Diffusion–Advection Equations on a Comb: Resetting and Random Search

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.


Introduction
In the standard theory of Brownian motion, the probability density function (PDF) P 0 (x, t) for finding a particle at position x at time t has a Gaussian form, for the initial conditions given by the Dirac delta function P 0 (x, t = 0) = δ(x − x 0 ) and for the natural (vanishing) boundary conditions at infinity, P 0 (±∞, t) = 0 and ∂ ∂x P(x, t)| x=±∞ = 0. Brownian motion is characterized by linear growth of the mean squared displacement (MSD), x 2 (t) = 2Dt, which means normal diffusion. However, in a large variety of transport phenomena in randomly inhomogeneous media, one observes deviation from this linear growth in time such that the MSD has a power-law dependence on time, x 2 (t) ∼ t α , which is a signature of anomalous diffusion (see, for example, [1]). When 0 < α < 1, it corresponds to subdiffusion, while α > 1 corresponds to superdiffusion. One of the well-known examples of anomalous diffusion is Brownian motion on a comb, which is governed by the Fokker-Planck equation [2,3] ∂ t P(x, y, t) = δ(y)D x ∂ 2 x P(x, y, t) + D y ∂ 2 y P(x, y, t), with the initial condition P(x, y, t = 0) = δ(x − x 0 )δ(y), and the boundary conditions for P(x, y, t) and ∂ ∂q P(x, y, t), q = {x, y}, are set to zero at infinity, x = ±∞, y = ±∞. Here, D x δ(y) and D y are diffusion coefficients along the x and y directions, respectively. The δ-function in the Fokker-Planck operator means that diffusion along the x direction (the so-called backbone) is allowed only at y = 0. Along the y direction (the so-called fingers), the particle performs normal diffusion. The MSD along the backbone has a power-law dependence on time x 2 (t) = D x √ D y t 1/2 Γ(3/2) , i.e., subdiffusion is observed in the system [2,3], while normal diffusion takes place along the fingers. Different generalizations of the comb geometry have been considered. For example, various diffusion processes have been considered in a comb with a finite finger length [4][5][6], diffusion on cylindrical [7,8] and circular combs [9][10][11], more complex branched structures [12], random comb models [13], and a comb with ramified teeth [14], as well as the problem of first encounters for two workers [15]. Diffusion processes in fractal mesh and grid structures have been considered as well: In this case, anomalous diffusion of a particle is affected by the fractal structure of the infinite numbers of backbones and fingers [16]. It has been shown that these models are useful for description of anomalous transport through porous solid pellets with various porous geometries [17]. Comb models are also applicable for describing diffusion in percolation clusters [2,18,19], anomalous transport of inert compounds in spiny dendrites [20][21][22], modeling electron transport in disordered nanostructured semiconductors [23,24], dispersive transport of charge carriers in two-layer polymers [25], percolative phonon-assisted hopping in twodimensional disordered systems [26,27], and anomalous diffusion of fluorescence recovery after photobleaching in a random-comb model [13]. Another interesting realization is that turbulent diffusion in a comb appears to be due to multiplicative noise [28,29].
Nowadays, one of the most explored problems in stochastic processes is the problem of stochastic resetting, meaning that a particle is reset to the initial (or any other) position from time to time. The one-dimensional Brownian motion with Markovian resetting with a constant resetting rate r was introduced by Evans and Majumdar [30]. It was shown that the solution for the PDF approaches a non-equilibrium steady state and, in the long-time limit, its MSD is saturated, x 2 (t) ∼ 1/r (also see the review paper [31] for more details). Moreover, Brownian motion in a two-dimensional comb in the presence of stochastic (Markovian) resetting can be solved analytically [32][33][34]. The marginal PDFs along both the backbone and fingers approach non-equilibrium steady states, and the MSDs are saturated according to the resetting rate: x 2 (t) ∼ 1/ √ r and y 2 (t) ∼ 1/r [32][33][34]. These models have been extended to diffusion processes with non-static resetting [34]. Stochastic resetting is a natural mechanism in various search processes, such as foraging [35], population dynamics [36], Michaelis-Menten enzymatic reactions [37], and human behavior of finding resources [38], to mention but a few. Resetting may also affect the first-passage properties and completion of the process. In particular, in the case of the one-dimensional Brownian search, the mean arrival time at the absorbing boundary becomes finite in the presence of resetting [30], while it is infinite in the absence of resetting [39]. The resetting dynamics of a Brownian particle under external potentials have been analyzed in detail, as well [40][41][42][43][44][45]. This issue can also be employed to understand resetting in molecular reaction systems.
