^2}}h\left({\displaystyle \frac{x}{t}}\right),$$

(60)

except for the minus sign of the current, with the probability density $h\left(u\right)$ of the velocities u. Note that the integral of the reciprocal Gaussian distribution may deviate from one, since positive velocities are possible, but they will not lead the tracer to reach the boundary. A common interpretation of this fact is to relocate all missing arrivals to infinity, thereby yielding a correct integral of one.

The simulation releases again ${10}^7$ tracers with the given Gaussian distribution of their velocities. Since some of the methods compared in Figure 5 cannot handle a constant correlation function, the process was additionally approximated with a very slowly decaying exponential correlation function (Ornstein–Uhlenbeck process) with an integral timescale of $T=R_1(\infty )=$ 10,000. The appropriate correlation functions $\rho \left(\tau \right)$ and the integrals $R_1\left(t\right)$ and $R_2\left(t\right)$ are shown in Figure 5a–c for both the process with constant traveling velocity as well as for the very slowly decaying process, proving that the differences between the two processes are marginal and that the other results are comparable.

Since there is no backflow, the currents, total arrival rates, and first arrivals in Figure 5d–f are all identical, except for the minus sign of the current. All previously introduced methods—Q₁, as given by Equation (2), Q₂, as given by Equation (5), Q₃, as given by Equation (50), J₁, as given by Equation (7), J₂, as given by Equation (53), P₁, as given by Equation (55), and P₂, as given by Equation (16)—yield identical results when based on the approximated process with slow exponential decay, whereas the empirical results are based on the simulation of the original process with constant velocity and constant correlation. However, no deviations are visible between the simulation and the prediction. Even the arrival rate with only negative velocities, as a rough approximation of the first-arrival rates, is valid for this test case, because no returns of the tracers to the boundary can occur. For comparison, the reciprocal Gaussian distribution as given by Equation (60) is shown in Figure 5f (method P₄), which is an exact solution for the process with the constant tracer velocity, with no visible deviations from the other methods based on the approximated process with a slow decay.

The maximum absolute deviations between the predictions and the simulations in Figure 5 are as follows: Q₁: $0.107\times {10}^{-3}$; Q₂: $0.107\times {10}^{-3}$; Q₃: $0.107\times {10}^{-3}$; J₁: $0.107\times {10}^{-3}$; J₂: $0.107\times {10}^{-3}$; P₁: $0.107\times {10}^{-3}$; P₂: $0.107\times {10}^{-3}$; P₄: $0.112\times {10}^{-3}$. The deviations between the various current predictions and those between the total arrival rates are negligible (<${10}^{-7}$). The deviation between P₂ and the reciprocal Gaussian distribution (P₄) as the ground proof for this process is $0.010\times {10}^{-3}$.

4.4. Artificial Correlation

In the last test case, a process is designed with an artificial correlation (see Figure 6a), causing relevant backflows. The integrals $R_1\left(t\right)$ and $R_2\left(t\right)$ are appropriate (Figure 6b,c). The fluctuating tracer velocity has a mean value of ${\mu }_u=0.4$ and a variance of ${\sigma }_u^2=0.36$. The simulated tracers are released at $x_0=20$, and the boundary is located at $x=0$.

The methods for predicting the current and the total arrival rate are Q₁, as given by Equation (2), Q₂, as given by Equation (5), and J₁, as given by Equation (7), which are numerically resolved integrals, as well as Q₃, as given by Equation (50) and J₂, as given by Equation (53), which are explicit solutions for the Gaussian distribution. Due to the complexity of the chosen correlation function, the shapes of the currents and total arrival rates are rich. However, all tested methods agree well with the results from a Monte Carlo simulation with ${10}^7$ numerical realizations of the process.

For this process, the approximation of the first-arrival rates by method P₁, as given by Equation (55), counting fluxes with negative velocities only, is suitable for small travel times only, where higher-order arrivals are still rare. However, due to the middle part of the correlation function with negative correlation and a certain time delay, substantial backflow occurs, with many tracer returns, and deviations of this approximation method from the first arrivals obtained from the simulation become evident in Figure 6f. Method P₂, as introduced here in Equation (16), yields suitable results for this test case as well.

The maximum absolute deviations between the predictions and the simulations in Figure 6 are as follows: Q₁: $0.101\times {10}^{-3}$; Q₂: $0.101\times {10}^{-3}$; Q₃: $0.101\times {10}^{-3}$; J₁: $0.084\times {10}^{-3}$; J₂: $0.084\times {10}^{-3}$; P₁: $2.461\times {10}^{-3}$; P₂: $0.152\times {10}^{-3}$. The deviations between the various current predictions and those between the total arrival rates are negligible ($<{10}^{-6}$ and $<{10}^{-5}$, respectively).

5. A Note on Anomalous Diffusivity

It has been shown that integrated linear stochastic processes, depending on the correlation function of their increments, can also lead to mean-square displacements that deviate from normal diffusion (see the function $R_2\left(t\right)$). With a long-lasting correlation function, e.g., a power-law dependence on time, a simple form of superdiffusive behavior can be temporarily obtained (see Figure 7), whereas short correlation functions yield normal diffusion after a brief initialization period. With an appropriately chosen correlation function with negative values after a short initialization phase, even a simple form of a subdiffusive behavior is possible. This holds only over a limited time range. In the long term, however, the correlations of all these processes vanish and, finally, normal diffusive behavior will be reached again. Furthermore, the enhanced or suppressed convection due to the correlation of step increments is one aspect of anomalous diffusivity, and it is not conclusive for other processes exhibiting anomalous diffusivity.

6. Conclusions

Currents (positive contributions from positive velocities and negative contributions from negative velocities), total arrival rates, and first-passage times of tracers through a boundary have been investigated. The distance traveled by the tracers from their release position follows a one-dimensional integrated stationary linear stochastic process with correlations and a mean drift. Except for stationarity, no restrictions are imposed on the correlation function or the spectral characteristics of the process, nor on the mean drift velocity, its variance, or the distance to the boundary.

General integral expressions are given for arbitrary velocity distributions. Currents and total arrival rates at a non-absorbing boundary are given as explicit solutions for Gaussian distributed velocities. The first-passage time probability density at an absorbing boundary is given as an implicit integral formulation, which is solvable numerically in an iterative procedure. No approximations are needed to derive the currents, total arrival rates, and first-passage probability densities. Therefore, the solutions have no systematic errors, except for (unfortunately, steadily growing) errors from imperfections of the numerical integration.

In four test cases with processes with different correlation functions, the introduced derivations have been tested successfully. Furthermore, with appropriate correlation functions, linear stochastic integrated processes can also mimic simple forms of anomalous diffusive behavior.