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Optimal escape from metastable states driven by non-Gaussian noise A. Baule1 and P. Sollich2 1School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK 2Department of Mathematics, King’s College London, London WC2R 2LS, UK (Dated: January 5, 2015) We investigate escape processes from metastable states that are driven by non-Gaussian noise. Using a path integral approach, we define a weak-noise scaling limit that identifies optimal escape pathsasminimaofastochasticaction,whileretainingtheinfinitehierarchyofnoisecumulants. This enables us to investigate the effect of different noise amplitude distributions. We find generically a reduced effective potential barrier but also fundamental differences, particularly for the limit when 5 thenon-Gaussiannoisepulsesarerelativelyslow. Hereweidentifyaclassofamplitudedistributions 1 thatcaninduceasingle-jumpescapefromthepotentialwell. Ourresultshighlightthathigher-order 0 noise cumulants crucially influence escape behaviour even in the weak-noise limit. 2 n a Escape from a metastable state underlies a large vari- The first two terms in Eq. (2) represent a constant drift J ety of phenomena in physics, chemistry, and biology [1] a and Gaussian noise of intensity D, capturing the ef- 2 and has accordingly received an extraordinary amount fect of continuous fluctuations. The term φ is the con- of attention since Kramers’ seminal work [2]. Recently, tribution of discontinuous jumps. This becomes evident ] h the problem has seen renewed interest in continuous sys- when ρ is interpreted as the density of independently c temsdrivenbynon-Gaussiannoise. Poissonianshotnoise andidenticallydistributedjumpamplitudes A andλ as e j m (PSN) has been investigated in the context of full count- the rate parameter of a Poisson process generating jump ingstatistics[3–10],wheretheescaperateine.g.Joseph- times t . G is then just the characteristic function of - j ξ t son junctions can be related to the current statistics. In Gaussian white noise with drift and superimposed zero- a the limit of weak noise intensity D, the Kramers rate mean PSN z(s)−(cid:104)z(s)(cid:105), where z(s)=(cid:80)n A δ(s−t ) t j=1 j j s assumes the exponential form r ∼= e−S/D, generalizing and n is Poisson distributed with mean λt. If ρ(A) is . t the classical Arrhenius result. The key challenge is to not normalizable, e.g. because of a power law divergence a m determine the action S in the presence of the infinite hi- ρ(A)∝|A|−α−1 for small A [31] with 0<α<2, it can- erarchy of noise cumulants that describes non-Gaussian not strictly be interpreted as an amplitude distribution. - d noise. Previous studies have truncated this hierarchy, Nevertheless, λρ(A)dA still gives the rate of jumps with n typicallyafterthecubicterm[3–8,11]. Analyticalresults size in the range [A,A+dA]. o retainingthefullhierarchyareavailableonlyforparticu- For our weak-noise limit we need to have a scale for c [ lar amplitude distributions, such as one-sided PSN with noise amplitudes, which we achieve by writing ρ(A) = exponentially distributed [12–17] or constant amplitudes ρ (A/A )/A in terms of a base distribution ρ (x). For 1 0 0 0 0 [9, 10, 18]. On the other hand, the exponential scaling φ this scaling translates into φ(g) = φ (A g), where φ v 0 0 0 4 in 1/D breaks down for escape driven by Lévy-stable is given by Eq. (3) with ρ replaced by ρ0. Table I lists 7 noise (LSN), as has been established both in simulations the amplitude distributions ρ (x) we consider, normal- 0 3 [19–26] and analytical work [27–29]. Here, we present a ized so that (cid:104)x2(cid:105) ≡ (cid:82) dxρ (x)x2 = 1. We also include 0 0 unified framework where exact results for the Kramers the case where the cumulants of the non-Gaussian noise 0 rate are obtained for a large class of non-Gaussian noise are truncated after some low order by expanding φ(g). . 