Another important topic in stochastic processes is the random search problem. Many studies on random searches in foraging theory with incomplete information have employed a Brownian search as a default strategy [46], while others have proposed Lévy flights as an efficient strategy for searching for sufficiently sparse targets [47], stating that the Lévy process is one of the most natural and optimal search strategies [48][49][50][51][52]. Various search strategies have been introduced and proposed, including different combinations of search processes [53][54][55][56][57].
The corresponding Fokker-Planck equation of a Brownian random search process for the non-normalized density function f (x, t) with a δ-sink of strength P fa reads [53][54][55] where D is a diffusion coefficient. One assumes here that the initial position is given at The δ-sink means that the random searcher positioned at the beginning at x = x 0 will be removed at the first arrival at x = X, i.e., f (x = X, t) = 0. Therefore, P fa (t) represents the first arrival time distribution (FATD) [53][54][55], which is obtained from Equation (3): which is a negative time derivative of the survival probability The FATD for the Brownian search is described by the Lévy-Smirnov density: with the long-time asymptotics, P fa (t) ∼ |X − x 0 |t −3/2 . Other important characteristics of searching are the search reliability and the efficiency. The search reliability is considered as the cumulative arrival probability [54]: which, for the Brownian search, is P = 1 (the searcher will find the target with the probability of one), while the search efficiency [54], represents the averaged inverse search time. Here, P fa (s) = L[P fa (t)](s) = ∞ 0 ℘ fa (t)e −st dt is the Laplace image of P fa (t). For a one-dimensional Brownian search, it is given by [54]: In a similar way, a Brownian random search on a comblike structure has also been considered. In this case, the initial position of the searcher is located at the backbone at (x, y) = (x 0 , 0), and the target is also located at the backbone at (x, y) = (X, 0), with a δ-sink δ(x − X)δ(y) of the strength ℘ fa (t) [58]. The FATD is given in terms of the Fox Hfunction with the long-time asymptotics reducing to the power law, ℘ fa (t) ∼ |X − x 0 |t −5/4 , while the search reliability equals one, and the efficiency becomes [58]: The inhomogeneous advection on the comb, where the motion along the backbone is interrupted by Brownian motion in the fingers, can be described by the Fokker-Planck operator It results in turbulent diffusion, which is characterized by the log-normal distribution and exponential growth of the MSD in time [29]. This behavior is analogous to one-dimensional geometric Brownian motion [29], which is used in the Black-Scholes model for option pricing [59,60]. The FATD is the Lévy-Smirnov distribution, and the process is suitable for searching for long-distance targets. Turbulent diffusion occurs due to a multiplicative noise, in contrast to the additive noise in Brownian diffusion with a drift. The impact of the resetting mechanism on turbulent diffusion is one of the main issues in this paper. This paper is organized as follows. In Section 2, the one-dimensional Brownian motion in the presence of a drift and resetting to the initial position of the particle is considered. As the main characteristics of the process, the PDF and the MSD are obtained. The analytical treatments of the corresponding Brownian search problem with the drift in the cases of a single target and two targets are presented. The main features of a subordination approachas the analytical tool used throughout the analysis-are described as well. The analytical results obtained for the FATD, the search reliability, and the efficiency are verified by numerical simulations. Brownian motion with drift in the presence of stochastic resetting on a two-dimensional comb is investigated in Section 3. Both analytical and numerical results are presented. The problem of a Brownian search with a drift on a two-dimensional comb in the cases of a single target and two targets is analyzed in detail. Section 4 is devoted to inhomogeneous advection with stochastic resetting on the comb. It is shown that three different scenarios for the MSD, depending on the resetting parameter, can be observed. These are: (i) exponential growth of the MSD in time, (ii) linear growth of the MSD in time, and (iii) saturation of the MSD. The results for the FATD, the search reliability, and the efficiency are presented as well. A generalization of the inhomogeneous advection search is also considered. A summary of the analysis is provided in Section 5.

