Sign-changes as a universal concept in first-passage time calculations

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by  Wilhelm Braun

Sign changes as a universal concept in first-passage time calculations
Wilhelm Braun1,∗ and Rüdiger Thul2
1Department of Physics, University of Ottawa, Ottawa, Canada K1N 6N5
2Centre for Mathematical Medicine and Biology, School of Mathematical Sciences,
University of Nottingham, Nottingham, NG7 2RD, UK
(Dated: January 4, 2017)
First-passage time problems are ubiquitous across many fields of study including transport pro-
cesses in semiconductors and biological synapses, evolutionary game theory and percolation. De-
spite their prominence, first-passage time calculations have proven to be particularly challenging.
Analytical results to date have often been obtained under strong conditions, leaving most of the
exploration of first-passage time problems to direct numerical computations. Here we present an
analytical approach that allows the derivation of first-passage time distributions for the wide class
of non-differentiable Gaussian processes. We demonstrate that the concept of sign changes natu-
7
rally generalises the common practice of counting crossings to determine first-passage events. Our
1
methodworksacrossawiderangeoftime-dependentboundariesandnoisestrengthsthusalleviating
0
common hurdles in first-passage time calculations.
2

n
a I. INTRODUCTION based on Brownian motion [39], most notably the ubiq-
J
uitousOrnstein-Uhlenbeckprocess(OUP)andgeometric

3 A frequent question in the study of stochastic systems Brownian motion.

is to determine when a particular event happens for the A distinct advantage of the Wiener–Rice approach is
]
C first time. Such first-passage time (FPT) problems are thatitoftengivesrisetocompactanalyticalexpressions,
N ubiquitous in the physical sciences, and applications in- which in turn provide great insight into the stochastic
cludetrappingreactions[1],earthquakes[2],goodness-of- systemunderinvestigation, ase.g. illustratedin[40,41].
.
o fit tests [3], habitat selection [4], dark matter halos [5], In this paper, we show how the concepts that under-
bi decisionmaking[6,7],diffusionincomplexenvironments lie the Wiener–Rice series can be generalised to non-
- [8, 9] and ion channel dynamics [10–12]. For a recent re- differentiable stochastic processes with arbitrary bound-
q view on the wider applications of FPTs see [13, 14]. Al- aries, thus translating the versatility and efficacy of the
[
though the concept of FPTs is easily stated, the actual Wiener–Rice approach to a much enhanced range of

1 computation poses substantial challenges. Researchers physically relevant dynamics.
v
cannowdrawonanextendedsuiteofmathematicaltech-
8
niquestoevaluateFPTs,rangingfromanalyticaldescrip-
4
tionsforspecialcasesincludingasymptoticexpansionsto II. SIGN CHANGING PROBABILITIES
6
0 semi-analytical approaches based on integral equations
0 to direct numerical simulations [15–31]. Notwithstand- Consider a stationary stochastic process X(t) and a
. ing these practical methods, the general solution to one time-dependent boundary S(t). Let N([0,t]) denote the
1
ofthemostfundamentalFPTproblemsisstillunknown: number of crossings of X(t) and S(t) in a fixed inter-
0
7 diffusion to an arbitrary boundary [24, 32–34]. This is val [0,t]. If X(t) is differentiable, the mean number of
1 even more remarkable since diffusion processes are often crossings of X(t) through S(t) in any given finite time
: used to approximate the dynamics of more complicated interval is always finite, i.e. E N([0,t]) < [36, 37].
v
{ } ∞
i stochastic systems [16, 35]. This entails that we can count the number of crossings,
X
The seminal work by Wiener and Rice [36, 37] has andinparticularthatthereisatmostonecrossingwithin
r been instrumental in advancing our understanding of a sufficiently small time interval of length ∆. When we
a
FPT problems. They derived a series representation of introduce the new process Z(t) = X(t) S(t), we see
−
theFPTdistributionforasufficientlysmoothstationary that Z(t) changes sign as soon as X(t) crosses through
stochastic process through a constant boundary. A com- S(t), and hence instead of counting crossings, we could
mon interpretation of this smoothness condition is that count sign changes. If X(t) is non-differentiable, Rice’s
thestochasticprocessisdifferentiableinthemean-square seminal work shows that E N([0,t]) = . While this
{ } ∞
sense. For a stationary stochastic process, differentiabil- seemscounterintuitive,infinitemeansarenotuncommon
ityarisesfromthebehaviourofitscovariancefunctionat in physics. A prominent example are power-law proba-
the origin. If it is differentiable there, the process is dif- bility distributions of the form p(x) x−α, α > 0 [42]
ferentiable in the mean square sense [38]. Interestingly, [43]. The divergence of E N([0,t]) d∼oes not allow us to
{ }
thisconditionisalreadyviolatedforanydiffusionprocess count the number of crossings or sign changes anymore.
