Probabilistic dynamics of some jump-diffusion systems

PHYSICAL REVIEW E 73, 026108 共2006兲

Probabilistic dynamics of some jump-diffusion systems Edoardo Daly* and Amilcare Porporato† Department of Civil and Environmental Engineering & Nicholas School of the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708, USA 共Received 29 August 2005; published 7 February 2006兲 Some exact solutions to the forward Chapman-Kolmogorov equation are derived for processes driven by both Gaussian and compound Poisson 共shot兲 noise. The combined action of these two forms of white noise is analyzed in transient and equilibrium conditions for different jump distributions and additive Gaussian noise. Steady-state distributions with power-law tails are obtained for exponentially distributed jumps and multiplicative linear Gaussian noise. Two applications are discussed: namely, the virtual waiting-time or Takàcs process including Gaussian oscillations and a simplified model of soil moisture dynamics, in which rainfall is modeled as a compound Poisson process and fluctuations in potential evapotranspiration are Gaussian. DOI: 10.1103/PhysRevE.73.026108

PACS number共s兲: 02.50.⫺r, 89.60.⫺k, 05.40.⫺a

I. INTRODUCTION

High-dimensional dynamical systems can be often studied in terms of a single representative variable and separating an internal deterministic dynamics from random noises accounting for external fluctuations 关1,2兴. The nature of such external forcings can be very different, according to their temporal regime and intensity distribution, and in some cases multiple forms of noise may coexist. A very typical case arises when fluctuations acting continuously in time are modeled using a Gaussian noise, which can be either additive or multiplicative 关2–4兴, while random jumps that occur instantaneously are described by means of a compound Poisson process 关5–8兴. Such systems, which will be referred to as jump-diffusion processes, have been analyzed in various fields. In stochastic finance, for example, they have been used to describe the joint action of small, frequent transactions and rare, large movements of money. The same models apply in general to problems related to queuing theory and storage models 关5,6兴. In physics, they have been adopted to study mechanisms of noise-driven transport—e.g., ratchettype models 关7,9,10兴. In neurophysiology, the membrane potential of a single neuron has been modeled by coupling an Ornstein-Uhlenbeck process 共Gaussian noise兲, which defines the nerve-cell voltage from its resting level, to a renewal process 共Poisson noise兲, which determines the spike trains of the action potential 关11–13兴. Jump-diffusion processes have also applications in population dynamics 关14兴, in hydrology to model soil water balance 共see 关15兴 and Sec. IV B兲, and in the general theory of stochastic processes 关16–18兴. Despite their wide interest, however, few analytical solutions have been proposed so far. The dynamics of the jump-diffusion system under analysis can be described using a Langevin equation x˙ = f„x共t兲… + g„x共t兲…␰共t兲 + I共t兲,

共1兲

mean and intensity ␴2—i.e., 具␰共t兲典 = 0 and 具␰共t兲␰共u兲典 = 2␴2␦共t − u兲—and I共t兲 is a compound Poisson noise, defined as N共t兲

I共t兲 = 兺 zk␦共t − tk兲,

where N共t兲 is a Poisson counting process with frequency ␭ 共e.g., the mean number of ␦ impulses per unit time兲 and 兵zk其 are the jump sizes distributed according to a probability density function b共z兲. When necessary, Eq. 共1兲 will be interpreted in the Stratonovich sense. Accordingly, Eq. 共1兲 corresponds to the forward Chapman-Kolmogorov equation 关1,16兴

⳵ ⳵ p共x,t兲 = − ⳵t ⳵x + ␴2 +

†

Electronic address: [email protected]

1539-3755/2006/73共2兲/026108共7兲/$23.00

冕

再冋

f共x兲 + ␴2g共x兲

册 冎

d g共x兲 p共x,t兲 dx

⳵2 关g共x兲2 p共x,t兲兴 − ␭p共x,t兲 ⳵x2

+⬁

␭b共y兲p共x − y,t兲dy,

共3兲

−⬁

which expresses the evolution of the transition probability density function 共PDF兲 of x, p共x , t兲. In the following, some special forms of Eq. 共3兲 will be analyzed. In particular, we will distinguish between additive 共Sec. II兲 and multiplicative 共Sec. III兲 Gaussian noise. In the first case, some examples of jump distributions 共e.g., two-sided exponential, gamma, and exponential distribution兲 are studied when the drift is either constant or linear, while the multiplicative noise is studied for linear drift and exponentially distributed jumps. Two possible applications of the results obtained in the previous sections are discussed in detail in Sec. IV.