One-Dimensional Brownian Motion with Drift
In this section, we consider resetting and search problems in the framework of a one-dimensional diffusion process with a drift. We show that this "simple" addition of a drift term in the corresponding equations leads to new physical effects, which are also based on well-known results of the one-dimensional diffusion-advection equation.
Therefore, to set the stage for the clear presentation of the analysis, we first offer a short overview of the results related to the one-dimensional diffusion-advection equation without stochastic resetting, which will be used later to find the corresponding results for more general problems. The corresponding Fokker-Planck equation with a constant velocity V reads [1] The initial condition P 0 (x, t = 0) = δ(x − x 0 ) and vanishing boundary conditions at infinity, P 0 (±∞, t) = 0 and ∂ ∂x P 0 (±∞, t) = 0, are imposed. In Laplace space, it reads The solution of Equation (11) is the Galilei-shifted Gaussian (see, for example, Ref. [1]), and in the following, we shall need its Laplace image, which reads From here, we find that the PDF is normalized, since and, respectively, The MSD is given by which means that the short-time diffusive behavior ( x 2 (t) 0 ∼ t) turns into ballistic motion in the long-time limit x 2 (t) 0 ∼ t 2 .

One-Dimensional Diffusion-Advection Equation with Stochastic Resetting
Now, we consider the diffusion-advection equation in the presence of stochastic resetting. The Fokker-Planck equation reads with the same initial condition P r (x, t = 0) = δ(x − x 0 ). Here, r is the rate of resetting to the initial position x 0 . The last two terms of the equation represent the loss of the probability from position x due to the reset to the initial position and the gain of the probability at x 0 due to resetting from all other positions, respectively. This equation means that between any two consecutive resetting events, the particle undergoes diffusion with a constant drift. From the Laplace transform of Equation (19), one finds Then, the inverse Laplace transform yields Equation (19) in the equivalent form where η(t) = e −rt and η(s) = 1 s+r . This equation can be solved by using a subordination approach [1,59,[61][62][63]. Equation (21) in Laplace space reads Let us present the solution of Equation (21) in the subordination form with the integral where P 0 (x, t) is the solution in Equation (13). Here, h(u, t) is the so-called subordination function. The latter is the PDF, which subordinates the process governed by Equation (21) to the process governed by Equation (13). By the Laplace transform of Equation (23), and by using the subordination function we find Performing the variable change s → 1/η(s) in Equation (12), we have Therefore, from Equations (25) and (26), we obtain Equation (22). Eventually, from Equations (14) and (25), we obtain the PDF in the presence of resetting, From here, one finds where P 0 (x, t) is defined by Equation (13), which is the solution of the corresponding Fokker-Planck equation without resetting. We also note that the case without the drift (V = 0) yields the known result for free diffusion with stochastic resetting [31]. From Equations (27) and (16), we find that the PDF P r (x, t) is normalized ( x 0 (t) r = 1), since From Equation (29) for the MSD, we find which results in Then, the long-time limit yields saturation of the MSD, while the short-time limit corresponds to the result without resetting, Equation (18). In the absence of the drift, the MSD reads [31].

Langevin Equation
We compare the analytical results against the ones obtained from direct numerical simulation of the dynamics by considering a Langevin equation in the presence of stochastic resetting to the initial position [44], where ζ(t) is a zero-mean Gaussian noise and V, D, and r are parameters that are used equivalently in the analytical case. Regarding the temporal evolution of the variance, ensembles of 10 4 particle positions were simulated considering a time step of ∆t = 0.01 across a time span of 10 3 in order to observe convergence of the processes. A graphical representation of the PDF is given in Figure 1 (left panel). The numerical results for the MSD, represented by dots, triangles, and squares in Figure 1 (right panel), show excellent agreement with the analytical results, represented by lines. A typical trajectory of a particle is shown in Figure 2.

