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Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations<x(t)x(s)>~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H unequal 1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(x,t) ~ R(x/t^H)/t^H. Our objective is to compute the scaling function R(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H = 1/2 + epsilon, to calculate the scaling function R(y) to first order in epsilon. We find that R(y) behaves as R(y) ~ y^phi as y ->0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y ->oo, with phi = 1 - 4 epsilon + O(epsilon^2) and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.


I. INTRODUCTION
Survival of a species of bacteria, translocation of DNA through a nano-pore, and diffusion in presence of an absorbing boundary are only few out of many situations, where the central question is the survival, or persistence of the underlying stochastic process. More precisely, persistence, or survival probability S(t) of a process is the probability that the process, starting from an initial positive position, stays positive over a time interval [0, t]. For many stochastic processes arising in non-equilibrium systems, persistence decays as a power law S(t) ∼ t −θ , where θ is called the persistence exponent [1]. For a simple Markov process such as one-dimensional Brownian motion, θ = 1/2 [2]. On the other hand, the exponent θ is non-trivial whenever the process is non-Markovian, i.e., has a memory. In addition to theoretical studies (for a brief review see [3]), the exponent θ has been measured in a number of experiments [4][5][6][7][8][9][10]. Even for Gaussian non-Markovian processes, θ is non-trivial [11]. For the latter processes that are close to a Markov process (i.e., whose correlators are close to that of a Gaussian Markov process) the exponent θ was computed perturbatively [12,13]. This perturbation theory has been used for various out-of-equilibrium systems, as the global persistence at the critical point of the Ising model in d = 4− dimensions [14], in simple diffusion close to dimension 0 [15], and in fluctuating fields such as interfaces [16][17][18].
A quantity that contains more spatial information than persistence S(t) is the probability density P + (x, t) of the particle at position x and at time t, given that it has survived (stayed positive) up to time t. To investigate P + (x, t), one can equivalently think of a process on the positive semi-infinite line [0, ∞] with absorbing boundary condition at the origin x = 0 (see Fig. 1). The question is, how does P + (x, t) depend on x? In other words, how does the presence of an absorbing boundary at the origin change the spatial dependence of the probability density of the particle at time t? In particular, it is clear that P + (x, t) must vanish as x → 0 and x → ∞. But how do they vanish there? One of the main messages of our paper is that for generic non-Markovian processes, P + (x, t) vanishes near its boundaries at x = 0 and x → ∞ in a non-trivial way, characterized by non-trivial exponents.
As the persistence S(t), the probability P + (x, t) can be computed exactly for a Gaussian Markov process, as e.g. one-dimensional Brownian motion. For non-Markovian processes, even if they are Gaussian, P + (x, t) was not known. In this work, we consider P + (x, t) for a class of one-dimensional Gaussian processes known as fractional Brownian motion (fBm), which are parametrized by their Hurst exponent H, with 0 < H < 1. The case H = 1/2 corresponds to ordinary Brownian motion, which is a Markov process, while for H = 1/2 the process is non-Markovian. The purpose of this paper is to develop a systematic perturbation theory to compute P + (x, t) for non-Markovian fBm's with H = 1/2+ , where is the expansion parameter for the perturbation theory. Here we present the result for P + (x, t) to O( ). It can be written as a combination of special functions, i.e. error and hypergeometric functions, see Eq. (10). To our knowledge, this is the first systematic (exact up to O( 2 )) calculation of P + (x, t) for fractional Brownian motion with H = 1/2.