In particular, the divergence holds for any t > 0, so we
cannot choose an infinitesimal time interval ∆ to over-
come it. Inturn, this preventsus from usingthe original
∗ [email protected] results by Wiener and Rice.
2
Progresscanbemadeherebyachangeofperspective.
We generally identify a crossing event in some interval
15 (a)
[t,t+∆] with Z(t) and Z(t+∆) having opposite signs,
X
i.e. we evaluate Z at the end points of the interval. For
sufficiently small ∆ and a differentiable stochastic pro- 10
cess, this is equivalent of a crossing anytime during the
interval and hence justifies the practice of evaluating Z 5 S
attheendpoints. Foranon-differentiablestochasticpro-
cess, we can still evaluate Z(t) and Z(t+∆) and check
0
for sign changes. The only difference is that this does
not inform us about the number of crossings, but about
0 2 4 6 8 10 12 14
theexistenceofat least onecrossing. Givenasufficiently
t
small ∆, this is the key information for practical calcu-
lations. Sign changes of Z(t) and Z(t+∆) can therefore
beusedtoidentifyFPTsforbothdifferentiableandnon- 5 (b) X
differentiable stochastic processes.
The versatility of sign changes in computing FPTs is 4
bestillustratedwithcasesthatcannotbesolvedwithex-
3
istinganalyticaltechniques,yetfrequentlyoccurinappli- S
cations. Figure 1 shows prototypical examples to which
2
current approaches cannot be applied since the bound-
aries are neither convex nor concave. These boundaries 1
can be found in areas as diverse as neuronal dynamics
0
(Fig. 1a) and molecular transition theory (Fig. 1b). It 0 5 10 15 20
is worth pointing out that the boundary in Fig. 1a orig- t
inates from transforming a FPT problem for a driven
stochastic system with constant threshold. In general, FIG. 1. An OUP (solid black line) with mean v (dashed
these problems cannot be solved analytically due to the black line) crosses through time dependent boundaries S(t)
explicittime-dependenceofthestochasticdynamics[44]. (green dashed line). The boundaries are (a) S(t) =
However, by shifting the time-dependence to the thresh- −5cos(t)exp(cid:0)−(0.1cos(t))2(cid:1)+tand(b)S(t)=(cid:80) α (2(t−
n n
old as we do here, the problem becomes analytically T ))4exp(−2(t−T ))H(t−T ) with α ∈ {0.8,0.75}, T ∈
n n n
tractable. The example in Fig. 1b illustrates motion {4,10}. Parameter values are (a) τ =1.0, v =10.0, D =2.0
in a complex energy landscape with a time-dependent and (b) τ =2.0, v =4.0, D =0.2. H denotes the Heaviside
metastable state [45, 46]. While this is most reminiscent step function.
of problems in physical chemistry, the same scenario oc-
curs in general phase space dynamics [47].