One-Dimensional Brownian Search with Drift
The random Brownian search with drift in one dimension is described by the Fokker-Planck equation [54] where f (x, t) is the non-normalized density function, and the last term in the equation is a δ-sink of strength P fa (t), which is considered as the FATD. From the condition f (x = X, s) = 0, one finds the FATD [39,54] which is the exponentially truncated Lévy-Smirnov probability density. The search reliability is [54,55] while the search efficiency has the form [54,55] For V = 0, one recovers the known result for the random Brownian search (8). The FATD and the search efficiency are depicted in Figure 3. In Figure 4, we present the survival probability obtained by numerical simulations (dots, triangles, and squares) in the framework of the Langevin equation approach and the numerical inverse Laplace transform in MATHEMATICA [64]. Next, we extend our analysis to the case of the Brownian search with drift in the presence of two sinks located at x = X 1 and x = X 2 , where X 1 < X 2 . The problem is governed by the Fokker-Planck equation where f (x = X 1 , t) = f (x = X 2 , t) = 0 and P fa,1 (t) + P fa,2 (t) = P fa (t) is the FATD. Without loss of generality, we consider V > 0. By the Fourier-Laplace transformation, we obtain By the inverse Fourier transform, we find Three cases of the initial position of the searcher are considered. These are: (i) x 0 < X 1 < X 2 , (ii) X 1 < x 0 < X 2 , and (iii) X 1 < X 2 < x 0 . Following the same approach as is used in the case of one sink, we find the FATD in the Laplace space as follows: Then, the search reliability reads The search efficiency for the case X 1 < x 0 < X 2 can be analyzed numerically, while the other two cases can be calculated exactly and correspond to the single-target problem.

Diffusion-Advection Equation on a Comb with Stochastic Resetting
The problem of the diffusion-advection equation on a comblike structure was introduced by Arkhincheev and Baskin in Ref. [2], and was also studied in Refs. [65,66] in terms of random walks with anisotropy that appear due to the presence of an external electrical field. We extend the model with stochastic resetting, which results in the Fokker-Planck equation (44) with the initial condition P r (x, y, where v is a constant velocity. For the Laplace image, one looks for the solution in the form which yields the following marginal PDF: Performing the Laplace transform of Equation (44) and taking into account Equation (46), one obtains By the inverse Laplace transform, we find the equation where η r (t) = L −1 (s + r) −1/2 = e −rt t −1/2 Γ(1/2) (for more details on tempered operators, see Ref. [67]). Disregarding resetting, and taking into account that η r=0 (t) = t −1/2 /Γ(1/2), one obtains where In the framework of the subordination approach, the Laplace image of the solution to Equation (49) is presented as follows: where P 0 (x, u) is the PDF (13). Then, the PDF (50) reads which is The case without resetting (r = 0) yields the PDF Eventually, the solution to Equation (48) reads From Equations (50) and (16), one can easily check that the PDF is normalized, since that is, x 0 (t) r = 1. For the MSD in the case of resetting, we find which, in the absence of resetting, reduces to Therefore, in the long-time limit, saturation of the MSD is due to the resetting, while in the short-time limit (when erf(x) ∼ 2z/ √ π and e −z ∼ 1 − z for z 1), we recover the result obtained for the resetting-free case in Equation (57).

Langevin Equation
To analyze the diffusion dynamics numerically, we use a system of Langevin equations in the presence of drift and stochastic resetting to the initial position [33,44]: y(t + ∆t) = y(0), with probability r∆t, where A(y) is a function that mimics the Dirac δ-function. To simulate A(y), diffusion across the x directions is permitted in a narrow strip with a width of 2 along the y axis such that the value of is of the same order of magnitude as the diffusion coefficients [69]. A graphical representation of the PDF is depicted in Figure 5 (left panel), while the MSD is presented in Figure 5 (right panel). A good agreement between the analytical and numerical results for the MSDs is obtained. The individual trajectory along the backbone in the presence of stochastic resetting is shown in Figure 6. The case without resetting is shown in Figures 7 and 8.