1 types. Our approach characterizes in particular the dy- In the following we focus on symmetric noise with zero 0 namics of the escape process and explains the difference drift (a = 0). A straightforward calculation from G 5 ξ in the scalings of PSN and LSN driven escape. thenshowsthatthenoisecovarianceisCov(ξ(t),ξ(t(cid:48)))= 1 v: We consider the time evolution of a single degree of (D+Dφ)δ(t−t(cid:48)), where Dφ =λ(cid:104)A2(cid:105)=λA20. freedom q under the influence of a conservative force, The starting point for our approach is a path integral i X with potential V, and noise ξ, q˙(t)=−V(cid:48)(q)+ξ(t). We forthepropagator,i.e.theprobabilityofarrivingatq at b r specify the noise ξ very generally as the derivative of time t when starting from q . We adopt an Itô conven- a a a process with stationary increments. It can then be tion and express the propagator as an inverse functional described via the characteristic functional Fourier transform, following [32]: Gξ[g]=(cid:104)ei(cid:82)0tdsξ(s)g(s)(cid:105)=e(cid:82)0tdsψ(g(s)), (1) (cid:90) (qb,t) (cid:90) (cid:104) g (cid:105) p(q ,t|q ) = D[q] D e−S[q,g]. (4) b a 2π where the characteristic function ψ has the form [30] (qa,0) ψ(g) = iga−Dg2/2+λφ(g) (2) The action functional is given by S[q,g] = (cid:90) (cid:82)tdsL(q(s),g(s)) with the underlying Lagrangian φ(g) = dA(cid:0)eigA−igA−1(cid:1)ρ(A). (3) 0 L(q,g)=ig(q˙+V(cid:48)(q))+Dg2/2−λφ(g). Nowintroduce 2 Distribution ρ (x) φ¯ (k)=φ (−ik)=(cid:104)ekx−kx−1(cid:105) 0 √ 0 0 Gaussian e−x2/2/ 2π φ¯ (k)=ek2/2−1 0,G Constant modulus [δ(x−1)+δ(x+1)]/2 φ¯ (k)=cosh(k)−1 0,const Gamma (α<2) |x|−1−αe−|x|/[2Γ(2−α)] φ¯ (k)=[(1−k)α+(1+k)α−2]/[2α(α−1)] 0,α Exponential (α=−1) e−|x|/4 φ¯ (k)=k2/[2(1−k2)] 0,exp Truncated φ φ¯ (u)= 1k2+bk4 0,trunc 2 (cid:82) TABLEI: Symmetricnoiseamplitudebasedistributionsρ studied. ForGammanoise,ρ (x)canbenormalizedto dxρ (x)= 0 0 0 1 for α<0, with the exponential a special case, but not for α>0. At generic amplitude scale A one has φ¯(k)=φ¯ (A k). 0 0 0 the scaling g →g˜/D, so that L=L˜/D with 0.0 Λ(cid:61)0 L˜(q,g˜) = ig˜(q˙+V(cid:48)(q))+g˜2/2−λDφ(g˜/D). (5) (cid:45)0.5 Φ (cid:72)a(cid:76) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) exp Intheabsenceoftheφ-component(λ=0),L˜isindepen- Φ (cid:45)1.0 G dent of D and we have the familiar scaling form of the Φ Gaussian action. This means that taking D → 0 yields s(cid:76) Α (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) (cid:42)(cid:72) (cid:45)1.5 theusualweak-noiselimitofthepathintegral. Forλ(cid:54)=0 q 2 (cid:72)b(cid:76) III weneedtoapplyasuitablescalingfor φ. Expandingthe II (cid:45)2.0 Φ exponential in Eq. (3) and using φ(g)=φ0(gA0) yields Φconst Ν 1 II trunc I φ(cid:18)g˜(cid:19) = A20(cid:104)x2(cid:105)(ig˜)2 + A30(cid:104)x3(cid:105)(ig˜)3 +... (6) (cid:45)2.5 (cid:236) meanpath 0 1 Μ 2 3 D D2 2! D3 3! 0.0 0.2 0.4 0.6 0.8 1.0 We now consider the scaling λ → λ˜/Dµ, A0 → s A˜ Dν, which makes the O(gñ) term of λDφ scale as FIG. 1: (Colors online) Optimal excitation paths for the dif- 0 D1−µ+n(ν−1). The exponents µ, ν define different scal- ferentφ0(seeTableI)inthepotentialV(x)=x4−6x2−2x+5, where q = −1.64, q = −0.17. Parameter values: λ = 1, ing regimes as D → 0, as shown in the inset of Fig. 1. a b A =1,α=1.2,b=1/2(correspondingtofourthordertrun- In regime I, all orders (n ≥ 2) in g˜ diverge as D → 0. 