tional bivariate probability function of X(t), we see that
The specific FPT problem that we will investigate
is to determine when a Gaussian process X(t), whose
(cid:90) ∞ (cid:90) S(t+∆)
initial value is drawn from its stationary distribution, p (tx )= dx dx f (x ,x x ). (2)
+− 0 1 2 2 1 2 0
crosses through S(t) from above for the first time, | |
S(t) −∞
i.e. we are interested in the random variable T =
inf t>0X(t) S(t) with a corresponding probability Note that Eq. (2) is always well defined and does not
dist{ributio|n (≤t). W}hile the theory presented here is rely on whether X(t) is differentiable or not. Since X(t)
valid for anyFGaussian process, we will illustrate our re- isGaussian, f2 isGaussianagain, andthecorresponding
sults with an OUP given its prominence across so many mean and covariance matrix can be obtained in closed
fields of study. The dynamics of theOUP is governed by form. By expanding the integrals in Eq. (2) to lowest
orders in ∆ following [48], we find
X v
X. = − dt+√DdW(t), (1) (cid:115)
− τ 1 ∆/τ (cid:20)[S(t) m (t)]2(cid:21)
p (tx )= exp − 1 , (3)
whereW(t)isastandardBrownianmotion,visthemean +− | 0 π 2[1−ρ(t)2] 2σ12(t)
of the OUP and τ, D > 0. The stationary variance and
ecoxrpr(elatti/oτn)c,oreeffispceicetnitveolfyt.heOUParegivenbyDτ/2and wanhderρe(mt)1a(tn)d=σ2v +den(xo0te−tvh)eρ(cto)r,reσl12a(tti)on=cσoe2ffi(1c−ienρt(ta)2n)d,
To−b|e|gin our analysis, we define the conditional sign the variance of X(t), respectively. Hence, p+−(tx0) =
change probability p (tx ) = P(Z(t) > 0,Z(t+∆) < σ(cid:112)∆/(πτ)f (S(t)x ), where f presents a cond|itional
+− 0 1 0 1
| |
0x ), i.e. the probability of a sign change in [t,t+∆] univariate Gaussian probability function. The value of
0
|
given a value of X(0) = x . Let f (x ,x x ) = x has been fixed but arbitrary so far. By integrating it
0 2 1 2 0 0
P(X(t ) = x ,X(t ) = x X(0) = x ) denote the|condi- outweobtaintheprobability (t)=(cid:82)∞p (tx)p(x)dx
1 1 2 2| 0 I1 0 +− |
3
(a) 10−2 (b)
(a) ✶✵✦✂
✭✄✮✶✵✦✁ (t)F10−3
10−3 ❋ 10−4
✶✵✦(cid:0)
t) 10−5
(1I ✷✳✵ ✷✳✺ ✸✳✵t ✸✳✺ ✹✳✵ 7.5 8.0 t 8.5 9.0
10−4 (c) (d)
10−3
(t)F10−4 (t)F10−3
10−5
2 4 6 8 10 12 14 10−5 10−4
t
5.5 6.0 6.5 7.0 11.0 11.5 12.0 12.5 13.0
t t
FIG. 3. FPT probability F(t) corresponding to Fig. 1a (a,b)
(b)
and Fig. 1b (c,d) obtained from MC simulations (solid grey
10−3
line), Eq. 4 (dashed green line) and Eq. 9 (green circles).
)
t
(
1I10−4 where (t) = (cid:80)∞ q (t)(k 1)!/(k n)! and q (t)
Jn k=n k − − n
denotes the probability of n crossings during the time
[0,t+∆] including one in [t,t+∆]. Therefore, (t) =
1
10−5 (cid:80)∞k=1qk(t) represents the probability of a crosJsing in
[t,t+∆] irrespective of how many other crossings have
4 6 8 10 12 14
occurred prior to t. In comparison, (t) measures the
t I1
probability of a sign change in [t,t+∆] regardless of the
behaviour of X(t) before t. Therefore, we expect (t)
1
FIG.2. CrossingprobabilityI1(t)forthetwocasesshownin to constitute a first order approximation 1(t) of I(t):
Fig.1obtainedfromMCsimulations(greencircles)andfrom (t) (t)= (t). This is confirmed inGFig. 3. F
1 1
Eq. (4) (black dashed line). F ≈G I
For ease of comparison, we provide separate panels
for the first two peaks of the FPT distribution for both
boundaries shown in Fig. 1. For all cases (t) lays
forasignchangeintheinterval[t,t+∆]. Sincebothp(x) G1
on top of MC results and only starts to overestimate
andp (tx)areGaussian,theintegralcanbeperformed