Before detailing our results, let us position them into a broader context: Fractional Brownian motion with H = 1/2 is relevant for polymer translocation through a nanopore. Consider a polymer chain composed of N monomers passing through a pore (translocation) from left to right, as drawn on Fig. 2. The dynamics of this translocation process has been investigated intensively due to its central role in understanding, e.g., viral injection of DNA into a host, or RNA transport through nano-pores, and mastering such applications as fast DNA or RNA sequencing through engineered channels [19][20][21][22]. The translocation coordinate s(t), namely the label of the monomer crossing the pore at time t, is key to quantitatively describing the translocation process [23][24][25][26], which begins when s = 1, and ends when s = N , i.e., when the first and the last monomer of the chain enter the pore, respectively, see Fig. 2. For large N , when the translocation is not yet complete, one can view s(t) as a stochastic process on the semi-infinite line with absorbing boundary conditions at s = 0. The absorbing boundary at s = 0 models that if the chain falls back to the left, i.e., on the starting side, it will diffuse away and not try again. The quantity P + (s(t) = x, t) then represents the probability that x monomers have translocated to the right at time t. To model the process s(t), one observes the following facts: (i) scaling arguments and numerical simulations show that s(t) is subdiffusive [27]; (ii) in absence of boundaries, numerical simulations indicate that s(t) is a Gaussian process [28]. Based on these observations it was proposed in Ref. [29] that a good candidate for s(t) is a fractional Brownian motion with H = 1/(1 + 2ν), where the exponent ν describes the growth of the radius of gyration with the number of monomers (R g ∼ N ν ) [30]. Thus for ν = 1/2, H < 1/2 and hence s(t) is generically a non-Markovian process, with absorbing boundary conditions at s = 0 and at s = N . Here we consider the limit of N → ∞. Thus our results for P + (x, t) of a fBm with H = 1/2 are directly relevant for polymer translocation.
Directions for further applications are numerous: Recently a relation was established between the statistics of avalanches associated with the motion of a driven particle in a disordered potential and persistence properties of the latter [31]. Higher-dimensional generalizations are avalanches of extended elastic objects, for which systematic field-theoretic treatments exist [32][33][34][35]. In few cases, no-hitting probabilities can be calculated for extended (non-directed) objects, as self-avoiding random walks avoiding extended objects [36]. Other approaches use real-space renormalization [37,38].
This article is organized as follows: Since some of the computations are rather technical, we first provide in Section II a brief summary of the main definitions and our principal results. In Section III, we introduce basic notations and reproduce the known results for H = 1 2 . Section IV explains the basic ideas of our perturbative approach, sketches the calculation, and discusses some of the subtle points. Our predictions are compared to numerical simulations in Section V. Conclusions are presented in Section VI. More technical points are relegated to two appendices: In Appendix A the correction to the action is derived. Appendix B contains the explicit calculation of the perturbation theory. Finally, Appendix C reviews the arguments for the scaling law φ = (1 − H)/H.

II. SUMMARY OF DEFINITIONS AND MAIN FINDINGS
Consider a particle, located at time t = 0 at the origin x = 0 and free to propagate on the real axis. For Gaussian processes, the probability to find the particle inside the interval (x, x + dx) at time t is given by where x 2 (t) is the particle's mean square displacement. A natural scaling variable is and most of the properties of the process are a function of this single variable. For example, the distribution prob-ability in Eq. (1) becomes In many problems the motion is confined to an interval, finite or semi-infinite. In presence of absorbing boundaries, the probability distribution of the particle position, subject to the condition that the particle has survived, has no longer a simple Gaussian form since it has to vanish at the boundaries. However, one can still express it as a function of the sole scaling variable y defined in Eq. (2), where x 2 (t) is the particle's mean square displacement in the unconstrained (without boundaries) process over the full real line. In particular, here we discuss the case where the particle can move on the positive semi-axis and is absorbed whenever x(t) < 0. We call P + (x, t) and R + (y) with y given in Eq.
