Hazard-selfsimilarity of diffusions’ first passage times

A recent study introduced a novel approach to the exploration of diffusions’ first-passage times (FPTs): selfsimilarity. Specifically, consider a general diffusion process that runs over the non-negative half-line; initiating the diffusion at fixed positive levels, further consider the diffusion’s FPTs to the origin. Selfsimilarity means that the FPTs are spanned by an intrinsic scaling of their initial levels. The recent study addressed two types of selfsimilarity: stochastic, scaling the FPTs in ‘real space’; and Laplace, scaling the FPTs in ‘Laplace space’. The Laplace selfsimilarity manifests an underlying sum-like structure. Shifting from the sum-like structure to a max-like structure—a-la the shift from the Central Limit Theorem to Extreme Value Theory—this study addresses a third type of selfsimilarity: hazard, scaling the FPTs in ‘hazard space’. A comprehensive analysis of hazard-selfsimilarity is established here, including: the universal distribution of the FPTs; the dramatically different statistical behaviors that the universal distribution exhibits, and the statistical phase transition between the different behaviors; the characterization of the generative diffusion dynamics, and their universal Langevin representation; and the universal Poissonian statistics that emerge when the initial levels are scattered according to the statistical steady-state of the generative diffusion dynamics. The analysis unveils the following linkages: of the universal distribution to the Gumbel, Gompertz, and Frechet laws; of the universal Langevin representation to diffusion in quadratic and logarithmic potentials; and of the universal Poissonian statistics to non-normalizable densities, to the maxima of the exponential law, and to the harmonic Poisson process.

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Commonly, one sets off from a specific diffusion process of interest, and then uses analytic and probabilistic methods to investigate FPTs of the process under consideration [31][32][33]. An altogether different approach is to set the focus on a certain FPTs structure of interest, and then to address the following go/no-go question: are there diffusion processes whose FPTs display the structure under consideration? If the answer is 'go', the following foundational questions arise: what is the statistical distribution of these FPTs? and what are the dynamics that generate diffusion processes with these FPTs?
The approach described above was recently introduced and explored in [34], with regard to: the FPTs to the origin of real diffusion processes whose dynamics are governed by Ito stochastic differential equations [35,36]; these dynamics are a widely-used model for the evolution of diffusion processes in the sciences [37][38][39]. Motivated by the profound notion of selfsimilarity [40][41][42], selfsimilar FPTs were investigated in [34]. Namely, FPTs structures in which: the FPT from an arbitrary positive level l to the origin is a scaled version of the FPT from the level 1 to the origin.
Two types of selfsimilarity were addressed in [34]. The first, termed stochastic selfsimilarity, scaled the FPTs directly in 'real space'. The second, termed Laplace selfsimilarity, scaled the FPTs in 'Laplace space'. Tracking random processes in Laplace and Fourier spaces is natural when the processes evolve via the summation of random variables; this evolution is prevalent, and its quintessential examples are random walks [43][44][45]. Analyzing the statistical behavior of sums of random variables is a principal pillar in probability theory and in statistical physics [46][47][48].
Shifting from aggregates to extremes, analyzing the statistical behavior of maxima of random variables is yet another principal pillar in probability theory and in statistical physics [48][49][50][51]. To illustrate the shift, consider the daily rainfall over a certain geographical area. Then, from the perspective of water management, interest is set on the aggregate of the daily rainfall. On the other hand, from the perspective of flood management, interest is set on the maximum of the daily rainfall. These different perspectives involve profoundly different universal statistical laws [48][49][50][51].
Extremes arise naturally in the context of FPTs. Indeed, consider a collection of motions and their respective FPTs-say to a certain target zone in the embedding space. Then, the minimal FPT is the 'collection's FPT': the first time that either of the motions reaches the target. And, the maximal FPT is the collection's 'completion time': the time it takes till all the motions reach the target. Extreme FPTs attracted significant research interest recently [24][25][26][27][28][29][30].
In this paper we shift from the aggregate perspective that inspired Laplace selfsimilarity to an extreme perspective that inspires a novel type of selfsimilarity: hazard selfsimilarity, which scales the FPTs in 'hazard space'. As Laplace and Fourier spaces are natural 'venues' to track sums of random variables, hazard space is a natural 'venue' to track maxima of random variables. Following a concise description of the underpinning setting (section 2), a 'motivational discussion' (section 3) shall: review the notions of stochastic selfsimilarity and of Laplace selfsimilarity; and introduce the shift from Laplace selfsimilarity to hazard selfsimilarity. Thereafter, a 'go' answer will be established regarding hazard selfsimilarity, and the two aforementioned foundational questions will be answered in detail (sections 4 and 5).
As shall be shown in section 4, hazard selfsimilarity yields a new universal statistical distribution of FPTs. This universal distribution has two parameters, one real and one positive. The sign of the real parameter turns out to have a dramatic effect on the statistical behavior of the hazard-selfsimilar FPTs, and a statistical phase transition takes place as the real parameter switches its sign. The universal distribution is related to the following statistical laws: when the real parameter is non-zero-to the Gumbel and Gompertz laws; and when the real parameter is zero-to the Frechet law. The Gumbel and Frechet laws are two (of the three) universal extreme-value laws that emerge via the 'Central Limit Theorem' of Extreme Value Theory [50,52].
The generative diffusion dynamics-i.e. the dynamics that generate diffusion processes with hazard-selfsimilar FPTs-shall be characterized in section 5. These generative diffusion dynamics include particular Langevin dynamics [53,54], from which: any other generative diffusion dynamics can be produced. Thus, the generative Langevin dynamics provide a universal representation for all generative diffusion dynamics. As will be shown, the generative Langevin dynamics are governed by a potential function that is composed of two components: quadratic and logarithmic.
A discussion regarding the hazard-selfsimilar FPTs, and their generative diffusion dynamics, shall be presented in section 6. The discussion will include an analogue of the 'first-passage duality' property [21] that is displayed by Laplace-selfsimilar FPTs [34]. Also, the discussion will characterize the statistical phase transition of the aforementioned universal distribution as: the intersection of stochastic selfsimilarity and hazard selfsimilarity.
The statistical steady state of the generative diffusion dynamics shall be analyzed in section 7. This steady state turns out to be characterized by a 'non-normalizable density'yet another topic that attracted significant interest recently [69][70][71][72][73][74][75]. Consequently, this steady state does not have a probabilistic interpretation; nonetheless, it does have a Poissonian interpretation [76,77]. Applying the latter interpretation, universal steady-state Poissonian statistics that emerge from the hazard-selfsimilar FPTs will be attained and studied in section 7.
The paper will conclude with a summary (section 8), and with an 'epilogue result' (section 9) asserting that: the notion of hazard selfsimilarity-which was based here on an underpinning max-like structure-cannot be based on an analogous underpinning min-like structure. The derivations of the key results established along the paper are detailed in the Methods (section 10).