+− | (t) for times very close and larger than the local max-
analytically, and we arrive at F
imum of the FPT distribution. This is analogous to the
(cid:114) ∆ Wiener–Riceexpansion. Forlatertimes,itbecomesmore
I1(t)= 8π2τ exp[e(t)](1+erf[f(t)]), (4) likely that the detected sign changes are not the first
ones and hence do not constitute a FPT. We therefore
where 2e(t) = ω(t)2σ2(t) (S(t) v)2/σ2(t), f(t) = count too many trajectories in (t). To alleviate this
(v ω(t)σ2(t))/(cid:112)2σ2(t1)and−ω(t)=−ρ(t)(v 1S(t))/σ2(t). problem, we need to subtract theGp1robability for the tra-
− 1 1 − 1
We compare the analytical expression for with di- jectories that we falsely included in (t). The leading
1 1
I G
rect Monte-Carlo (MC) simulations in Fig. 2 for the two correctiontermto (t)accountsforalltrajectoriesthat
1
G
choices of S(t) shown in Fig. 1. The agreement is ex- have sign changes in [t,t+∆] and [s,s+∆] for s < t
cellent, capturing the multimodal character of the cross- [40]. Let p (s,tx ) = P(X(s) > S(s),X(s+∆) <
+−+− 0
|
ing probability induced by the non-monotonicity of S(t) S(s+∆);X(t) > S(t),X(t+∆) < S(t+∆)x ) denote
0
|
andspanningmorethanthreeordersofmagnitude, with theprobabilityforsignchangesintwodisjunctintervals,
probabilities reaching values smaller than 10−5. then
p (s,tx )=
+−+− 0
III. FPT PROBABILITIES |
(cid:90) ∞ (cid:90) S(t)+∆
dx dx f (x ,x x )p (sx ,x ,x ), (6)
3 4 2 3 4 0 +− 0 3 4
| |
From a conceptual point of view, (t) corresponds to S(t) −∞
1
I
the first term (t) of the FPT expansion derived by
Wiener and RicJe1. Their original results for the full FPT wherep+−(s|x0,x3,x4)isdefinedanalogouslytoEq.(2),
theonlydifferencebeingthathereweconditiononthree
distribution (t) that measures crossings in the interval
F instead of one value of X(t). By expanding Eq. (6) to
[t,t+∆] can be expressed as
lowest order in ∆ and using Eq. (3), we find
(cid:88)∞ ( 1)n
(t)= − (t), (5) ∆
F n=0 n! Jn+1 p+−+−(s,t|x0)=σ2πτf2(S(t),S(s)|x0). (7)
4
10−4 12 12
(a) 10 (a) 10 (b)
8 8
6 6
4 4
2 2
t) 0 0
(s, 10−5 −20 1 2 3 4 5 6 7 8 −20 1 2 3 4 5 6 7 8
2 t t
I
10−2 (c) 10−2 (d)
10−6 (t)F10−3 (t)F10−3
2.7 2.8 2.9 3.0 3.1 10−4
s 10−4
1.65 1.70 1.75 1.80 1.60 1.65 1.70 1.75 1.80
t t
(b) FIG. 5. (a,b) An OUP (solid black line) with mean v
10−4 (dashedblackline)crossesthroughatimedependentbound-
ary S(t)=−40cos(t)exp(cid:0)−(4cos(t))2(cid:1)+1.5t (green dashed
) line). Parameter values are τ = 0.2, v = 7 and (a) D = 2.0,
t
(s,210−5 ((ba)) aDnd=(b1)0.ob(tca,din)eFdPfrTomprMobCabsiliimtiueslatFio(nt)sc(osorrliedspgornedyinlignet)o,
I
Eq. 4 (dashed green line) and Eq. 9 (green circles).
10−6
approximation of (t) as compared with (t). Note
1
5.3 5.4 5.5 5.6 5.7 F G
that (t) generally captures the rising phase of the
s G2
FPT distribution very well, even including the max-
imum in Fig. 3b. This behaviour is consistent with
FIG.4. CrossingprobabilityfortwodisjunctintervalsI2(s,t) other, although technically more limited, series expan-
for the two cases shown in Fig. 1 for a fixed time t indicated sions [49, 50]. While (t) tends to overestimate (t),
1
by the solid vertical black line at t = 3.1 (a) and t = 5.75 (t) underestimates thGe true FPT statistics when iFt de-
2
(b)obtainedfromMCsimulations(greencircles)andEq.(8) G
viatesatlargertimes. Thisisexpectedsincewesubtract
(black line).
too many trajectories in the computation of (t) [40],
2
G
and can be remedied by including higher order terms in
our expansion.