(2) the normalized probability distribution and the scaling function of the problem in presence of an absorbing boundary at the origin, In contrast to the free case, the functional form of R + (y) is not the same for all Gaussian processes, but depends on the precise nature of the latter. Here we study a particular class of processes, the fractional Brownian motion (fBm), for which the autocorrelation function in absence of boundaries is where H with 0 < H < 1 is the Hurst exponent. For H = 1/2, the fBm identifies with Brownian motion where D is the diffusion constant. Note that only for H = 1/2, the Gaussian process x(t) is Markovian. For other values of H, the process is non-Markovian. For Brownian motion (H = 1/2), the form of R + (y) can be obtained using the method of images (see Section III), The superscript (0) identifies the case H = 1/2. For other values of H, due to the non-Markovian nature of the process, the method of images no longer works and the computation of R + (y) becomes a challenging problem. In this paper we compute this function, using a perturbative approach for H = 1/2 + , to first order in . The final result is W (y) = 1 6 y 4 2 F 2 1, 1; where γ E is Euler's constant, 2 F 2 a hypergeometric function, and erfi the imaginary error-function. We can write a convergent series-expansion where each line is equivalent to the corresponding line in Eq. (10). Both sums converge for all y, but problems of numerical precision appear for y > 7. In that region, one can use the asymptotic expansion At y = 7, the difference between (11) and (12) is smaller than 10 −6 .
We obtain, at first order in , the asymptotic expansions of R + (y), These asymptotics can be recast into where the two exponents φ and γ are at first order in given by In a recent publication [29] (reviewed in Appendix C), a general scaling relation, valid for arbitrary self-affine processes with stationary increments, was proposed between the exponent φ, the persistence exponent θ, and the Hurst exponent H, For fBm, it is known rigorously that θ = 1 − H [16]. This result predicts that for fBm, One of the objectives of this paper was to verify this scaling relation up to O( ) in a perturbation theory around H = 1/2. Using H = 1/2 + , one expects for fBm. This is in agreement with our result (15), putting the scaling arguments on a firmer footing. It is interesting to note that the scaling function R + (y) given in Eq. (4) has, at least to O( ), the same leading large-y behavior ∼ e −y 2 /2 as in the unconstrained case (4). This behavior can be understood by a simple heuristic argument: far from the boundary the process is not "aware" of the latter. Our calculation reveals that the process nevertheless knows about the boundary, and R + (y) has a subleading power-law prefactor y γ where γ is a new (independent) exponent, whose result to order is given in Eq. (15).
Our analytical results are then verified via numerical simulations for H = 2/3.

III. PRELIMINARIES: BROWNIAN CASE (H = 1/2)
To simplify notations, we set D = 1 in the following. The final result (9), expressed in the variable y, is of course independent of this choice.
The spreading of a Brownian particle is given by the Fokker-Planck equation The propagator Z + (x 0 , x, t) times dx gives the probability to find the Brownian particle inside the interval (x, x + dx) at time t, knowing that the particle was at x 0 at time t = 0. With absorbing boundary conditions at the origin we have, using the method of images This propagator is not a probability distribution because it is not normalized. Its normalization, the so-called survival probability, gives the probability that the particle is not yet absorbed by the boundary at x = 0. The survival probability vanishes when x 0 → 0; however, in that limit, the probability distribution for the non-absorbed particles remains welldefined: Another quantity with a finite limit for x 0 = 0 is This allows to write the probability P (0) Using in Eq. (24) the scaling variable defined in (2), y = x/ √ 2t, we recover (8). Eq. (24) is simpler than Eq. (22) because the x 0 dependence is discarded from the beginning. We will use this definition to compute Z + (x, t) for H = 1/2 + .

IV. PERTURBATION THEORY (H = 1/2)
The process x(t) is Gaussian for all values of H, but it is Markovian only for H = 1/2. For all other values of H, the process is non-Markovian and this makes the problem difficult to solve. Our idea is to expand around H = 1/2. In a first step, we construct an action, which calculates expectation values of the Gaussian process x(t), with bulk expectation values (6). In a second step, we obtain the propagator with absorbing boundary conditions at x = 0. In a third step we calculate the probability P + (x, t) perturbatively, using the action constructed in step 1. In the fourth step, we put together all pieces and interpret our result.