where f共x兲 and g共x兲 are deterministic functions of the state variable x共t兲, ␰共t兲 is a Gaussian ␦-correlated noise of zero

*Electronic address: [email protected]

共2兲

k=1

II. ADDITIVE GAUSSIAN NOISE

In a variety of problems, external fluctuations can be taken into account by simply adding a noise source to the deterministic component, f共x兲—that is, assuming g共x兲 = 1 in Eq. 共1兲 关5,10兴. Provided that f共x兲 attains a finite value when 026108-1

©2006 The American Physical Society

PHYSICAL REVIEW E 73, 026108 共2006兲

E. DALY AND A. PORPORATO

x tends to ±⬁, while p共x , t兲 and its derivatives tend to zero, Eq. 共3兲 can be Fourier transformed as

⳵ * p 共s,t兲 = is兵f共x兲p共x,t兲其* − ␴2s2 p*共s,t兲 + ␭关b*共s兲 − 1兴p*共s,t兲, ⳵t

while the cumulants of order higher than 2 are ␬m = 共a + m − 1兲!␭t / 关共a − 1兲!␥m兴. The PDF is therefore asymmetrical 共e.g., positive skewness兲, because of the asymmetry in the jumps, and the kurtosis is

共4兲

␤=3+

where p 共s , t兲 is the characteristic function of p共x , t兲, i.e., *

p*共s,t兲 =

冕

+⬁

共5兲

eixs p共x,t兲dx,

−⬁

and 兵h共x , t兲其 indicates the Fourier transform of the function inside the brackets—i.e., h*共s , t兲. The two particular cases of constant and linear drift are now studied in detail, using simple jump distributions b共z兲 that allow one to obtain exact solutions of Eq. 共3兲. *

A. Constant drift

Assuming constant drift, f共x兲 = k, the characteristic function of the PDF becomes p*共s , t兲 = p*共s , 0兲p*1共s , t兲p*2共s , t兲, with p*共s , 0兲 = exp共ix0s兲, given that p共x , 0兲 = ␦共x − x0兲, and p*1共s,t兲

= exp关共iks − ␴ s 兲t兴, 2 2

p*2共s,t兲 = exp兵␭关b*共s兲 − 1兴t其.

共6兲

Therefore the solution can be represented by the convolution between the transient PDF of the Wiener process with drift, whose characteristic function is p*共s , 0兲p*1共s , t兲 关8兴, and the PDF of the compound Poisson process. A similar result was obtained in 关17兴 in the absence of drift. The first case we consider is that of jumps extracted from a two-sided exponential distribution, also called a Laplace or Pascalian distribution, b共z兲 = ␥ exp共−␥兩z兩兲 / 2 共␥ ⬎ 0兲, so that p2共s , t兲 = exp关−␭s2t / 共␥2 + s2兲兴. In this case, the mean and variance vary linearly in time as ␮t = x0 + kt and vart = 2t共␴2 + ␭ / ␥2兲, respectively. Odd cumulants are zero—i.e., p共x , t兲 is symmetric—while even cumulants of order higher than 2, as expected, are not affected by the Gaussian noise and read ␬m = m!␭t / ␥m. Furthermore, the kurtosis is equal to 关19兴

␤=3+

6␭/␥ , t共␭/␥2 + ␴2兲2 4

共7兲

which means that for finite time the distribution has tails that are heavier than a Gaussian. It is easy to see that in the limiting case of no Gaussian noise 共i.e., ␴2 = 0兲 and both ␭ and ␥2 tending to +⬁ with a finite ratio, the solution tends to a Gaussian distribution 关20兴. A quite general choice for positive definite jumps 共z ⬎ 0兲 is the Gamma distribution, i.e., b共z兲 = ␥aza−1 exp共−␥z兲 / ⌫共a兲, where ⌫共·兲 is the gamma function 关19兴 and ␥ ⬎ 0 and a ⬎ 0 are two parameters. In this case, p*2共s,t兲

冋

册

␥t = exp − ␭t + , 共␥ − is兲a a

共8兲

and the mean and variance of x depend on time as ␮t = x0 + 共k + a␭ / ␥兲t and vart = t关2␴2 + a␭共1 + a兲 / ␥2兴, respectively,