Since p is a Gaussian, we can again integrate out
+−+− A particularly useful property of our approach is that
thedependenceonx toobtaintheprobability (s,t)=
(cid:82)∞ 0 I2 it allows for explorations beyond the small noise limit.
p (s,tx)p(x)dx for sign changes in two disjunct
0 +−+− | Figure 5 shows results when the noise strength D is in-
intervals [s,s+∆] and [t,t+∆] in closed form
creased by a factor of 5 above an already moderate noise
level. Asexpected,theFPTdistributionsshifttosmaller
∆ exp[E(s,t)](1+erf[F(s,t)]
I2(s,t)=σ24π2τ G(s,t)) . (8) times for stronger noise, but the the agreement between
numerics and analytics persists.
We provide details for E(s,t), F(s,t) and G(s,t) in Forthecomputationof 1(t)and 2(s,t)weintegrated
I I
the Appendix. Note the structural similarity between outthedependenceontheinitialvaluesoftheOUP.This
Eqs. (4) and (8). In Fig. 4 we plot (s,t) as a func- could be done analytically since the OUP was initially
2
tion of s for fixed values of t. ThereIis almost no dif- Gaussiandistributed. Asthetheorystandsitcanequally
ference between the analytical expression and direct MC welldealwithhighlyskewedinitialdistributionsorsharp
simulations, with the agreement ranging over three or- initialvaluesase.g. usedin[51]. Asanillustrationofthis
ders of magnitude in Fig. 4b. For a first-passage event point,wechosex0 [v+2,v+2.1],whichcorrespondsto
∈
at time t, second crossings can happen at any moment a strongly localised initial distribution. Figure 6 shows
s<t. Therefore, we obtain the second order approxima- results for 1(t) and (t). For the analytical results, we
I F
tviaolnueGs2o(ft)sotfhFat(ta)rebycosmummemnisnugraIt2e(sw,itt)ho∆veranadllspuobstsriabclet uofsetdhep(dxi0s)tri=buδt(iovn+o2f −x0x.0)Wdeuoebtsoertvheerneaasroronwabslyupgpooordt
this probability from (t): agreementbetweennumericalandanalyticalresults. Im-
1
G portantly,thetheoryisabletocapturethesmallpeakin
(cid:88) the FPT distribution just before t=2. The main reason
(t)= (t) (n∆,t), (9)
2 1 2
G I − I for the difference between the analytical and numerical
n
results lies in the MC error. Indeed, the discrepancy
where we respect the strict inequality n∆ < t. Fig- between the analytical and numerical value for (t) in
1
I
ure 3 shows that (t) provides a significantly improved Fig. 6a at times t 2 and t 14 decreases with more
2
G ≈ ≈
5
ulation are numerically expensive. While we illustrated
our approach with the OUP, it works for any Gaussian
process since equations (3) and (6) do not assume a spe-
cific form of the correlation coefficient ρ(t). What the
time-dependenceofρ(t)determinesisthescalingof (t)
1
I
and (s,t) with ∆. Interestingly, for the OUP with
2
I
ρ(t) exp( t), we find (t) √∆ when t > ∆/2, a
1
∼ −| | I ∼
conditionthatisalmostalwayssatisfiedduetothesmall-
ness of ∆. By changing the correlation coefficient such
that the stochastic process becomes differentiable, e.g.
the widely used ρ(t) exp( t2), we recover the original
∼ −
findings by Wiener and Rice [48].
To illustrate the versatility of our approach, we chose
examplestowhich, tothebestofourknowledge, current
analytical methods cannot be applied since the time de-
pendent boundaries are neither convex nor concave. It
might therefore be tempting to consider the stochastic
process Z(t) as introduced in Sec. II, which is the differ-
ence between the stochastic process X(t) and the time
dependent boundary S(t). Consequently, the FPT prob-
lemforZ(t)involvesaconstantboundaryat0. However,
the stochastic differential equation for Z(t) is explicitly
time-dependent, which again renders the problem ana-
lytically intractable under general conditions [44].