A. Step 1: The Action
For all H, x(t) is a Gaussian process, therefore the statistical weight of a path x(t ) without any boundary is proportional to exp(−S[x]) where the action S[x] is quadratic in x and given by Note that we use standard field-theoretic notation, noting f (x) a function of the variable x, and S[x] a functional, depending on the function x(t ), with 0 < t < t. The kernel G(t 1 , t 2 ) of the action is related to the autocorrelation function of the process via For H = 1/2, the action is simple. In this case, setting D = 1, Using the result (A7) in Eq. (25), we recover the standard Brownian action For a generic value of H the kernel G(t 1 , t 2 ) becomes non-local. For H = 1/2 + one can write where is the action (28) and S (1) [x] has been computed in Appendix A Note that we have introduced a regularization for coin- It is however sufficient to check that the law (6) is correctly reproduced, and that the final result is cutoff independent. B.
Step 2: The Propagator with an Absorbing Boundary For a generic value of H, the propagator Z + (x 0 , x, t), denoting the probability that the particle reaches x at time t, starting from x 0 at time 0, and staying positive over the interval [0, t], can be written using standard path integral notation as Here Θ[x] is an indicator function that is 1 if the path x(t ) stays positive over the interval [0, t] and 0 otherwise. The action S[x] is given in (25). In the limit x 0 → 0, we expect, as in the Brownian case (H = 1/2), the propagator to vanish as x φ0 0 where the yet unknown exponent φ 0 depends on H. Note that for H = 1/2, φ 0 = 1 (see Eq. (23)). For H = 1/2 + , we expect that φ 0 = 1+a 1 +O( 2 ), where a 1 is yet unknown. Analogous to Eq. (23) for H = 1/2 we define Z + (x, t) as Using the expansion of the action given in Eq. (29) and φ 0 = 1 + a 1 , we write to leading order in x 0 where Z (0) We will see that for The main achievement of this paper is the calculation of Z (1) + (x, t) defined in Eq. (34). This calculation is rather involved, both conceptually and technically. Therefore, we will relegate several technical calculations to Appendix B. Eq. (34) can be devided into three pieces: × lim x0→0 1 x 0 × 1 x 0 The first term, Z A + (x, t) is simple, and is evaluated in appendix B 1. We now come to the evaluation of the contribution Z B + (x 0 , x, t), defined in Eq. (37). In Fig. 4 we show a path which contributes to Z B + (x 0 , x, t). The sum of all these paths is a product of transition probabilities. Explicitly, it reads, ordering t 1 < t 2 , which gives an extra factor of 2 compared to (37): (20). The factors D(x 1 ,x 1 ) and D(x 2 ,x 2 ) take into account the terms ∂ t1 x(t 1 ) and ∂ t2 x(t 2 ) in the action S 1 [x].
Finally, we have sett 1 = t 1 , andt 2 = t 2 , since we have taken the limit of their differences to 0. In order to perform the six integrations in Eq. (38) it turns out to be convenient to evaluate its Laplace transform, Z B + (x 0 , x, s). From now on, we will always denote with f (s) the Laplace transform of a function f (t), defined as This Laplace transform leads to two important simplifications: The first simplification is that now the nested time-integrals over t 1 and t 2 become a product. To see this, we remind that if f 1 and f 2 are two functions which depend on t, then the Laplace transform of their convo-lution is simply the product of their Laplace transforms, This consideration generalizes to 3 and more times. We obtain for the Laplace transform of (38) The second simplification is even more important, and is most easily understood on the example of the bulk propagator Its Laplace-transform is While integrals over x > 0 involving (B19) give errorfunctions, which are hard to integrate further, the same integrals over (44) remain similar exponential functions; the only complication is that one has to distinguish between x smaller or larger than y.