0 cation of φ ). Thin lines: paths directly sampled from the exp In regime II, there are always some higher orders that weight(4)withgintegratedout,forD=0.01. (a)Meansam- diverge as D → 0, while in regime III all orders scale to ple path compared with theory for φ . (b) Different scaling α zero as D → 0 so that one effectively recovers the case regimesofnon-Gaussiannoisegivenbythescalingexponents λ=0. For the particular combination ν = 1(µ+1) with µ, ν. 2 µ > 1 (red line) only the g˜2 term remains in Eq. (5) as D → 0. The non-Gaussian noise strength D ∝ D → 0 φ here, so this is a valid weak noise-limit but one that re- The boundary conditions are q(0) = q , q(t) = q , and a b duces to effective Gaussian noise. Only for µ = ν = 1 accordingly we write the optimal action as S(q ,t;q ). b a do all orders in g˜ remain in Eq. (5) as D → 0. This is Comparing Eq. (9) with Eq. (8) shows that it can be ex- the scaling we adopt: it represents a genuine weak-noise pressedasafunctionalofg∗(t)only. Onphysicalgrounds limit of our generic noise, since Dφ ∝ D → 0 while the we require real solutions for q∗ and hence purely imagi- infinite hierarchy of noise cumulants is retained. naryg∗,sowedefinek(s)=ig∗(s)andgetfortheaction After scaling, we obtain for the path integral (cid:90) (qb,t) (cid:90) (cid:20) g˜ (cid:21) S(q ,t;q )=(cid:90) tds(cid:20)1k2+λ(cid:0)kφ¯(cid:48)(k)−φ¯(k)(cid:1)(cid:21) (11) p(q ,t|q ) = D[q] D e−S[q,g˜]/D, (7) b a 2 b a 2πD 0 (qa,0) where φ¯(k)=φ(g)=φ(−ik). Evaluating φ(g) for imagi- with Lagrangian nary arguments in this way requires a continuation into L˜(q,g˜) = ig˜(q˙+V(cid:48)(q))+g˜2/2−λ˜φ˜(g˜) (8) thecomplexplaneandsoanonzeroradiusofconvergence and φ˜(g˜) = φ (gÃ˜ ). We now drop the tildes as only forφ(g),aconditionthatismetifρ(A)hastailsthatare 0 0 no heavier than exponential. One could formulate the rescaled variables are used below. As D → 0 we can solution of (9,10) in terms of a Hamiltonian H with the calculate p(q ,t|q ) within a saddle-point approach as b a p(q ,t|q ) ∼= e−S[q∗,g∗]/D where ∼= indicates the asymp- conjugate momentum ∂L/∂q˙ = ig: L = igq˙−H(q,ig). b a totic behaviour for small D. The optimal paths q∗, g∗ TheEL-equationsarethenjustHamilton’sequationsde- scribing motion with a conserved H. are determined from the condition δS = 0, which leads Let us now consider specifically escape from a to the coupled Euler-Lagrange (EL) equations metastable state q , located at the minimum of the a q˙ = −V(cid:48)(q)+ig−iλφ(cid:48)(g) (9) metastablebasinofV,tothetopofthepotentialbarrier g˙ = V(cid:48)(cid:48)(q)g, (10) at q (>q ). For Gaussian noise, the path integral solu- b a 3 tion of this problem is essentially analogous to the quan- 1.0 tummechanicaltunnelingproblemtreatedinasemiclas- sicalapproximation[33](seealso[34–36]foradiscussion 0.8 inthestatisticalmechanicscontext). Thetheoryoflarge deviations gives the dominant scaling of the escape rate 1 0.6 S m 0 r in the weak-noise limit as [37, 38] or n S r ∼=e−S(qb;qa)/D, S(qb;qa)= infS(qb,t;qa). (12) 0.4 0.5 t≥0 0.2 0 Theoptimalpaththatprovidesthelowest(infinimumin (cid:45)2 (cid:45)1 0 1 2 Eq. (12)) action is achieved for t → ∞ [33]. For λ = lnA0 0.0 0, S(qb;qa) can be determined analytically as twice the (cid:45)4 (cid:45)2 0 2 4 