In the past, non-differentiable stochastic processes
have often been dealt with by transforming them into
FIG. 6. (a) I1(t) obtained from MC simulations (green differentiable versions via smoothing. This is usually
circles) and Eq. 4 (red dashed line). (b) FPT distribu- done by either convolving the non-differentiable stochas-
tion obtained from MC simulations (grey lines) and using
tic prcess with a kernel or by filtering the correlation
for p(x ) = δ(v +2−x )(red). The boundary is given by
S(t) =0−40cos(t)exp(cid:0)−(04cos(t))2(cid:1)+1.5t, and the parame- function in frequency space [53, 54]. This requires firstly
to find appropriate kernels or filters, and secondly to de-
ter values of the OUP are τ =10.0, v=10.0, D=2.0.
termine a bandwidth or cut-off frequency. This is not
a trivial task, and often choices can only be justified
a posteriori. From a practical point of view, this is a
MC realisations and hence is not a systematic error of
severe limitation. In contrast, the approach presented
the approach.
here works directly with the original non-differentiable
stochastic process and thus avoids any of these compli-
cations.
IV. DISCUSSION AND CONCLUSION On theoretical grounds, we expect that the dominant
contribution of is in determining the rising phase of
1
I
In the present work, we have derived an analyti- the peaks of the FPT distribution. This is well captured
cal approach to FPT calculations for stationary non- bytheresultsshowninthisstudy. Recently, theconcept
differentiable Gaussian processes. Using the concept of of few encounters has been introduced [55], and it was
sign changes, the results presented here overcome the of- shown that the left part of the peaks of the FPT distri-
ten strong limitations of existing analytical techniques, bution is essential. Given the compact expression for 1,
I
such as convexity of the boundary, time scale separa- our results may prove useful in studying few encounters
tions or crossings being rare [24, 37, 49, 52], and al- in more detail.
low us to construct high-fidelity approximations of the While we have focussed on stationary stochastic pro-
full FPT statistics, which are notoriously hard to com- cesses, an intriguing avenue for future research con-
pute. Our approach is based on the ideas of Wiener cernsthegeneralisationofourfindingstonon-stationary
andRicetoappropriatelycountrandomtrajectoriesand stochastic processes. The starting point is Eq. (3), but
thus demonstrates that their seminal concept for differ- now both the mean and the standard deviation become
entiable Gaussian processes can be generalised to non- time-dependent. Results analogous to Eqs. (4) and (8)
differentiable stochastic processes. Given that the com- wouldhighlighttheconceptualeleganceofsignchanging
putation of (t) is entirely based on the evaluation and probabilitiesandmoreoverdemonstratethatsignchang-
2
G
summation of Gaussian distributions, it is numerically ing probabilities are a powerful and practical concept
fast and allows us to accurately describe events even for investigating FPT and general level crossing [56, 57]
whentheyoccurwithlowprobability,forwhichMCsim- problems.
6
ACKNOWLEDGMENTS and
WewouldliketothankPaulMatthewsforveryinsight- α(s,t)= Ss−v(1−ρ2s) + St−v(1−ρ2s) 2ρ˜ [S S
ful discussions and Dave Parkin for computational sup- σ2 σ2 − σ σ s t
s t s t
port. WB was partly supported by a Vice-Chancellor’s +S v(ρ 1)+S v(ρ 1)+v2(1 ρ ρ +ρ ρ )],
s t t s t s s t
Scholarship for Research Excellence at the University of − − − −
2(v(1 ρ )ρ S ρ ) 2(v(1 ρ )ρ S ρ )
Nottingham and NSERC Canada. β(s,t)= − s s− s s + − t t− t t
σ2 σ2
s t
2ρ˜
[ S ρ S ρ +vρ +vρ 2vρ ρ ] ,
Appendix: Details for Eq. (8) − σ σ − s t− t s t s− s t
s t
ρ2 ρ2 2ρ˜
We here provide details for the sign changing prob- γ(s,t)= σs2 + σt2 − σ σ ρsρt,
ability in two disjunct intervals, Eq. (8). The func- s t s t
tions E(s,t) and F(s,t) are given by E(s,t) = A(s,t)+
(cid:112) while G(s,t) reads as
B2(s,t)/Γ(s,t) and F(s,t)=B(s,t)/ Γ(s,t) with
(cid:113)
v2 α(s,t) G(s,t)= (1 ρ˜2)σ2σ2+σ2(ρ2σ2+ρ2σ2 2ρ˜σ ρ σ ρ ),
A(s,t)=−2σ2 − 2(1 ρ2), − s t s t t s − s s t t
−(cid:101) ρ(s t) ρ(s)ρ(t)
v β(s,t) ρ(cid:101)(s,t)= (cid:112) − −(cid:112) .