Setting
This paper addresses the FPTs, to the origin, of a general diffusion process that runs over the non-negative half line. To that end the following setting-which is identical to the setting in [34]-is used. The temporal axis is t ⩾ 0, and differentiation with respect to the temporal variable t is denoted · (·) = ∂ ∂t (·). The spatial axis is x ⩾ 0, and differentiation with respect to the spatial variable x is denoted (·) ′ = ∂ ∂x (·). The position of the diffusion process at the time point t is X (t), and the dynamics of the diffusion process are governed by the Ito stochastic differential equatioṅ [36]. The functions V (x) and D (x) that appear in equation (1) manifest, respectively: a position-dependent velocity, which is real valued; and a position-dependent diffusion coefficient, which is positive valued. The source of randomness that 'drives' the diffusion dynamics is a Wiener process W (t), a.k.a. Brownian motion [78,79]; the temporal derivativeẆ (t) of the Wiener process is Gaussian white noise. Initiating the diffusion process from the positive level X (0) = l, the focus is henceforth set on the FPT to the origin (2) [2]. Namely, T (l) is the time it takes the diffusion process to get from the positive level l to the spatial origin 0. As diffusion processes have continuous trajectories [31], the FPT T (l) is well defined indeed. The FPT's distribution function is denoted P (t; l) = Pr[T (l) ⩽ t]. Namely, P (t; l) is the probability that-initiating at the temporal origin from the level l-the diffusion process will reach the spatial origin up to time t. The distribution function P (t; l) is governed by the backward Fokker-Planck-a.k.a. the forward Kolmogorov-partial differential equatioṅ [32]. The boundary conditions of this partial differential equation are: P (0; l) = 0 for all l > 0 (as it takes the diffusion process a positive time to get from the positive level l to the spatial origin); and P (t; 0) = 1 for all t ⩾ 0 (as, trivially, T (0) = 0 with probability one). Last, along this paper a scaling function ϕ (u) (u ⩾ 0) is a smooth function that satisfies the following properties: it is monotone increasing from ϕ (0) = 0 to ϕ (∞) = ∞, and it passes through the point ϕ (1) = 1. Perhaps the simplest example of a scaling function is the linear function ϕ (u) = u. In turn, the linear scaling function is a special case of the power scaling function: ϕ (u) = u ϵ , where ϵ is a positive exponent.

Motivation
The source of randomness that 'drives' the diffusion dynamics-Brownian motion-displays a host of profound statistical properties [79]. Two principal such properties, and the motivations that they provide to the exploration of selfsimilar FPTs, shall now be discussed.
Brownian motion is a selfsimilar process [41], i.e. its trajectory is a fractal object that is invariant with respect to a certain spatio-temporal scale transformation. The selfsimilarity of Brownian motion induces the following intrinsic scaling of the Brownian-motion positions: (1), where the equality is in law, and where the scaling function is ϕ (t) = √ t.
Namely, the Brownian-motion position at time t is a scaled version of the Brownian-motion position at time 1. Shifting from Brownian motion to the above FPTs, the following scaling property was introduced and explored in [34]: where the equality is in law, and where ϕ 1 (l) is a general scaling function. The scaling property of equation (4) was termed stochastic selfsimilarity. Brownian motion is also a Levy process [80], i.e. its increments are stationary random variables, and its non-overlapping increments are independent random variables. Consequently, the Brownian-motion position at an integer time t is the sum of t IID copies of the Brownianmotion position at time 1, and hence: where θ is a real Fourier variable. In fact, this Fourier-transform formula holds for all times t (rather than for only integer times t).
Shifting from Brownian motion to the above FPTs, and from the Fourier transform to the Laplace transform, the following Laplace log-scaling property was introduced and explored in [34]: The log-scaling property of equation (5) was termed Laplace selfsimilarity. For levels l such that ϕ 2 (l) is an integer, Laplace selfsimilarity has the following 'integer meaning': the FPT T (l) is equal, in law, to the sum of ϕ 2 (l) independent copies of the FPT T (1). Levy processes evolve in two markedly different ways [80]. One way is continuously, and the only Levy process with a continuous trajectory is Brownian motion. The other way is via jumps, and there is a multitude of Levy processes with pure-jump trajectories (see, for example, [81]). Given a Levy process that evolves via positive jumps, set M (t) to be the maximal jump up to time t. For an integer time t, the random variable M (t) is the maximum of t IID copies of the random variable M (1), and hence: Pr[M (t) ⩽ x] = Pr[M (1) ⩽ x] t . In fact, this distributionfunction formula holds for all times t (rather than for only integer times t).
Shifting from Levy processes with positive jumps to the above FPTs, the following distributional log-scaling property shall be explored here: The log-scaling property of equation (6) is termed hazard selfsimilarity. For levels l such that ϕ (l) is an integer, hazard selfsimilarity has the following 'integer meaning': the FPT T (l) is equal, in law, to the maximum of ϕ (l) independent copies of the FPT T (1). Equations (4)-(6) manifest markedly different scaling structures regarding the FPT T (l). The stochastic selfsimilarity of equation (4) manifests scaling in 'real space'. The Laplace selfsimilarity of equation (5) manifests scaling in 'Laplace space'; this scaling is akin, in structure, to the notion of ultra diffusion in the context of random motions [82]. The hazard selfsimilarity of equation (6) manifests scaling in 'hazard space'; this scaling is based on the notion of the backward hazard rate. Hazard rates-a.k.a. failure rates-are widely applied in survival analysis [83][84][85] and in reliability engineering [86][87][88]. The notion of the backward hazard rate shall be described in section 4.2 below, and a backward-hazard-rate formulation of equation (6) shall be presented in equation (8) below.
Appearing in the Laplace selfsimilarity of equation (5), the function L(θ) is a logarithmic representation of the Laplace transform of the FPT T (1). Analogously, appearing in the hazard selfsimilarity of equation (6), the function R (t) is a logarithmic representation of the distribution function of the FPT T (1). As shall be explained in section 4.2 below, the negative gradient −Ṙ (t) of the function R (t) is the backward hazard rate of the FPT T (1).
The structural resemblance between the Laplace selfsimilarity and the hazard selfsimilarity-manifested by the right-hand sides of equations (5) and (6)-is self-evident. The probabilistic-interpretation resemblance between the Laplace selfsimilarity and the hazard selfsimilarity-manifested by their 'integer meanings'-is also self-evident. As Levy processes facilitate the extension of random walks from a discrete temporal axis (t = 0, 1, 2, . . .) to a continuous temporal axis (t ⩾ 0), equations (5) and (6) facilitate the extension of the discrete probabilistic interpretations-of Laplace selfsimilarity and of hazard selfsimilarity-to continuous selfsimilarity structures. Stochastic selfsimilarity and Laplace selfsimilarity were explored in [34]; hazard selfsimilarity shall be explored here.