To evaluate (42), we now have to calculate the Laplacetransforms of its factors: Finally, the term in brackets in Eq. (42) can be rewritten, using a Fourier decomposition for Z Note that the time integral in the second line of Eq. (46) is diverging at small times. Since the path integral is defined as discretized in time, a natural approach consist in discretizing this integral, with a step-size τ . This would indeed be the only possible approach for stronger divergences, like 1/t 2 . However, since our integral is only logarithmically diverging, we can take an easier path, by using a small-time cutoff τ : We note that the regularization by discretization gives the same result apart from the term −γ E . We will check later that it only contributes to the normalization, which will drop from the final result.
Collecting the results of Eqs. (45) and (46) in Eq. (42), and doing the remaining space-derivatives we find Performing the space-integrations, we find Note that this is (rescaling k → √ sk) The next step is to invert this Laplace transform which is performed in Appendix B 3. D.
Step 4: The Probability P+(x, t) The final result for Z where I(z) is defined in Eq. (B53). The first line is arranged as to not contribute to the normalization, whereas c(t) is independent of z and will not appear in the final conditional probability. γ E is Euler's constant. The probability distribution, P + (x, t), to find a non-yet-absorbed particle in the interval (x, x + dx) can be computed following the lines of Eq. (24) to order as Note that the term proportional to c(t) cancels in normalized objects such as P + (x, t). Therefore, we obtain where R This gives for Eq. (53) up to terms of order 2 This is the final result announced in equation (9), with I(y) calculeted in (B53) and below.

V. COMPARISON TO NUMERICS
In this Section, we compare our analytical results with numerical simulations. More specifically, we consider the super-diffusive process with H = 2 3 .

A. Methodology of simulations
We aim to sample a fBm processes x(t) at discrete times t 1 = 1, t 2 = 2, . . . , t L = L. The covariance matrix of {x 1 , . . . , x i , . . . , x L } coincides with the autocorrelation function of the original fBm process in Eq. (6), setting D = 1, The L × L covariance matrix C is symmetric and has positive eigenvalues; it is thus possible to find a matrix A, positive and symmetric, such that C = A 2 . Matrix A is called the square root of C. One can simulate paths of a fBm using the standard procedure for Gaussian correlated processes: (i) Determine A, the square root of C. (ii) Each path x = {x 1 , . . . , x i , . . . , x L } is given by the matrix multiplication x = A η. The vector η = {η 1 , η 2 , . . . , η L } is a set of L independent Gaussian numbers with unitary variance and zero mean. It is easy to check that these paths are characterized by the correct covariance matrix (56).
Unfortunately this procedure is time consuming, as for step (i) it requires the full diagonalization of C. Better results are obtained by making use of the stationarity of the increments ξ i = x i − x i−1 (we set x 1 = ξ 1 ). Using Eq. (56) we can computeC, the covariance matrix of the increments, where k = 0, . . . , L − i, andC i+k,i =C i,i+k . The matrix C is symmetric and positive definite like the matrix C, but it also is a Toeplitz matrix. For Toeplitz matrices efficient numerical methods allow to avoid the full diagonalization ofC. In particular, the Levinson algorithm (for a practical implementation of Levinson's algorithm see [39] and [40]) is suitable for first passage problems, as it recursively generates the increment ξ i+1 given ξ 1 , ..., ξ i . The points of the fBm path are given by x i = i j=1 ξ j . In our simulation we are interested only in positive paths (x i > 0 for all i). The Levinson method allows to discard negative paths whenever a x i < 0 is generated, without building the full path.

B. Simulation results
For each positive path we record the final position x L . The histogram of the rescaled variable y := x L /(2L H ) is the scaling function R + (y). The results for H = 2/3 and the Markovian case H = 1/2 are presented on Figs. 5 and 6. For small y the scaling function, R + (y) behaves as a power-law, with an exponent φ. For H = 1/2 we expect φ = 1, for H = 2/3 we expect φ = 1/2. Inspired by our perturbative calculation we predict that for y → ∞, R + (y) behaves like ∼ y γ e −y 2 /2 . In order to facilitate the comparison, we define the scaling function The numerical data for the scaling funtion r + (y) defined in Eq. (58) are shown on Fig. 7 for H = 2/3. They clearly show two distinctive power-law behaviors: For small y this power law is ∼ y φ with φ = 1 2 , predicted by the scaling relation φ = 1−H H . For large y a larger exponent γ = 0.7 ± 0.03 is measured. This is consistent with the perturbative calculation, which suggests γ > φ for H > 1/2 and γ < φ for H < 1/2.