B(s,t)= , 1 ρ(t)2 1 ρ(s)2
σ2 − 2(1 ρ(cid:101)2) − −
−
(cid:20) (cid:21)
1 γ(s,t)
Γ(s,t)=2 + , For notational convenience, subscripts refer to time ar-
σ2 1−ρ(cid:101)2 guments, e.g. σt =σ(t)=σ(cid:112)1−ρ2(t) and St =S(t).
[1] A. J. Bray and R. Smith, Journal of Physics A: Mathe- [17] C. Kuehn, Multiple Time Scale Dynamics, 1st ed., Ap-
matical and Theoretical 40, F235 (2007). plied Mathematical Sciences No. 191 (Springer-Verlag,
[2] M.V.Matthews,W.L.Ellsworth, andP.A.Reasenberg, 2015).
BulletinoftheSeismologicalSocietyofAmerica92,2233 [18] S. Redner, A Guide to First-Passage Processes, 1st ed.
(2002). (Cambridge University Press, 2010).
[3] R.ChicheporticheandJ.-P.Bouchaud,Phys.Rev.E86, [19] L. Alili, P. Patie, and J. Pedersen, Stochastic Models
041115 (2012). 21, 967 (2005).
[4] T. T. Per Fauchald, Ecology 84, 282 (2003). [20] A. F. Atiya and S. A. K. Metwally, SIAM Journal on
[5] M. Maggiore and A. Riotto, The Astrophysical Journal Scientific Computing 26, 1760 (2005).
711, 907 (2010). [21] B.Lindner,A.Longtin, andA.Bulsara,NeuralComput.
[6] V. Srivastava and N. E. Leonard, in American Control 15, 1761 (2003).
Conference (2015) pp. 2113–2118. [22] E. Urdapilleta, Phys. Rev. E 83 (2011).
[7] J. A. R. Marshall, R. Bogacz, A. Dornhaus, R. Planqué, [23] J. Touboul and O. Faugeras, Adv. Appl. Prob. 40, 501
T. Kovacs, and N. R. Franks, Journal of The Royal So- (2008).
ciety Interface 6, 1065 (2009). [24] L.SacerdoteandM.T.Giraudo,inStochastic Biomath-
[8] S.Condamin,O.Bénichou, andM.Moreau,Phys.Rev. ematical Models,editedbyM.Bachar,J.J.Batzel, and
Lett. 95, 260601 (2005). S. Ditlevsen (Springer-Verlag Berlin Heidelberg, 2013)
[9] T.Guérin,N.Levernier,O.Bénichou, andR.Voituriez, pp. 99–148.
Nature 534, 356 (2016). [25] H. C. Tuckwell, J. Appl. Prob. 13, 39 (1976).
[10] G. L. Gerstein and B. Mandelbrot, Biophys. J. 4, 41 [26] T.TaillefumierandM.O.Magnasco,Proceedingsofthe
(1964). National Academy of Sciences 140, E1438 (2013).
[11] I. Goychuk and P. Hänggi, Proceedings of the National [27] M.R.EvansandS.N.Majumdar,Phys.Rev.Lett.106,
Academy of Sciences 99, 3552 (2002). 160601 (2011).
[12] R. Thul and M. Falcke, Europhysics Letters 79, 38003 [28] S. Reuveni, Phys. Rev. Lett. 116, 170601 (2016).
(2007). [29] J. Newby and J. Allard, Phys. Rev. Lett. 116, 128101
[13] A. J. Bray, S. N. Majumdar, and G. Schehr, Advances (2016).
in Physics 62, 225 (2013). [30] D. Holcman and Z. Schuss, Multiscale Modeling & Sim-
[14] R. Metzler, G. Oshanin, and S. Redner, eds., First- ulation 12, 1294 (2014).