6 height of the energy barrier, ∆V =V(q )−V(q ). Since b a lnΛ D = 2T for thermal noise, Eq. (12) thus recovers the FIG. 2: (Colors online) The normalized action S = norm Arrhenius result [1]. In this case, the optimal escape or S(q ;q )/(cid:0)2∆V/(1+λA2)(cid:1) with A = 1 obtained for the b a 0 0 “excitation” path is the time-reverse of a deterministic same potential as in Fig. 1. The reference value 2∆V/(1+ relaxation path from qb to qa [34–36]. λA20) is the action for the Gaussian limit of φ, i.e. truncated For λ (cid:54)= 0, deterministic relaxations with g∗ = 0 still after the quadratic term. The φα curve is dashed where optimal paths cannot be found from the EL equations because solve the EL equations and have zero action, but their they involve jumps. Inset: S as a function of lnA for norm 0 time-reversal no longer gives the excitation paths. This φ and φ , at fixed D =λA2 =1. For large A (small λ) is clear from the predictions for different amplitude dis- theexpcurvesαconverge to Sφ =(cid:82) d0qmin(2V(cid:48)(q),1/A0). 0 0 tributions in Fig. 1, which we have confirmed by direct path sampling. The optimal escape paths have the characteristic instanton shape: for large t the system spends most of its time close to q and q while the actual bar- of the noise (averaged again over small dt) in the large a b rier transition is sharply localized in time. The key ob- deviation limit D → 0. The function π(ξ) thus general- servationisthattheinstantonshapevarieswithφ,while izes the simple quadratic ξ2/2 appearing in the Wiener the deterministic relaxation and its time reverse are en- measure exp[−(cid:82) dsξ(s)2/(2D)] for Gaussian noise. tirely independent of φ and λ. Moreover, we observe Onecannowthinkofq(t)asapathinthe(q,v)-plane, (cid:82) that the optimal action S(q ;q ) is reduced compared with v = q˙. Then the action reads S = dqπ(v + b a to the Gaussian limit of the noise for a range of small V(cid:48)(q))/|v|andforeachq wecanfindv =q˙ simplyasthe λ values: the non-Gaussian noise makes escape faster. minimum of π(v+V(cid:48)(q))/|v|. We do not need to enforce (cid:82) Differencesbetweenamplitudedistributionsbecomepro- the total time constraint t= dq/|v| as we want t→∞ nouncedespeciallyinthelimitλ→0: theGaussiancase and the integral automatically diverges at both ends for is approached continuously with the low-order truncated paths between stationary points of V. The trivial global φ and for constant and Gaussian distributed noise am- minimum is v = −V(cid:48)(q), which describes deterministic plitudes, though the approach is extremely slow for the relaxation. For an excitation from qa to qb >qa we need latter (Fig. 2). For exponential and Gamma noise, the v >0, on the other hand. The condition for a minimum action is discontinuous at λ = 0: as λ → 0 it converges of π(v+V(cid:48)(q))/v with respect to v for excitation paths to a value considerably smaller than 2∆V. Puzzlingly, can be cast in the form for α > 0 the small λ regime appears inaccessible, with V(cid:48)(q) = β(k)/k, (13) q∗ becoming complex below some threshold. In order to understand these surprising observations, v = β(cid:48)(k)−V(cid:48)(q) (14) weproceedanalyticallyandintegrateoutg directlyfrom using basic properties of Legendre transforms. Here, k Eq.