Hazard-selfsimilar FPTs
As set above, P (t; l) = Pr[T (l) ⩽ t] denotes the distribution function of the FPT T (l). Substituting equation (6) into equation (3), a calculation yields the following differential equatioṅ The derivation of equation (7) is detailed in section 10.
With regard to the function R (t), equation (7) manifests an ordinary differential equation in the temporal variable t. Hence, the terms appearing in square brackets in equation (7) must be independent of the spatial variable l. Specifically, as indicated in equation (7): the coefficient of the term R (t) is a constant α, which is a real number; and the coefficient of the term R (t) 2 is a constant β, which is a positive number. So, the ordinary differential equation (7) admits the formulationṘ (t) = R (t) [α − βR (t)].
As noted above, the boundary conditions of the partial differential equation (3) assert that P (0; l) = 0 (for all l > 0). In turn, equation (6) implies the boundary condition R (0) = ∞ for the function R (t). With this boundary condition satisfied, Table 1 specifies the solution R (t) of the ordinary differential equation (7); the derivation of the solution is detailed in section 10. Table 1 also specifies the solution's asymptotic behavior in the temporal limit t → ∞. The solution's parameters are the constants indicated in equation (7): the real number α, and the positive number β. As evident from Table 1, the sign of the real parameter α has a dramatic effect on the solution R (t), and this solution displays a phase transition as the parameter α changes its sign (from negative to positive, and vice-versa).

First-passage statistics: distribution, phase transition, and randomness
The solution R (t) detailed in Table 1 yields the distribution function of equation (6). This distribution function 'inherits' the solution's parameters, α and β, as well as the solution's dependence on the sign of the parameter α. Namely, the sign of the real parameter α has a Table 1. Depending on the sign of the parameter α, the table specifies the following information. Solution column: the solution R (t) to the ordinary differential equatioṅ , with the boundary condition R (0) = ∞. Asymptotics column: the solution's asymptotic behavior in the temporal limit t → ∞ (in this column ∼ denotes asymptotic equivalence).

Parameter
Solution dramatic effect on the statistics of the FPT T (l), and these statistics exhibit a phase transition at the zero parameter value α = 0. Induced by the asymptotic behavior that is detailed in Table 1, the dramatically different statistical behaviors of the FPT T (l) are as follows. Commonly, the inherent randomness of random variables is gauged in an analog way, i.e. via continuous measures such as the standard deviation and the Shannon entropy. As proposed by Mandelbrot [89], the inherent randomness can also be gauged in a digital way, i.e. via discrete categorizations. Based on the convergence/divergence of moments and of moment generating functions, such a categorization was presented in [90]. Analogously to the Saffir-Simpson digital scale for hurricanes, and to the DEFCON digital scale for defense readiness, this categorization comprises five escalating degrees of randomness [90]: #1 infra mild; #2 mild; #3 borderline; #4 wild; #5 ultra wild.
According to the categorization of [90], the inherent randomness of the FPT T (l) is determined by the parameter α as follows: ultra-wild when α > 0; wild when α = 0; and mild when α < 0. In effect, the parameter α is a quantitative gauge of the inherent randomness of the FPT T (l). Indeed, on the one hand: the more positive the parameter α-the greater the probability Pr[T (l) = ∞], and hence the 'wilder' the FPT's tail behavior. And, on the other hand: the more negative the parameter α-the greater the radius of convergence |α| of the FPT's moment generating function, and hence the 'milder' the FPT's tail behavior.

First-passage statistics: density, backward hazard rate, and hazard-selfsimilarity
The density function of the FPT T (l) is the temporal derivative of its distribution function, Depending on the value of the parameter α, the table specifies the following statistics of the FPT T (l): its backward hazard rateṖ (t; l) /P (t; l); its mode t mode (l); and its quantile function Q (u; l). With regard to the quantile function, the admissible values of the quantile variable u are as follows: Hence, the ratio of the density function to the distribution function isṖ (t; l) /P (t; l) = −ϕ (l)Ṙ (t). This ratio manifests the backward hazard rate of the FPT T (l), which is specified in Table 2.
The FPT's backward hazard rate has the following probabilistic meaning: it is the likelihood that the FPT be realized at the time point t, given the information that it was not realized after the time point t. In other words, setting the time point t to be the 'present', the backward hazard rate is: the likelihood that the FPT occurs now-at the 'present'-given that it does not occur in the 'future' 1 .
With the notion of the backward hazard rate at hand, the reason why the log-scaling property of equation (6) was termed hazard selfsimilarity is now evident. Indeed, equation (6) admits the following backward-hazard-rate formulation: Namely, hazard selfsimilarity manifests scaling in 'hazard space': the backward hazard rate of the FPT T (l) (the ratio appearing on the left-hand side of equation (8)) is a scaled version of the backward hazard rate of the FPT T (1) (the ratio appearing on the right-hand side of equation (8)). The distribution function P (t; l) and the backward hazard rateṖ (t; l) /P (t; l) imply the following pair of facts regarding the density functionṖ (t; l) of the FPT T (l). Firstly, this density function vanishes at the temporal origin, as well as at the temporal infinity:Ṗ (0; l) = 0 anḋ P (∞; l) = 0. Secondly, the temporal shape of this density function is unimodal.
Specifically, with respect to the temporal variable t, the density functionṖ (t; l) displays the following shape: it is monotone increasing in the temporal range 0 < t < t mode (l); and it is monotone decreasing in the temporal range t mode (l) < t < ∞. Thus, over the positive temporal range 0 < t < ∞, the maximal value of the density functionṖ (t; l) is attained at the time point t mode (l)-the mode of the FPT T (l), which is specified in Table 2.
In addition to the backward hazard rateṖ (t; l) /P (t; l) and to the mode t mode (l), Table 2 also specifies the quantile function Q (u; l) of the FPT T (l). Namely, the quantile function is the inverse function-with respect to the temporal variable t-of the FPT's distribution function: Pr[T (l) ⩽ Q (u; l)] = u. In particular, the quantile Q( 1 2 ; l) is the FPT's median:

Generative diffusion dynamics
Having analyzed the hazard-selfsimilar FPTs in the previous section, we now turn to analyze the corresponding generative diffusion dynamics, i.e.: the diffusion dynamics that generate diffusion processes with hazard-selfsimilar FPTs. As noted above, diffusion dynamics are governed by: the position-dependent velocity V (x); and the position-dependent diffusion coefficient D (x).
As indicated in equation (7), the generative velocity and diffusion coefficient are coupled to the scaling function ϕ (l). Indeed, the coefficient of the term R (t) in equation (7) couples together the generative velocity, the generative diffusion coefficient, and the scaling functiondoing so via the relation where α is a real number. And, the coefficient of the term R (t) 2 in equation (7) couples the generative diffusion coefficient and These relations imply that-in terms of the scaling function ϕ (l)-the generative velocity and diffusion coefficient admit the following formulations: and Namely, the FPTs display the hazard selfsimilarity of equation (6) if and only if: the velocity is that of equation (9); and the diffusion coefficient is that of equation (10).

Special scaling functions
Using equations (9) and (10), Table 3 details the generative velocities and diffusion coefficients that correspond to four specific scaling functions: linear ϕ (l) = l; quadratic ϕ (l) = l 2 ; power ϕ (l) = l ϵ , where ϵ is a positive exponent; and logarithmic ϕ (l) = ln (1 + sl) / ln (1 + s), where s is a positive scale. As shall be described below, the linear and quadratic scaling functions, as well as a particular logarithmic scaling function, are of special importance. The special importance of the power scaling function shall be described at the end of section 7.2.

Linear scaling.
With regard to given generative diffusion dynamics, consider a stochastic process whose position at the time point t isX (t) = ϕ [X (t)]. Namely, the positions X (t) of the diffusion process are observed via the scaling function ϕ (l) of its hazard-selfsimilar FPTs-thus producing the positionsX (t). A calculation using Ito's formula implies that the stochastic process with positionsX (t) is a diffusion process, and that its diffusion dynamics are as follows: they generate hazard-selfsimilar FPTs with the linear scaling function (see the linear row of table 3). The derivation of this result is detailed in section 10. So, the linear scaling function emerges from any generative diffusion dynamics via the transformation of positions X (t) → ϕ [X (t)]. Table 3. Four specific scaling functions-linear, quadratic, power, and logarithmic. The columns detail these scaling functions, as well as their corresponding velocities and diffusion coefficients. In the Power row: ϵ is a positive exponent. Evidently, the linear and the quadratic scaling functions are special cases of the power scaling function (with respective exponents ϵ = 1 and ϵ = 2). In the Logarithmic row: s is a positive scale; In this case the diffusion dynamics are governed by a Langevin stochastic differential equation [53,54]. Setting D (x) ≡ d in equation (10) yields the ordinary differential equation ϕ ′ (x) 2 = d β ϕ (x). In turn, imposing the scaling-function properties implies that: the resulting solution of this differential equation is the quadratic scaling function (see the quadratic row of table 3). So, the quadratic scaling function characterizes generative Langevin dynamics.

Logarithmic scaling.
Yet another special case of diffusion dynamics is when the velocity vanishes, V (x) ≡ 0. In this case the resulting diffusion process is a martingale [33]. Setting V (x) ≡ 0 in equation (9) yields the ordinary differential equation ϕ ′ ′ (x) = α β ϕ ′ (x) 2 . In turn, imposing the scaling-function properties implies that: the resulting solution of this differential equation is a logarithmic scaling function with scale s = exp(− α β ) − 1 (see the logarithmic row of table 3); this solution is a proper scaling function if and only if the parameter α is negative. So, a particular logarithmic scaling function characterizes generative martingale dynamics.

Generative Langevin dynamics
Consider two different generative diffusion dynamics: one-as above-with positions X (t), and with scaling function ϕ (l); the other with positions X * (t), and with scaling function ϕ * (l). In turn, the stochastic processes whose positions areX (t) = ϕ [X (t)] andX * (t) = ϕ * [X * (t)] are identical in law. Indeed, as argued above, these stochastic processes are identical diffusion processes: they are both governed by the diffusion dynamics that generate hazard-selfsimilar FPTs with the linear scaling function (see the linear row of table 3). Consequently, the former diffusion process (with positions X (t)) can be generated from the latter diffusion process (with positions X * (t)) via the following transformation of positions: So, from any given diffusion process with hazard-selfsimilar FPTs we can transit to any other such diffusion process. In particular, we can set the 'base' diffusion process-the one with positions X * (t)-to be: the diffusion process that generates hazard-selfsimilar FPTs with the quadratic scaling function (see the quadratic row of table 3). As shown above, the generative diffusion dynamics of this 'base' diffusion process are Langevin [53,54]: their diffusion coefficient is constant, D * (x) = d (where d is a positive number); and, in turn, their velocity x echoes the statistical phase transition of the hazard-selfsimilar FPTs. On the one hand, in the parameter range α > 0 the generative Langevin velocity is monotone increasing from V * (0) = −∞ to V * (∞) = ∞, and it passes from negative values to positive values at the level x * = 2d/α. Consequently, below the level x * the generative Langevin velocity is negative, and hence it 'pushes' towards the spatial origin. And, above the level x * the generative Langevin velocity is positive, and hence it 'pushes' towards infinity; this 'push-to-infinity' causes the FPT T (l) to be infinite with a positive probability. On the other hand, in the parameter range α ⩽ 0 the generative Langevin velocity is always negative, and hence it always 'pushes' towards the spatial origin; this 'always-negative velocity' causes the FPT T (l) to be finite with probability one.
Langevin dynamics are often represented via their underlying potential-a function whose negative gradient is the velocity [53,54]. Up to an additive constant, corresponding to the generative Langevin velocity V * (x) is the following generative Langevin potential: Evidently, this Langevin potential is composed of two components-one quadratic, and one logarithmic.
The quadratic component −α · x 2 /4 of the generative Langevin potential manifests the potential's asymptotic behavior in the spatial limit x → ∞. The coefficient of the quadratic component, −α, is free to assume any real value. Langevin dynamics with a positive quadratic potential generate the Ornstein-Uhlenbeck diffusion process [55][56][57].
The logarithmic component d · ln (x) of the generative Langevin potential manifests the potential's asymptotic behavior in the spatial limit x → 0. The coefficient of the logarithmic component, d, is specific-it is the constant diffusion coefficient of the generative Langevin dynamics. Langevin dynamics with a logarithmic potential attracted significant interest in the recent years [58][59][60][61][62][63][64][65][66][67][68]. At the statistical phase transition of the FPTs-the zero parameter value α = 0-the generative Langevin potential reduces to its logarithmic component, The generative Langevin potential also echoes the statistical phase transition of the hazardselfsimilar FPT. On the one hand, in the parameter range α > 0 the generative Langevin potential has an inverted U shape: it is monotone increasing from U * (0) = −∞ over the range 0 < x < x * ; and it is monotone decreasing over the range x * < x < ∞ to U * (∞) = −∞. On the other hand, in the parameter range α ⩽ 0 the generative Langevin potential has a funnel shape: it is monotone increasing from U * (0) = −∞ to U * (∞) = ∞.