Brownian Search with Drift on a Comb
In this section, we consider a Brownian random search process on a comblike structure in the presence of an external drift along the backbone (constant external force). The Fokker-Planck equation then reads with the initial condition f (x, y, t = 0) = δ(x − x 0 )δ(y). The δ-sink means that the searcher is annihilated by reaching the point (x, y) = (X, 0). According to the definition in Equation (4), the FATD results from double integration where S(t) = ∞ −∞ ∞ −∞ f (x, y, t) dx dy is the survival probability. By the Laplace transform of Equation (61), we find Again, using the substitution we find By the Fourier transform with respect to x, we find and thus, which is the solution of the equation The inverse Fourier transform of Equation (65) with respect to k gives . From the condition we conclude that it corresponds to f 1 (x = X, t) = 0, i.e., f 1 (x = X, s) = 0. Thus, for the FATD in Laplace space, we have We also note that the FATD (68) can be directly obtained from the FATD for the onedimensional search with drift (see Equation (34) while the efficiency is given by For v = 0, we recover the known result (9). The FATD and the efficiency for the random search with drift on the comb are depicted in Figure 9. In Figure 10, we give the survival probability obtained by numerical simulations of the Langevin equation and the numerical inverse Laplace transform obtained by MATHEMATICA [64].
One can also consider the Brownian search with drift on a comb in the presence of two sinks located on the backbone at x = X 1 and x = X 2 , where X 1 < X 2 . The corresponding Fokker-Planck equation is with the initial condition f (x, y, t = 0) = δ(x − x 0 )δ(y). We consider v > 0, while the case for v < 0 can be treated in a similar way. Following the same approach suggested for the one-sink problem, we consider the marginal PDF f 1 (x, t) = ∞ −∞ f (x, y, t). Then, its Laplace image reads Due to the presence of two sinks, the absorbing condition is f (x = X 1 , y = 0, t) = f (x = X 2 , y = 0, t) = 0, and correspondingly, f 1 (x = X 1 , t) = f 1 (x = X 2 , t) = 0. We will also consider three cases of the initial positions of the searcher: (i) x 0 < X 1 < X 2 , (ii) X 1 < x 0 < X 2 , and (iii) X 1 < X 2 < x 0 . For the FATD ℘ fa (t) = ℘ fa,1 (t) + ℘ fa,2 (t), we obtain: Therefore, the search reliability is The search efficiency for the case X 1 < x 0 < X 2 can be analyzed numerically, while the other two cases can be calculated exactly, and the results are the same as for the single-target problem on the comb.

Inhomogeneous Advection with Stochastic Resetting
Inhomogeneous advection on a comb in the presence of stochastic resetting is described by the following equation: with the initial condition P(x, y, t = 0) = δ(x − x 0 )δ(y). The case without resetting was considered in Ref. [29]. By the Laplace transform, we have The solution can be represented in the form P(x, y, s) = g(x, s) e − √ s+r D |y| . Then, the marginal PDF along the backbone is p 1 (x, s) = ∞ −∞ P(x, y, s) dy = 2 D s+r g(x, s), and the equation for the marginal PDF reads The inverse Laplace transform yields where η(t) = L −1 (s + r) −1/2 = e −rt t −1/2 Γ(1/2) . In order to find the solution of this equation, we use the subordination approach. Let us consider the standard inhomogeneous advection equation which, in Laplace space, reads The solution of the equation for x > x 0 in Laplace space is [29] From the subordination approach, we obtain where we used The inverse Laplace transform yields the solution where is the solution of Equation (77) without resetting [29]. From Equation (84), the Laplace image of the MSD reads and the inverse Laplace transform yields Here, E α (z) = ∑ ∞ n=0 z n Γ(αn+1) is the one-parameter Mittag-Leffler function, and its Laplace image is L[E α (at α )] = s α−1 s α −a [68]. For the large argument, the Mittag-Leffler function reads D t (see Ref. [68]). Therefore, this yields the long-time limit of the MSD, which has three different regimes: (i) exponential growth in time for r < v 2 /D, (ii) linear growth with time for r = v 2 /D, and (iii) saturation for r > v 2 /D.