Passage Phenomena and Their Applications (WorldSci- [31] M. J. Chacron, B. Lindner, and A. Longtin, Phys. Rev.
entific, Singapore, 2014). Lett. 92, 080601 (2004).
[15] A. J. F. Siegert, Phys. Rev. 81 (1951). [32] L.M.Bujorianu,Stochastic Reachability Analysis of Hy-
[16] P. C. Bressloff, Stochastic Processes in Cell Biology, brid Systems (Springer, London, 2012).
1st ed., Interdisciplinary Applied Mathematics No. 41 [33] A. G. Nobile, E. Pirozzi, and L. M. Ricciardi, “On the
(Springer-Verlag, 2014). estimation of first-passage time densities for a class of
7
Gauss–Markov processes,” in EUROCAST 2007, LNCS [46] W. Ebeling, Y. Romanovsky, and L. Schimansky-Geier,
4379, edited by R. M.-D. et al. (Springer-Verlag, Berlin eds., Lectures on the Stochastic Dynamics of Reacting
Heidelberg, 2007) pp. 146–153. Biomolecules (World Scientific, 2003).
[34] M.Tamborrino,MathematicalBiosciencesandEngineer- [47] S.Wiggins,IntroductiontoAppliedNonlinearDynamical
ing 13, 613 (2016). Systems and Chaos (Springer, 2003).
[35] C. Gardiner, Stochastic Methods. A Handbook for the [48] J.A.McFadden,JournaloftheRoyalStatisticalSociety,
Natural and Social Sciences, 4th ed. (Springer, 2009). Series B 29, 489 (1967).
[36] S.O.Rice,BellSystemTechnicalJournal22,282(1944). [49] J.DurbinandD.Williams,JournalofAppliedProbabil-
[37] H.CramérandM.R.Leadbetter,StationaryandRelated ity 29, 291 (1992).
Stochastic Processes,1sted.(JohnWileyandSons,New [50] W. Braun, P. C. Matthews, and R. Thul, Phys. Rev. E
York, London, Sydney, 1967). 91 (2015).
[38] A. Papoulis and S. U. Pillai, Probability, Random Vari- [51] T. Schwalger, F. Droste, and B. Lindner, Journal of
ables,andStochasticProcesses(McGraw-Hill,NewYork, computational neuroscience (2015).
2002). [52] M. Schindler, P. Talkner, and P. Hänggi, Phys. Rev.
[39] I. I. Gikhman and A. V. Skorokhod, The Theory of Lett. 93, 048102 (2004).
Stochastic Processes I (Springer Verlag, 2004). [53] R. L. Stratonovich, Topics In the Theory of Random
[40] T. Verechtchaguina, I. M. Sokolov, and L. Schimansky- Noise, Vol. 2 (CRC Press, 1967).
Geier, Phys. Rev. E 73 (2006). [54] L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta,
[41] L. Badel, Phys. Rev. E 84, 041919 (2011). and S. Santucci, Physics Letters A 158, 9 (1991).
[42] M. Newman, Contemporary Physics 46, 323 (2005). [55] A. Godec and R. Metzler, Physical Review X 6, 041037
[43] For α ≤ 2, the mean of the probability distribution di- (2016).
verges,yetthedistributionoftheintensityofsolarflares [56] J.-M. Azaïs and M. Wschebor, Level Sets and Extrema
or the distribution of family names are well captured by of Random Processes and Fields (Wiley, Hoboken, New
power laws with α≤2 [58, 59]. Jersey, 2009).
[44] J.TouboulandO.Faugeras,JournalofPhysiology-Paris [57] A. Feuerverger, P. Hall, and A. T. Wood, Journal of
101, 78 (2007). Time Series Analysis 15, 587 (1994).
[45] P. Hänggi, P. Talkner, and M. Borkovec, Reviews of [58] E.T.LuandR.J.Hamilton,AstrophysicalJournal380,
Modern Physics 62, 251 (1990). L89 (1991).
[59] D. H. Zanette and S. C. Manrubia, Physica A 295, 1
(2001).