Discussion
In this section we present several remarks regarding the hazard-selfsimilar FPTs, and their generative diffusion dynamics. The remarks are with regard to the different statistical behaviors of the FPTs: the regime of mild randomness, characterized by the negative parameter range α < 0; the regime of ultra-wild randomness, characterized by the positive parameter range α > 0; and the phase transition between the two regimes, the zero parameter value α = 0. Table 4. Logarithmic representations for three statistical distributions: that of the hazard-selfsimilar FPTs, with a negative parameter α < 0; the Gumbel extreme-value law; and the Gompertz law. In the table F (u) andF (u) = 1 − F (u) denote, respectively, distribution and survival functions. Also, in the table, c is a positive coefficient and s is a positive scale. Table 4 provides logarithmic representations for three statistical distributions. The first is for the statistical distribution of the hazard-selfsimilar FPTs, with a negative parameter α < 0. The second is for the Gumbel extreme-value law, and the third is for the Gompertz law. The Gumbel and Gompertz laws are discussed below, as well as the relations of the hazard-selfsimilar FPTs (with α < 0) to these laws. Extreme value theory establishes three universal laws for the extremes-the maxima and the minima-of collections of independent and identically distributed random variables [49][50][51][52]. One of the three universal laws is Gumbel, and the logarithmic representation of the Gumbel law for maxima is specified in Table 4. On the one hand, the Gumbel law is supported on the entire real line. On the other hand, the statistical distribution of the hazard-selfsimilar FPTs (with α < 0) is supported on the positive half-line. As evident from the logarithmic representations of Table 4, the latter statistical distribution 'adjusts' the underpinning Gumbel structure as follows: it replaces the Gumbel term exp(su) (which is defined over the range −∞ < u < ∞) by the term [exp(su) − 1] (which is defined over the range 0 < u < ∞). 2 So, the statistical distribution of the hazard-selfsimilar FPTs (with α < 0) can be perceived as an 'adjusted' Gumbel law.

Mild randomness
The Gumbel law for minima is, in effect, a 'mirroring' of the Gumbel law for maxima. Conditioning the Gumbel law for minima to have positive values, it yields the Gompertz law [91]. This law is used by demographers and by actuaries as a statistical model for the lifespans of adults [92][93][94][95]; it emerges via branching search [96]; and it emerges via the stochastic 'Moore clock' [97]. The logarithmic representation of the Gompertz law is specified in Table 4. On the one hand, the logarithmic representation of the Gompertz law is based on its survival function. On the other hand, the logarithmic representation of the statistical distribution of the hazard-selfsimilar FPTs (with α < 0) is based on their distribution function. As evident from the logarithmic representations of Table 4, the latter statistical distribution replaces the Gompertz term [exp(su) − 1] by its reciprocal, the term 1/[exp(su) − 1] (both these terms are defined over the range 0 < u < ∞). So, the statistical distribution of the hazard-selfsimilar FPTs (with α < 0) can be perceived as an 'Inverse Gompertz' law.

Ultra-wild randomness
The statistical distribution of the hazard-selfsimilar FPTs-with a positive parameter α > 0has an atom at infinity. Namely, as shown above, these hazard-selfsimilar FPTs (with α > 0) are infinite with a positive probability. Conditioning the hazard-selfsimilar FPTs (with α > 0) to be finite has an interesting effect on their statistical distribution. Indeed, given the information {T (l) < ∞}, the conditional statistical distribution of the FPT T (l) coincides with the (unconditional) statistical distribution of a 'α -mirrored' FPT: the FPT T (l) with the negative parameter −α. The derivation of this result is detailed in section 10. From the perspective of the generative Langevin dynamics, the conditioning effect has the following 'potential manifestation': replace the generative Langevin potential U * (x) = −|α| · x 2 4 + d · ln (x) (with a negative quadratic component) by the generative Langevin potential U * (x) = |α| · x 2 4 + d · ln (x) (with a positive quadratic component).
A similar conditioning effect takes place in the case of inverse Gauss FPTs [21]. With regard to Laplace-selfsimilar FPTs-which display the Laplace log-scaling property of equation (5)-the two following facts were established in [34]: the universal statistical distribution of the Laplace-selfsimilar FPTs is inverse Gauss; and the generative Langevin dynamics (of Laplace-selfsimilar FPTs) are governed by a linear Langevin potential. From the perspective of these generative Langevin dynamics, the conditioning effect (regarding the universal inverse Gauss distribution of Laplace-selfsimilar FPTs) has the following 'potential manifestation': switch the slope of the linear Langevin potential from negative to positive, while retaining the slope's absolute value.

Phase transition
As noted above, extreme value theory establishes three universal laws for the maxima of collections of independent and identically distributed random variables [49][50][51][52]. One of the three universal laws is Frechet, which is supported on the positive half-line. The distribution function of the Frechet law is F (u) = exp[−1/(su) γ ], where: s is a positive scale; and γ is a positive exponent. It follows from Table 1  The statistical phase transition of the hazard-selfsimilar FPTs has the following characterization: it is the intersection of equations (4) and (6). Indeed, consider the FPT T (l) to satisfy the stochastic selfsimilarity of equation (4), as well as the hazard selfsimilarity of equation (6). Equation (4) implies that P (t; l) = P[t/ϕ 1 (l) ; 1]; in turn, equation (6) further implies the relation ϕ (l) R (t) = R[t/ϕ 1 (l)]. It is evident from Table 1 that this relation holds if and only if R (t) = 1/(βt)-in which case the two scaling functions coincide, ϕ (l) = ϕ 1 (l). Thus, as argued: the zero parameter value α = 0 characterizes the aforementioned 'intersection of selfsimilarities'.