Langevin Equation
In the case of inhomogeneous advection on a comb, completely different results are obtained due to the multiplicative noise. The microscopic approach to the process without resetting is described by the Langevin equation: where ζ(t) is a Gaussian noise. Therefore, y(t) is a random Brownian motion, so A(y) is a function of the random variable. Without any restriction of generality, we may use A(y) = δ(y), which corresponds to the inhomogeneous advection motion on the comb.
To validate the analytical results, an ensemble of 10 6 particles was simulated based on the system of Langevin Equations (89) and (90), which, in the presence of resetting, are where the time step is ∆t = 0.01. It should be admitted that for the multiplicative noise, one cannot use the numerical approximation of the Dirac δ-function considered in Section 3. For numerical purposes, one can use another approximation of the Dirac δ-function (see Ref. [4]). Consequently, we use a zero-mean Gaussian function A(y) = 1 √ 2πσ exp(−y 2 /[2σ 2 ]) in the limit σ → 0. A graphical representation of the PDF is shown in Figure 11 (left panel), while the MSD obtained analytically and by numerical simulations is shown in Figure 11 (right panel). A typical trajectory along the backbone in the presence of stochastic resetting is shown in Figure 12.  Dashed regions are introduced for these resetting events to be more visible.

Inhomogeneous Advection Search on a Comb
For the completeness of the analysis, we consider an inhomogeneous advection search on the comb, or the so-called turbulent diffusion search. The corresponding Fokker-Planck equation reads [29] Following the standard procedure described in Section 3.2, we find the FATD in Laplace space as follows: The inverse Laplace transform of Equation (94) for X > x 0 gives the Lévy-Smirnov distribution [29]: The search reliability equals one (P = 1), while the efficiency is Therefore, the turbulent diffusion search is more efficient than the Brownian search for long-distance targets, but the searcher should have a prior knowledge of the direction of the target, which is not the case for the Brownian search. The FATD and the efficiency are shown in Figure 13. The survival probability is shown in Figure 14.   (95) and (96), the corresponding inhomogeneous advection search problem cannot be considered for the initial position of the searcher at the origin (x 0 = 0). To consider a random search with the initial condition at x 0 = 0, we add some constant advection v 0 = v , where > 0. In this case, x 0 = 0 is no longer a singular point for the space derivative. The search equation reads

Remark 1. As observed above in Equations
Following the same procedure for the FATD described above, we obtain The reliability becomes P = 1, and the efficiency becomes which is also valid for x 0 = 0. However, the searcher moves only in the advection direction.
Following the same procedure for the FATD at X > x 0 > 0, we obtain (101) The search reliability is P = ℘ fa (s = 0) = 1. The inverse Laplace transform of Equation (101) yields Therefore, the efficiency has the form Remark 3. We note that in the limit λ → 1, the FATD (101) and the efficiency (103) become where we use L'Hôpital's rule, and thus, we recover the corresponding results ( (94) and (96)) for the inhomogeneous advection search.

Remark 4.
For λ = 0, in Equation (101), for the FATD, we obtain ℘ fa (s) = e − 2 √ D v √ s (X−x 0 ) . This case corresponds to homogeneous advection on the comb, and thus, the FATD can be obtained from Equation (68) for D x → 0. Namely, we have The efficiency reads which has the same behavior as the efficiency for the one-dimensional Brownian search (see Equation (8)).

Summary
We present an overview of various drift-diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both the Brownian diffusion search with drift and the inhomogeneous advection search on the comb structures were analyzed. The analytical results were verified by numerical simulations in terms of coupled Langevin equations for the comb structures. The subordination approach was one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.
In conclusion, consideration of the comb model (or comb geometry) is an important issue for investigation of the interplay between diffusion, drift, and geometry. For example, further modification of the comb model, like a fractal tartan [16,[70][71][72], can be an interesting task for understanding of the impact of fractal geometry on the fractional transport, as well as for the experimental implementation and the technological design and development of sparse sensor arrays [73,74].

Conflicts of Interest:
The authors declare no conflict of interest.