(7). Intheweaknoise-limitthiscanagainbedoneby has to be found from Eq. (13) and then gives v using saddle point integration and gives an action for q alone. Eq. (14). This implicitly defines a function v = q˙ = (Technicallywediscretizeintosmalltimeintervalsdtand Ξ(V(cid:48)(q)) and hence characterizes the shape of the exci- takedt→0afterD →0.) Definingβ(k)=k2/2+λφ¯(k), the resulting action is S[q] = (cid:82) dsπ(q˙(s) + V(cid:48)(q(s))). tation path. Moreover, we obtain the action simply as Here π(·) is the Legendre transform of β(·), i.e. π(f) = (cid:90) qb S = dqk(q), (15) max[kf −β(k)] with the maximum taken over the range of k where φ¯(k) remains non-singular. Note that the qa function π is not equivalent to the Hamiltonian H since where k(q) is the solution of Eq. (13). Eqs. (13–15) re- we have integrated out the momenta ig. In fact, since produce existing results for special cases. In the Gaus- q˙+V(cid:48)(q)=ξfromtheoriginalequationofmotion,theac- sian case (λ = 0) one has β(k) = k2/2; thus k(q) = tionS[q]=(cid:82) dsπ(ξ(s))givestheweightofanytrajectory 2V(cid:48)(q) and v = V(cid:48)(q), the expected time reverse of 4 the relaxation path. For escape driven by one-sided the integral covers the q-range of the jump. Our con- PSN without a Gaussian component, as investigated in clusion is therefore that for case (c) the optimal paths Refs.[12,13,16,18],wehaveβ(k)=λA2k2/[2(1−A k)] will have jumps even for nonzero noise rates λ, provided 0 0 for exponential and β(k) = λ(cid:0)eA0k−A0k−1(cid:1) for con- that Vm(cid:48)ax > V0(cid:48); for Gamma noise with α > 0 one has stant amplitudes. This yields k(q) = 2V(cid:48)(q)/(λA2 + V(cid:48) ={1+λA2(2α−2)/[α(α−1)]}/(2A ),whichasincase 0 0 0 0 2A V(cid:48)(q)) [12, 13, 16] for the exponential case and k(q) (b) scales with 1/A if we keep D =λA2 constant. The 0 0 φ 0 as the solution of k = ln(1+k(A +V(cid:48)(q)/λ))/A [18] presenceofajumpincertainparameterregimesindicates 0 0 for the constant case, respectively. that there is competition between escape via a sequence To use Eq. (13) to understand the observations in of small amplitude noise steps, and waiting for an atyp- Fig. 2 we distinguish three cases: (a) φ¯ has no singular- ically large (O(1)) noise pulse to kick the system close ities for real k, which includes our scenarios φ , φ , to the barrier. The latter is preferred when the noise trunc G φ ; (b) φ¯ has singularities for real k beyond which amplitudes have an exponential tail with large enough const it is not defined, and diverges as these singularities are scale A , and when non-Gaussian noise pulses arrive at 0 approached; this happens for φ or more generally φ a sufficiently small rate to make successful accumulation exp α with α<0, with singularities at k =±1/A ; (c) φ¯ has of many small pulses less likely. ± 0 singularities but remains bounded on approaching them, One can push the analysis further, e.g. to show that as for our φ with α>0. Now write Eq. (13) as in the limit λ → 0 at constant A the function Ξ(·) and α 0 hence the excitation paths become identical for Gamma λ= k (V(cid:48)(q)−k/2)≡R(k) (16) andexponentialamplitudedistributions,i.e.independent φ¯(k) of the exponent α. The limiting shape is Ξ(V(cid:48)(q)) = V(cid:48)(q) for V(cid:48)(q) < 1/(2A ), and Ξ(V(cid:48)(q)) = ∞ other- 0 The RHS R(k) diverges as 1/k for k → 0 and decreases wise; thus the optimal excitation paths consist of ini- as k increases. In case (a) defined above, it then hits tial and final segments of time-reversed relaxations, con- zero at k = kG = 2V(cid:48)(q). The limit λ → 0 thus gives nected by a jump, and the resulting action is just S0. back purely Gaussian behaviour (λ = 0) as we found. For large A one then has S ≈ (q −q )/A and the 