6a + 11a2 + 6a3 + a4 ␭, t关共a + a2兲␭ + 2␥2␴2兴2

共9兲

which is again greater than 3. It is interesting to note that in the particular case when a = 1—that is, when the gamma distribution reduces to an exponential one, all the even cumulants correspond to those of the previous case, Eq. 共7兲, with jumps distributed as a two-sided exponential PDF. It is clear that the constant drift only affects the means of the distributions but not their shape. In fact, using the coordinate transformation ␩ = x − kt and ␶ = t, the problem with constant drift can be reduced to a sum of noises without any deterministic component in the new variables ␩ and ␶. This latter case was already discussed in part in 关17兴 within a more general context. B. Linear drift

Another paradigmatic case is that of linear drift, f共x兲 = −kx 共k ⬎ 0兲, which, in the absence of the Poisson noise, is the well-known Ornstein-Uhlenbeck process. A general solution of the problem with both the noises can be written again as a product of characteristic functions p*共s , 0兲p*1共s , t兲p*2共s , t兲, with now

冋

p*1共s,t兲 = exp ix0se−kt +

再冕 再冕

p*2共s,t兲 = exp

s

册

␴2s2 −2kt 共e − 1兲 , 2k

␭ * 关b 共u兲 − 1兴du ku

⫻ exp −

s exp共−kt兲

冎

冎

␭ * 关b 共u兲 − 1兴du , ku

共10兲

where u is a dummy variable. Assuming b共z兲 to be a two-sided exponential distribution, the function p*2共s , t兲 becomes p*2共s,t兲 =

冉

␥2 + s2e−2kt ␥2 + s2

冊

␭/2k

,

共11兲

which leads to mean and variance ␮t = x0e−kt and vart = 关1 − exp共−2kt兲兴关共␭ + ␥2␴2兲 / 共k␥2兲兴, respectively. Odd cumulants are zero 共i.e., symmetric PDF兲, while even cumulants higher than 2 are given by

␬m = 共1 − e−mkt兲

共m − 1兲!␭ . k␥m

共12兲

In the absence of Gaussian noise 共e.g., ␴ = 0兲, the steady-state PDF, obtained by inverting p*共s , 0兲p*2共s , t兲 关21兴 for t → ⬁, reads

026108-2

PHYSICAL REVIEW E 73, 026108 共2006兲

PROBABILISTIC DYNAMICS OF SOME JUMP-…

p共x兲 =

2␯兩x兩−␯␥1−␯K␯共␥兩x兩兲 , 冑␲ ⌫ 1 − ␯ 2

冉 冊

冉

␥2 + s2e−2kt ␥2 + s2

冊

␭/2k

⫻ exp兵i␭关tan−1共s/␥兲

− tan−1共se−kt/␥兲兴其,

共14兲

p*1共s , t兲

while reads as in Eq. 共10兲. The mean in this case varies as ␮t = x0e−kt + 共1 − e−kt兲␭ / 共k␥兲, the variance is vart = 关1 − exp共−2kt兲兴关共␭ + ␥2␴2兲 / 共k␥2兲兴, and the other cumulants are expressed by Eq. 共12兲. This means that the even cumulants of the process with jumps distributed as one-side and two-sided exponential PDF’s coincide. In the special case when ␴ = 0, the process is always positive and provided that ␭ / k 艌 1 the steady-state PDF is, as well known, a gamma distribution 关24,25兴 ␭/k

p共x兲 =

␥ x␭/k−1e−␥x . ⌫共␭/k兲

共15兲

The state-dependent effect of external Gaussian fluctuations is commonly described by the so-called multiplicative noise 关2–4兴. Differently from the additive noise, the multiplicative one may introduce substantial changes in the properties of the system, such as noise-induced phase transitions 关2兴 and the appearance of power-law tails 关18,22,26–28兴. The case of noise with state-dependent jumps is also interesting 关29兴, but it will not be discussed here. Assuming a multiplicative Gaussian noise and random jumps exponentially distributed, Eq. 共3兲 becomes

+ ␴2 +␭

再冋

f共x兲 + ␴2g共x兲

冕

册 冎

d g共x兲 p共x,t兲 dx

␥e−␥共x−z兲 p共z,t兲dz,

0

which can be written in the form 关1兴

+␭

冕

x

册

⳵ d g共x兲 p共x,t兲 − ␴2 关g2共x兲p共x,t兲兴 dx ⳵x

e−␥共x−z兲 p共z,t兲dz.