Steady state
So far, the initiation of the diffusion dynamics was deterministic. Namely, the diffusion dynamics were initiated from a fixed positive level l. We shall now shift the diffusion-dynamics initiation from deterministic to stochastic. Specifically, the generative diffusion dynamics (of hazard-selfsimilar FPTs) shall now be initiated from their statistical steady state. This section will explore the effect of steady-state initiation on the resulting FPTs.

Non-normalizable densities
Consider diffusion dynamics that are governed by the Ito stochastic differential equation (1). The statistical steady state of the diffusion dynamics is characterized by the 'zero-flux' solution of the Fokker-Planck partial differential equation that corresponds to equation (1) [98]. Specifically, the zero-flux solution is π Diff (x) = k exp [K (x)] /D (x), where: k is a positive constant; and K (x) is a function whose derivative is the ratio of the velocity to the diffusion coefficient, In general, the zero-flux solution can be either integrable´∞ 0 π Diff (x)dx < ∞, or non-integrable´∞ 0 π Diff (x)dx = ∞. In the integrable case k is set to be a normalizing constant (i.e. such that´∞ 0 π Diff (x)dx = 1), and then π Diff (x) manifests the probabilistic steady-state density of the diffusion dynamics. In the non-integrable case π Diff (x) is a "non-normalizable density"-a topic that attracted significant interest recently [70][71][72][73][74][75].
As established above, the velocity and the diffusion coefficient of the generative diffusion dynamics are given, respectively, by equations (9) and (10). Consequently, the zero-flux solution that corresponds to the generative diffusion dynamics is The derivation of equation (12) is detailed in section 10. When the scaling function is linear, ϕ (l) = l, then equation (12) yields π Diff (x) = k exp( α β x) 1 x ; evidently, this specific zero-flux solution is non-normalizable. Applying the change-of-variables y = ϕ (x) to the zero-flux solution of equation (12) implies that ∞ 0 π Diff (x)dx =´∞ 0 k exp( α β y) 1 y dy. Namely, the area under the (general) zero-flux solution of equation (12) is the same as: the area under the (specific) zero-flux solution that corresponds to the linear scaling function-which is infinite. So, the zero-flux solution of equation (12) is always non-normalizable. Nonetheless, in this subsection we shall treat this solution as if it was a proper density function.
Rather than initiating the generative diffusion dynamics deterministically (from a fixed positive level l), we initiate the dynamics stochastically-doing so from their statistical steady state. To that end, the fixed initial position l shall now be randomized according to the 'density function' π Diff (x) of equation (12). In turn, this randomization yields-formally-the temporal 'density function' π FPTs (t) =´∞ 0Ṗ (t; l) π Diff (l)dl, where: as above,Ṗ (t; l) is the density function of the FPT T (l).
On the one hand, a calculation using equations (6) and (12) implies that π FPTs (t) = kβṘ (t) /[α − βR (t)]. On the other hand, as noted above, the ordinary differential equation (7) admits the formulationṘ (t) = R (t) [α − βR (t)]. Thus, we obtain that The derivation of equation (13) is detailed in section 10. Equation (13) asserts that, up to a multiplicative constant, the temporal 'density function' π FPTs (t) is identical to the function R (t)-which was detailed in Table 1. About the temporal origin the behavior of the function R (t) is asymptotically equivalent to that of the temporal function 1/t. Consequently, as the 'density function' π Diff (x), also the 'density function' π FPTs (t) is non-normalizable.

Poissonian interpretation
As explained above, the function π Diff (x) of equation (12) and the function π FPTs (t) of equation (13) are both non-normalizable densities. Hence, these functions do not have probabilistic interpretations. Yet, these functions do have Poissonian interpretations-to be presented in this subsection.
Consider a positive-valued function λ (u) that is defined over the positive half-line (0 < u < ∞). A random collection of positive points forms a Poisson process-with intensity λ (u)when the two following properties hold [99]. Poisson property: the number of points residing in the interval (a, b) (where 0 < a < b < ∞) is a Poisson random variable with mean´b a λ (u) du. Independence property: the numbers of points residing in disjoint intervals are independent random variables. With the notion of Poisson processes at hand, we are all set to describe the Poissonian interpretations of the above 'density functions': π Diff (x) of equation (12); and π FPTs (t) of equation (13).
Rather than considering a single diffusion process, consider countably many diffusion processes that run in parallel. On the one hand, these processes are mutually independent, and they are all governed by the same generative diffusion dynamics-with the velocity and the diffusion coefficient of equations (9) and (10). On the other hand, the processes are initiated from different levels, and these levels are set as follows: they form a Poisson process with the intensity π Diff (x) of equation (12). In turn, this specific scattering of the initial levels yields the following steady-state behavior [76,77]: at any time point t, the positions of the diffusion processes (that run in parallel) also form a Poisson process with intensity π Diff (x). So, the initial Poissonian scattering-which was set at the temporal origin-is maintained all along the temporal axis. Now, further consider the FPTs, to the spatial origin, of the diffusion processes (that run in parallel). As established above with regard to a diffusion process that initiates from the positive level l, the density function of its FPT isṖ (t; l). Consequently, the displacement theorem of the theory of Poisson processes asserts that [99]: the collection of the FPTs forms a Poisson process with the intensity π FPTs (t) =´∞ 0Ṗ (t; l) π Diff (l)dl. The calculation of the intensity π FPTs (t)which produced equation (13)-was described in the previous subsection.
So, the setting of countably many generative diffusion processes that run in parallel provides a Poissonian interpretation to both the 'density functions' π Diff (x) and π FPTs (t). Indeed, the 'density function' π Diff (x) of equation (12) is a spatial Poissonian intensity that yields a steady-state behavior of the diffusions' positions. And, the 'density function' π FPTs (t) of equation (13) is the temporal Poissonian intensity of the diffusions' hazard-selfsimilar FPTsafter having scattered the diffusions' initial positions according to the Poissonian intensity π Diff (x).
As asserted in Table 1, at the statistical phase transition (α = 0) the function R (t) is harmonic (R (t) = 1/(βt)). In turn, equation (13) implies that: at the statistical phase transition (α = 0) the temporal Poissonian intensity of the diffusions' hazard-selfsimilar FPTs is also harmonic, π FPTs (t) = k/t. Poisson processes that are governed by harmonic intensities display a host of remarkable and unique statistical properties [100].
Power scaling. We end this subsection with delivering on a promise made in section 5.1describing the special importance of the power scaling function: ϕ (l) = l ϵ , where ϵ is a positive exponent (see the power row of Table 3). To that end, consider the statistical phase transition, the zero parameter value α = 0. Setting α = 0 in equation (12) yields the spatial Poissonian intensity π Diff (x) = kϕ ′ (x)/ϕ (x). With regard to this intensity, the following relation holds: ϕ (l) = l ϵ if and only if π Diff (x) = ϵk/x. So-at the statistical phase transition-the power scaling function characterizes a harmonic Poissonian steady state of the generative diffusion processes (that run in parallel).