0 0 b a 0 For case (b) the same is true if V(cid:48)(q) is small enough excitation path becomes a jump directly from q to q . a b for kG to lie in the allowed range [k−,k+]. For larger In the limit A0 → ∞ our Gamma noise retrieves LSN V(cid:48)(q), R(k)goestozeroalreadyask →k+ becauseφ(k) where ρ(A) ∝ |A|−1−α. Our results show that the effec- diverges, so that in the limit λ → 0 we get the non- tive barrier governing the escape rate then vanishes, and Gaussian solution k = k+ = 1/A0. As then also φ(cid:48)(k) henceclarifywhytheexponentialscalingwith1/D must diverges, the corresponding trajectory q∗(s) must have break down in this case as found both numerically [19– a section with infinite slope, i.e. a jump. For λ → 0, 26]andanalytically[27–29]. Notethatourargumentsfor k(q) has the value kG = 2V(cid:48)(q) or k+ = 1/A0 depend- allcases(a,b,c)aboveimplythatthesolutionofEq.(16) ing on which is smaller, giving for the action (15): S → alwaysobeysk <k =2V(cid:48)(q),andhencetheaction(15) S0 = (cid:82) dqmin(2V(cid:48)(q),1/A0). This is clearly lower than is lowered by the nGon-Gaussian noise, S < 2∆V. Fig. 2 the Gaussian limit S = 2∆V. Therefore, non-Gaussian suggests that even S <2∆V/(1+λA2), i.e. the effective 0 noise of infinitesimal rate λ discontinuously lowers the barrier is smaller than for the equivalent Gaussian noise. effective barrier for escape. This effect occurs for large Insummary,wehavedefinedaweaknoisescalinglimit enoughnoiseamplitudes,specifically2A V(cid:48) >1where 0 max that allows the effects of non-Gaussian noise to be fully V(cid:48) = maxV(cid:48)(q) for q ∈ [q ,q ]. We emphasize that max a b accounted for. We have identified a class of amplitude truncation of the cumulant hierarchy at any finite order, distributions that lead to optimal escape paths contain- which gives polynomial φ¯, always brings us back to case ing jumps; these encompass the entire path for large (a) and cannot reproduce even qualitatively the discon- noise amplitudes. We have shown that even very rare tinuous behaviour for λ→0. non-Gaussian noise (λ → 0) can discontinuously lower Incase(c),smallλrequiresk →kG duetothebound- the effective barrier for escape, an effect that cannot be edness of φ¯. However, if k+ < kG, then this point is captured by approximations that truncate the hierarchy outside the allowed range of k, and so for small enough of noise cumulants. It would be highly interesting to λ there is no solution for k in Eq. (13), in agreement observe such non-Gaussian escape events in an experi- with our observation for Gamma noise with α > 0 in mental system, e.g. in Josephson junctions [39, 40]. We Fig. 2. Small λ here means λ< φ¯(kk++)(V(cid:48)(q)−k+/2), or remark that our path integral approach can be extended equivalently V(cid:48)(q) > V(cid:48) ≡ β(k )/k . In the range of q to give the prefactor inEq. (12), byincluding the fluctu- 0 + + wherethisistrue,onecannowcheckthatπ(v+V(cid:48)(q))/v ation determinant and regularizing multi-instanton con- is monotonically decreasing for v > 0, reaching the tributions [34, 35, 41]. Application of our approach to limit k for v → ∞: the optimal velocity is infinite, systemswithmorethanonedegreeoffreedomcouldalso + Ξ(V(cid:48)(q))=∞, and there must be a jump in the optimal answeropenquestionssuchastheeffectofnon-Gaussian (cid:82) path. To the action this then contributes k dq where noise on the selection of transition configurations [42]. + 5 [24] A. V. Chechkin, O. Y. Sliusarenko, R. Metzler, and J. Klafter, Phys. Rev. E 75, 041101 (2007). [25] H.Chen,J.Duan,X.Li,andC.Zhang,AppliedMathe- [1] P.Ha¨nggi,P.Talkner,andM.Borkovec,Rev.Mod.Phys. matics and Computation 218, 1845 (2011), ISSN 0096- 62, 251 (1990). 