共18兲

0

Given the mathematical complexity of Eq. 共16兲, transient solutions are not considered here, but the attention is focused on the steady-state conditions, in which case the current is constant. Hereinafter, it will be assumed that either a natural boundary or a reflecting barrier is present at x = 0, so that the probability current vanishes in steady-state conditions 关2兴. We analyze in particular the case of linear drift, f共x兲 = −kx, and linear multiplicative Gaussian noise, g共x兲 = x. In this case, the vanishing probability current leads to the ordinary integro-differential equation J共x兲 = 共␴2 − k兲xp共x兲 − ␴2

d 2 关x p共x兲兴 + ␭ dx

冕

x

e−␥共x−z兲 p共z兲dz = 0.

0

共19兲 Laplace transforming the previous relation and using the condition 关xp共x兲兴x=0 = 0 gives

␴ 2s

d2 d ␭ 2 ˜ ˜p共s兲 − ˜p共s兲 = 0, 2 p共s兲 + 共␴ − k兲 ds ds ␥+s

共16兲

共20兲

where ˜p共s兲 is the Laplace transform of p共x兲. The solution of Eq. 共20兲 is 关30兴

冉 冊 s ␥

2F 1共 ␣

+ 1, ␤ + 1,2;1 + s/␥兲,

共21兲

where C1 is a constant determined by the condition ˜p共0兲 = 1, 2F1共· , · , · ; · 兲 is the hypergeometric function 关19兴 and

␣=

− k + 冑k2 + 4␭␴2 , 2␴2

␤=

− k − 冑k2 + 4␭␴2 . 2␴2

共22兲

Going back to the original variable x 关31兴, the solution reads +␣ 共␥x兲, p共x兲 = Ce−␥xx␣−1L−−␣␤−1

共23兲 Lm n 共·兲

⳵2 2 关g 共x兲p共x,t兲兴 − ␭p共x,t兲 ⳵x2 x

冋

J共x,t兲 = f共x兲 + ␴2g共x兲

˜p共s兲 = C1 1 +

III. MULTIPLICATIVE NOISE

⳵ ⳵ p共x,t兲 = − ⳵t ⳵x

共17兲

where the probability current J共x , t兲 is

where ␯ = 共1 − ␭ / k兲 / 2, K␯共·兲 is the modified Bessel function of the second kind, and ⌫共·兲 is the gamma function 关19兴. The PDF is symmetric, and when ␯ ⫽ 0 it decays as a power-exponential distribution 关22兴, since for 兩x兩 → + ⬁ it tends to zero as 兩x兩␭/共2k兲 exp共−␥兩x兩兲. At the origin, it has a finite value when ␯ ⬎ 0—i.e., ␭ ⬍ k—while it goes to +⬁ when k ⬍ ␭. In particular, when ␯ = 0—i.e., ␭ = 2k—the PDF has the same distribution as the one of the jumps that is a two-sided exponential PDF, p共x兲 = ␥e−␥兩x兩 / 2; this fact was already noted in 关23兴. Using a one-sided exponential distribution for positive jumps, the case of linear drift leads to p*2 =

⳵ ⳵ p共x,t兲 = − J共x,t兲, ⳵t ⳵x

共13兲

is the genwhere C is a constant of normalization and eralized Laguerre polynomial 关19兴. A remarkable property of p共x兲 is that, since it decays at infinity as a power law—e.g., 2 p共x兲 ⬃ x−共1+k/␴ 兲 when x → + ⬁ 关19兴—its extremes are scale invariant 关22,26–28兴. Figure 1 represents in a log-log plot the tails of the PDF’s for different values of the noise strength, showing the power-law decay for high values of x with slope that tends to −1 as ␴ increases. Interestingly, in the absence of Gaussian noise, the PDF loses this property and Eq. 共23兲 reduces to the gamma distribution of Eq. 共15兲. On the con-

026108-3

PHYSICAL REVIEW E 73, 026108 共2006兲

E. DALY AND A. PORPORATO

tiplicative Gaussian noise, stabilizes the system generating intermittent bursts with power-law tails, with a mechanism that is similar to the ones discussed in 关18,22,26兴. In the limiting case where the Poisson process has very frequent events 共e.g., high values of ␭兲 and small jumps 共e.g., high values of ␥兲, the Langevin equation tends to x˙ = ␭ / ␥ − kx + x␰t, which was studied in 关27兴; the corresponding steadystate PDF 关Eq. 共23兲兴 tends to 2

p共x兲 = Cx−2−k/␴ exp关− ␭/共␥␴2x兲兴,

共24兲

which can be normalized provided that k ⬍ −␴2.