Minimal and maximal FPTs
Denote by P FPTs the Poisson process of the diffusions' hazard-selfsimilar FPTs, whose intensity is π FPTs (t). In this subsection we shall investigate the minimal and maximal points of the Poisson process P FPTs , i.e.: the first/smallest FPT and the last/largest of the FPT-among the FPTs of the generative diffusion processes that run in parallel.
As noted above, about the temporal origin the behavior of the function R (t) is asymptotically equivalent to that of the temporal function 1/t. Consequently, equation (13) implies that the intensity π FPTs (t) is not integrable at the temporal origin. In turn, this non-integrability implies that: the Poisson process P FPTs has infinitely many points at the vicinity of the temporal origin [99], and hence it does not have a minimal point.
As specified in Table 1, about the temporal infinity the function R (t) displays markedly different asymptotic behaviors. When the real parameter α is either positive or zero (α ⩾ 0) then the function R (t) is not integrable at the temporal infinity-and thus neither is the intensity π FPTs (t). In turn, this non-integrability implies that: the Poisson process P FPTs has infinitely many points at the vicinity of the temporal infinity [99], and hence it does not have a maximal point 3 .
When the real parameter α is negative (α < 0) then the function R (t) is integrable at the temporal infinity, and thus so is the intensity π FPTs (t). In turn, this integrability implies that: the points of the Poisson process P FPTs do not accumulate at the vicinity of the temporal infinity, and hence the process does have a maximal point-which we denote T max .
The maximal FPT T max is no larger than the positive time t if and only if the Poisson process P FPTs has no points in the temporal ray (t, ∞). The latter event occurs with probability exp[−´∞ t π FPTs (τ )dτ ]. Calculating the integral´∞ t π FPTs (τ )dτ , we obtain that the distribution function of the maximal FPT T max is: The derivation of equation (14) is detailed in section 10. The term [1 − exp (−|α|t)] appearing in equation (14) manifests the distribution function of an Exponential random variable with mean 1/|α|. Consequently, when the constant k is an integer then equation (14) has the following probabilistic interpretation: the maximal FPT T max is equal, in law, to the maximum of k independent copies of an exponential random variable with mean 1/|α| . Interestingly, the distributional structure of the maximal FPT T max (with respect to the positive parameter k) is similar to the distributional structure of the hazardselfsimilar FPT T (l) (with respect to the positive level l).
Consider a collection of stochastic motions taking place in some space, and further consider their respective FPTs to a certain target zone. The minimal FPT-the first/smallest of the FPTs-is the time it takes the 'fastest' motion to reach the target. Conversely, the maximal FPT-the last/largest of the FPTs-is the time it takes the 'slowest' motion to reach the target.
Following a path set in [34], this paper addressed maxima of FPTs from an altogether different perspective: selfsimilarity. Specifically, consider a general Ito diffusion that runs over the non-negative half-line. Initiating the diffusion from the positive level l, further consider the diffusion's FPT to the origin T (l). Loosely speaking, the foundational research question of this paper was: when is the FPT T (l) equal, in law, to the maximum of a certain number (ϕ (l)) of independent copies of the FPT T (1)? The precise formulation of the foundational research question was via the notion of hazard selfsimilarity (equation (6)). The analysis established here explored hazard-selfsimilar FPTs, as well as the diffusion dynamics that generate Ito diffusions with such FPTs.
The analysis unveiled and presented the universal statistical distribution of hazardselfsimilar FPTs. This universal distribution was shown to be governed by the function R (t) (detailed in Table 1). Depending on the real parameter α of the function R (t), the universal distribution was shown to display markedly different statistical behaviors: a 'mild' degree of randomness when α is negative; a 'wild' degree of randomness when α is zero; and an 'ultra-wild' degree of randomness when α is positive. The parameter α turned out to be a quantitative gauge of the inherent randomness of the universal distribution: the more negative α-the 'milder' the distribution; and the more positive α-the 'wilder' the distribution In the 'mild' regime (α < 0), the universal distribution is characterized by a logarithmic representation ( Table 4) that is related to the Gumbel extreme-value law, as well as to the Gompertz law. In the 'ultra-wild' regime (α > 0), the universal distribution is as follows: with a positive probability, it yields an infinite value; and, with the complementary probability, it yields statistics that are identical to the 'mild' regime with parameter −|α|. At the phase transition between the two regimes (α = 0), the universal distribution coincides with a particular Frechet extreme-value law-the one who's 'reciprocal' is the exponential law.
The universal distribution, which emerges from the notion of hazard selfsimilarity, has linkages to the two statistical laws that were shown to emerge in [34]: the Inverse-Gauss law-the universal statistical distribution of Laplace-selfsimilar FPTs (equation (5)); and the Inverse-Gamma law-the universal statistical distribution of stochastic-selfsimilar FPTs (equation (4)). Indeed, the finite-infinite interplay between the two regimes (α < 0 and α > 0) of this paper's universal distribution is akin to a finite-infinite interplay of the inverse-Gauss law-termed 'first-passage duality' in [21]. And, the phase transition of this paper's universal distribution-with its particular Frechet law-is the intersection of two selfsimilarities: hazard selfsimilarity (equation (6)) and stochastic selfsimilarity (equation (4)).
Beyond the universal distribution, the analysis here also pinpointed the class of the generative diffusion dynamics. Namely, the analysis unveiled and presented the diffusion dynamicscharacterized by specific velocities and diffusion coefficients (equations (9) and (10))-of Ito diffusions with hazard-selfsimilar FPTs. In particular, the analysis pinpointed the generative Langevin dynamics, whose diffusion coefficient is constant. Moreover, the analysis asserted that any generative diffusion dynamics can be produced from the generative Langevin dynamics via a transformation of diffusion positions.
The generative Langevin dynamics were shown to be characterized by a 'composite' Langevin potential function comprising of two components (equation (11)): a quadratic component with the real coefficient α; and a logarithmic component whose coefficient is the (constant) diffusion coefficient of the generative Langevin dynamics. The quadratic component dominates the generative Langevin dynamics at large values, and the logarithmic component dominates the generative Langevin dynamics at small values. The different statistical behaviors of the universal distribution are related to the quadratic component of the Langevin potential as follows: the 'mild' regime (α < 0) is characterized by a '∪ shape' of this component; the 'ultra-wild' regime (α > 0) is characterized by a'∩ shape' of this component; and at the phase transition (α = 0) this component vanishes.
Addressing the statistical steady state of the generative diffusion dynamics gave rise to a 'non-normalizable density' (equation (12)). Consequently, the generative diffusion dynamics do not have a steady state in a probabilistic sense. Nonetheless, they do have a steady state in a Poissonian sense-in which countably many generative diffusion dynamics run in parallel. Using the Poissonian steady-state interpretation, and considering the FPTs of the countably many diffusions, the analysis showed that: these FPTs form a universal Poisson process with an intensity that, up to a multiplicative constant, is equal to the function R (t) (detailed in Table 1).
Depending on the real parameter α of the function R (t), the universal Poisson process of FPTs was also shown to display markedly different behaviors. In the 'mild' regime (α < 0), the universal Poisson process has a maximal FPT that-loosely speaking-is equal in law to: the maximum of a certain number of independent random durations governed by an Exponential law with mean 1/|α| (equation (14)). In the 'ultra-wild' regime (α > 0), the universal Poisson process has infinitely large FPTs, and hence it does not have a maximal FPT. At the phase transition (α = 0), the universal Poisson process is the harmonic Poisson process [100], and it does not have a maximal FPT.