3003. [2] H. Kramers, Physica 7, 284 (1940), ISSN 0031-8914. [26] T. Gao, J. Duan, X. Li, and R. Song, SIAM Journal on [3] J. Ankerhold, Phys. Rev. Lett. 98, 036601 (2007). Scientific Computing 36, A887 (2014). [4] B. Huard, H. Pothier, N. Birge, D. Esteve, X. Waintal, [27] P. Imkeller and I. Pavlyukevich, Journal of Physics A: and J. Ankerhold, Annalen der Physik 16, 736 (2007), Mathematical and General 39, L237 (2006). ISSN 1521-3889. [28] P.ImkellerandI.Pavlyukevich,StochasticProcessesand [5] E. V. Sukhorukov and A. N. Jordan, Phys. Rev. Lett. their Applications 116, 611 (2006), ISSN 0304-4149. 98, 136803 (2007). [29] P. Imkeller, I. Pavlyukevich, and T. Wetzel, The Euro- [6] H. Grabert, Phys. Rev. B 77, 205315 (2008). pean Physical Journal Special Topics 191, 211 (2010), [7] T. Novotny´, Journal of Statistical Mechanics: Theory ISSN 1951-6355. and Experiment 2009, P01050 (2009). [30] R. Cont and P. Tankov, Financial Modelling with Jump [8] E. V. Sukhorukov and A. N. Jordan, Physica E: Low- Processes (Chapman and Hall/CRC, 2003). dimensionalSystemsandNanostructures42,550(2010). [31] I. Koponen, Physical Review E 52, 1197 (1995). [9] M. I. Dykman, Phys. Rev. E 81, 051124 (2010). [32] P. Ha¨nggi, Zeitschrift fu¨r Physik B Condensed Matter [10] I. A. Khovanov and N. A. Khovanova, Phys. Rev. B 89, 75, 275 (1989). 085419 (2014). [33] H.Kleinert,Pathintegralsinquantummechanics,statis- [11] A. Baura, M. K. Sen, G. Goswami, and B. C. Bag, The tics, polymer physics, and financial markets (World Sci- Journal of Chemical Physics 134, 044126 (2011). entific, 2009). [12] C.VandenBroeckandP.Ha¨nggi,Phys.Rev.A30,2730 [34] B.Caroli,C.Caroli,andB.Roulet,JournalofStatstical (1984). Physics 26, 83 (1981). [13] J. M. Sancho, Phys. Rev. A 31, 3523 (1985). [35] U. Weiss, Phys. Rev. A 25, 2444 (1982). [14] J. Masoliver, Phys. Rev. A 35, 3918 (1987). [36] A. J. Bray and A. J. McKane, Phys. Rev. Lett. 62, 493 [15] E.Hernańdez-Garcia,L.Pesquera,M.A.Rodriguez,and (1989). M. San Miguel, Phys. Rev. A 36, 5774 (1987). [37] M.I.FreidlinandA.D.Wentzell,RandomPerturbations [16] J. M. Porra` and J. Masoliver, Phys. Rev. E 47, 1633 of Dynamical Systems (Springer, New York, 1998), 2nd (1993). ed. [17] F. Laio, A. Porporato, L. Ridolfi, and I. Rodriguez- [38] H.Touchette,PhysicsReports478,1 (2009),ISSN0370- Iturbe, Phys. Rev. E 63, 036105 (2001). 1573. [18] L. Billings, M. I. Dykman, and I. B. Schwartz, Phys- [39] A. V. Timofeev, M. Meschke, J. T. Peltonen, T. T. ical Review E (Statistical, Nonlinear, and Soft Matter Heikkila¨, and J. P. Pekola, Phys. Rev. Lett. 98, 207001 Physics) 78, 051122 (pages 4) (2008). (2007). [19] P. D. Ditlevsen, Phys. Rev. E 60, 172 (1999). [40] Q. Le Masne, H. Pothier, N. O. Birge, C. Urbina, and [20] J.-D. Bao, H.-Y. Wang, Y. Jia, and Y.-Z. Zhuo, Phys. D. Esteve, Phys. Rev. Lett. 102, 067002 (2009). Rev. E 72, 051105 (2005). [41] H.C.LuckockandA.J.McKane,Phys.Rev.A42,1982 [21] A. V. Chechkin, V. Y. Gonchar, J. Klafter, and R. Met- (1990). zler, EPL (Europhysics Letters) 72, 348 (2005). [42] M. I. Dykman, P. V. E. McClintock, V. N. Smelyanski, [22] B. Dybiec, E. Gudowska-Nowak, and P. Ha¨nggi, Phys. N.D.Stein,andN.G.Stocks,Phys.Rev.Lett.68,2718 Rev. E 73, 046104 (2006). (1992). [23] B. Dybiec, E. Gudowska-Nowak, and P. Ha¨nggi, Phys. Rev. E 75, 021109 (2007).