A more accurate comparison between the numerical data and the perturbation theory is possible. Our perturbative result given in Eq. (9) is equivalent to r + (y) = y 1 + W (y) + O( 2 ) . In order to compare to numerics, we use While the two expressions are equivalent to order , the latter (59) has the merit to resum the logarithms for small and large y into the power-law behavior where the exponents are the order-results For H = 2/3, i.e. = 1/6, we predict a scaling ∼ y γ , γ = 2 3 , using (61). Note that the curve drawn is exatly the asymptotic behavior of our analytical result (59), using (10), thus also the amplitude and not only the exponent are estimated. This can more clearly be seen on Fig. 8, where the solid (blue) line represents the theoretical order-prediction, and the dashed line the asymptotic behaviors given in Eq. (60).
Conversely, relation (59) can be used to extract W (y) from r + (y), This relation should work the better, the smaller is. Using our numerical results for H = 2 3 , we obtain the curve presented on Fig. 9. The agreement is quite good for 1 ≤ y ≤ 2.5. It breaks down for larger y due to numerial problems. For y < 1, the deviations can be attributed to the large value of . . The dashed lines are the predicted asymptotic behaviors, ∼ y φ (for small y) and ∼ y γ (for large y). Superimposed are the simulation data shown on Fig. 7. Note that there is no fitting parameter. The deviation at small y is due to the fact that is rather large, so the order-slope φ ≈ 1 − 4 = 1 3 is smaller than the exact result φ = 1−H H ≡ 1/2− 1/2+ , which evaluates to 1 2 . For large y, but smaller than 3, the effective cutoff of the simulation, both amplitude and slope are correctly predicted. from the numerical data for H = 2 3 , using relation (62). The agreement is quite good for 1 ≤ y ≤ 3. It breaks down for y > 3 due to numerial problems. For y < 1, the deviations can be attributed to the large value of .

VI. CONCLUSIONS
In this article, we develop a systematic scheme to calculate the corrections to the universal scaling function R + (y) for fractional Brownian motion, in an = H − 1 2 expansion. We compute the full scaling function R + (y) to first order in . In particular we find that R + (y) behaves as R + (y) ∼ y φ as y → 0 (near the absorbing boundary), while R + (y) ∼ y γ exp(−y 2 /2) as y → ∞ (far from the boundary), with, at the first order in , φ = 1 − 4 + O( 2 ) and γ = 1 − 2 + O( 2 ). For small our results confirm the scaling relation found in Ref. [29]: R + (y) ∼ y φ with φ = θ/H. For fractional Brownian motion it is known that θ = 1 − H, so that φ = (1−H)/H ≈ 1−4 +. . .. Far from the boundary, i.e. for large y, the leading behavior R + (y) ∼ exp(−y 2 /2) recovers the Gaussian propagator (4) in absence of boundaries; our approach shows that R + (y) has a subleading power law prefactor y γ , where γ is a new (independent) exponent.
Our numerical simulations show that the predictions of the asymptotic behavior of R + (y) hold at H = 2/3. In particular the two exponents γ and φ have been measured and shown in Fig. 7.
Let us stress that few results are known about non-Markovian processes in presence of boundaries. Perturbation theory developed in this paper can provide substantial new insight here. The method is versatile and can in principle be extended to the calculation of other quantities such as the propagator for a process confined to a finite interval with absorbing boundaries, or alternatively with other, e.g. reflecting boundary conditions. Particularly interesting for applications would be the hitting probability Q(x, L), the probability that a generic stochastic process starting at x and evolving in a box [0, L] hits the upper boundary at L before hitting the lower boundary at 0 [41]. In the context of polymer translocation, the hitting probability is the probability that a finite polymer chain will ultimately succeed in translocating through a pore.