Epilogue
As noted above, the study of extreme FPTs addresses both minima and maxima. Heredue to the max-based definition of hazard selfsimilarity (equation (6))-only maxima were addressed. This arises the question: can a min-based definition of hazard selfsimilarity be devised? And if the answer is affirmative, then the follow-up question is: are there non-trivial FPTs that display min-based hazard selfsimilarity? As shall be argued below, the answer to the question is positive, whereas the answer to the follow-up question is negative.
A min-based definition of hazard selfsimilarity is The left-hand side of equation (15) is the survival function of the FPT T (l). The functionφ (l) appearing on the right-hand side of equation (15) is an inverted scale function,φ (l) = 1/ϕ (l). And, the functionR (t) appearing on the right-hand side of equation (15)  Equation (15) is a 'minima counter-part' of equation (6). For levels l such that ϕ (l) is an integer, equation (15) has the following 'integer meaning': the FPT T (1) is equal, in law, to the minimum of ϕ (l) independent copies of the FPT T (l).
In terms of the distribution function P (t; l) = Pr[T(l) ⩽ t] and of the corresponding forward hazard rate (see footnote #1), equation (15) admits the formulation . (16) Equation (16) is a 'minima counter-part' of equation (8). As equation (8), also equation (16) manifests scaling in 'hazard space'. Namely, equation (16) has the following meaning: the forward hazard rate of the FPT T (l) (the ratio appearing on the left-hand side of equation (16)) is a scaled version of the forward hazard rate of the FPT T (1) (the ratio appearing on the right-hand side of equation (16)). Analyzing equation (15) analogously to the analysis of equation (6)-which was carried out in section 4-yields no meaningful solution. Indeed, solving for the functionR (t) of equation (15) is analogous to solving for the function R (t) of equation (6). However, while the boundary condition of the function R (t) is R (0) = ∞, the boundary condition of the func-tionR (t) isR (0) = 0. This boundary-condition difference has profound consequences. On the one hand, solving for the function R (t) yields the non-trivial solution that was detailed in Table 1; and this solution induces a rich statistical structure which was investigated along this paper. On the other hand, solving for the functionR (t) yields the trivial solutionR (t) ≡ 0; and this solution implies that the FPT T (l) is infinite with probability one, Pr[T(l) = ∞] = 1.
So, the notion of hazard selfsimilarity yields non-trivial FPTs via the max-based definition of equation (6), and yields trivial FPTs via the min-based definition of equation (15). This is the reason why only the former definition of hazard selfsimilarity was addressed and explored in this paper. (7) Consider the distribution function

Derivation of equation
Differentiating this distribution function with respect to its temporal variable t yieldṡ Differentiating this distribution function with respect to its level variable l yields and Substituting the derivatives of equations (18)-(20) into the partial differential equation (3) implies that In turn, equation (21) implies equation (7). (7) The ordinary differential equation (7) implies thaṫ

Solution of equation
As noted above, α is a real constant, and β is a positive constant. Consider the case α = 0. In this case equation (22) can be written as followṡ Integrating equation (23) implies that where c 0 is a real integration constant. In turn, it follows from equation (24) that where c = exp(−c 0 ). The boundary condition R (0) = ∞ implies that c = β, and hence we obtain that Consider the case α = 0. In this case equation (22) can be written as follows Integrating equation (27) implies that 1 R (t) = βt + c 0 , where c 0 is a real integration constant. The boundary condition R (0) = ∞ implies that c 0 = 0, and hence we obtain that Note that equation (29) follows from equation (26) via the limit α → 0.

Linear scaling
Consider a stochastic process whose positions are Y (t) = ϕ [X (t)]. Combined together, the Ito stochastic differential equation (1) and Ito's formula [36] imply thaṫ As argued above-for generative diffusion dynamics-the following couplings hold. The generative velocity, the generative diffusion coefficient, and the scaling function are coupled via the following relations: V (l) ϕ ′ (l) + D (l) ϕ ′ ′ (l) = αϕ (l), where α is a real number; and D (l) ϕ ′ (l) 2 = βϕ (l), where β is a positive number. Substituting these relations into equation (30) yieldṡ In turn, as Y (t) = ϕ [X (t)], equation (31) implies thaṫ So, the stochastic process whose positions are Y (t) = ϕ [X (t)] is a diffusion process with the following diffusion dynamics: the linear velocity V (x) = αx; and the linear diffusion coefficient D (x) = βx.

Derivation of equation (14)
Considering the real parameter α to be negative (α < 0), we calculate the probability that the Poisson process P FPTs has no points in the temporal ray (t, ∞). As the intensity of the Poisson process P FPTs is π FPTs (t), the probability is: So, in order to calculate the probability of equation (46), we need to calculate the integral appearing in equation (46). Using equation (13) and Table 1, note that: In turn, taking the limit b → ∞ implies that ∞ a 1 u (u + 1) du = ln a + 1 a .

Data availability statement
No new data were created or analyzed in this study.