In the more general framework of anomalous diffusion, the presence of boundaries has been especially studied for non-Gaussian processes. For instance, Lévy flights are Markovian superdiffusive processes whose increments obey a Lévy stable (symmetric) law of index 0 ≤ µ ≤ 2. The Hurst exponent is H = 1/µ [42]. By virtue of the Sparre Andersen theorem [43], the persistence exponent is θ = 1/2, independent of µ. The Laplace Transform of the scaling function R + (y) has been computed in [44] for a generic value of µ. A scaling analysis of this Laplace Transform shows that R + (y) behaves as R + (y) ∼ y 1/(2µ) as y → 0 (this in in agreement with the scaling relation φ = θ/H), while far from the boundary the Lévy-stable behavior is recovered.
An increasing interest is devoted to Gaussian processes with self-affine anomalous displacements x 2 (t) ∼ t 2H with 0 < H < 1 [16,[45][46][47]49]. Our current results apply only to fractional Brownian motion, i.e. self-affine Gaussian processes defined by the autocorrelation function (6). In particular for fBm it is known that (i) the process has stationary increments, (ii) θ = 1−H, and (iii) φ = θ/H. For all other Gaussian processes with Hurst exponent H, (i) the increments are non-stationary, (ii) θ = 1 − H and we particularly emphasize that, (iii) no scaling relation is known between φ and θ (unlike in fBm where φ = θ/H). Among such processes it is possible to show that the one, defined by the autocorrelation function describes the subdiffusive behavoir of a tagged monomer in an elastic interface which initially was flat [16]. For this process the persistence exponent is known only to first order in [16], whereas neither the exponents φ, nor γ are known analytically. It would be interesting to determine the full scaling function R + (y) for this process within our perturbative framework.
It remains to compute the term G (1) . Integrating by parts one has using that the first two terms in (A3) do not contribute since they only depend on one of the times. The derivative is The second term is not well-defined. We decide to introduce a regularization for coinciding times t = t → ln |t − t | = ln τ where τ > 0 should be thought of as the time-discretization of the path-integral. Let us first give the final result, before commenting on this approximation: This yields for the action We see that the only possibly ambiguous term, the term of order ln τ , is proportional to the zeroth-order action , thus equivalent to a change in the diffusion constant D. Thus its effect is easy to check in the final result, when looking at observables in a domain unaffected by the boundary.
This term is easily evaluated. Indeed, Eq. (36) can be recast in the following form In going from the first to the second line we have used the expression of the propagator in the Brownian case in Eq. (20), introducing the factor of a from the observation that the latter appears together with x 2 , and readjusting the normalization.
In terms of the variable z = x/ √ 2t this gives (23) and The integration over k We splitZ B + (x 0 , x, s) into two parts Z B + (x 0 , x, s) =Ĩ 1 (x 0 , x, s) +Ĩ 2 (x 0 , x, s) (B4) The expansion of this term for small x 0 must be done with care; when x 0 acts as a regulator, one cannot simply expand in it. We claim, and show below that In order to prove this, we group the four terms in (B5) into two times two terms; the first combination is where the divergent contribution is This expansion in x 0 is justified since e −x 0 √ s x0 stands outside the integrand, thus does not act as a regulator.
The second contribution to (B5) is Since x x 0 , we can Taylor-expand cos(k(x + x 0 ) √ s) and cos(k(x − x 0 ) √ s), leading to The contributions proportional toĨ div 1 cancel between (B10) and (B13), and we arrive at the decomposition (B7).
On the other hand, the only contribution to the normalization of the propagator Z + (x, t) comes from c(t). Since Z (0) + (x, t) integrated over x equals 1, we conclude that the survival-probability is in agreement with θ = 1 − H. This is a non-trivial check of our calculations.