IV. APPLICATIONS FIG. 1. Log-log plot showing the heavy tails of the PDF in the case of f共x兲 = −kx and g共x兲 = x with positive exponentially distributed jumps. The slope of the curves 共e.g., −1 − k / ␴2兲 tends to −1 with increasing ␴ 共in the figure ␴ varies from 0.1 to 0.8 with steps of 0.1兲.

trary, when the Poisson noise is turned off, the system is the well-known multiplicative Gaussian process the solution of which is a log-normal distribution with mode exponentially decaying to zero 共the process does not reach a steady state兲. Figure 2 shows two time series 共obtained following the numerical scheme reported in 关32兴兲 and the corresponding PDF’s, Eq. 共23兲. Figure 2共b兲, in particular, represents a very intermittent case, in which the drift is zero and both the rate of occurrence and the mean size of the jumps are very low, so that the dynamics is mainly driven by the Gaussian noise. However, it is the presence of the Poisson process that allows the system to reach a steady state by providing a repulsion from the origin. Thus, the Poisson noise, coupled to the mul-

A. Virtual waiting-time process

The so-called Takàcs process appears in queuing and storage problems 关8兴 and in stochastic input-output systems, as discussed in 关5,6兴. Assuming, for example, that the dynamics of the stock level x共t兲 is characterized by a constant decay rate k plus a superposition of continuous small inflows and outflows, modeled as a Wiener process, and large instantaneous inflows, described by a Poisson process with rate ␭ and with size exponentially distributed with mean 1 / ␥, the probability current in stationary conditions 关e.g., Eq. 共18兲兴 is J共x兲 = − kp共x兲 − ␴2

⳵ p共x兲 + ␭ ⳵x

冕

x

e−␥共x−z兲 p共z兲dz = 0.

0

共25兲 Solving the equation by Laplace transform leads to the solution 关5兴 p共x兲 =

冋

冉

− k − ␥␴2 − a k − ␭/␥ 2 共k − ␥ ␴ + a兲exp x 2 ␴ 2a 2 ␴ 2a

冉

− 共k − ␥␴2 − a兲exp

− k − ␥␴2 + a x 2 ␴ 2a

冊册

,

冊

共26兲

where a = 冑共k − ␥␴2兲2 + 4␭␴2. For the system to be stable and reach a steady state, the condition ␭ / ␥ ⬍ k must hold 关8兴. Examples of two different PDF’s for different ␴ are shown in Fig. 3. As ␴ decreases, the value of the PDF at zero moves towards higher values, and for ␴ = 0 the PDF becomes exponential with an atom of probability in x = 0, which is the classic solution of Takàcs 关8兴. B. Daily soil moisture dynamics

FIG. 2. Examples of time series and corresponding PDF’s for linear drift, f共x兲 = −kx, and multiplicative noise, g共x兲 = x. Parameters are 共a兲 k = 0.05, ␭ = 0.25, 1 / ␥ = 1.5, ␴ = 0.1 and 共b兲 k = 0, ␭ = 0.01, 1 / ␥ = 0.1, ␴ = 0.25.

Simple but realistic models capturing the essential dynamics of the terrestrial water balance are important to analyze the linkage between soil moisture and the climate, soil, and vegetation system. Although in previous studies 关25,33–35兴 the only stochastic component in soil moisture dynamics was assumed to be rainfall intermittency, in what follows we provide a first attempt to include also fluctuations in potential evapotranspiration. Considering the soil as a constant storage capacity w0, the daily soil moisture balance at a point is expressed as 关15,25兴

026108-4

PHYSICAL REVIEW E 73, 026108 共2006兲

PROBABILISTIC DYNAMICS OF SOME JUMP-…

FIG. 3. Different PDF’s for the virtual waiting-time problem for noise intensities ␴ = 0.1 共solid line兲 and 0.4 共dashed line兲 and corresponding time series. Common parameters: ␭ = 0.1, 1 / ␥ = 1, and k = 0.5.

w0

dx˙ = I„x共t兲,t… − E„x共t兲,t…, dt

共27兲

where x is the effective relative soil moisture defined between 0 共e.g., dry soil兲 and 1 共e.g., soil saturation兲 关25,33兴. Infiltration I(x共t兲 , t) is the amount of rainfall entering into the soil, while E(x共t兲 , t) is the evapotranspiration rate. Although Eq. 共27兲 is modeled in continuous time, the soil water balance—e.g., Eq. 共27兲—is interpreted at the daily time scale 关15,25,33,34兴. At this scale, precipitation can be idealized as a compound Poisson process, whose events, occurring with rate ␭, carry a random amount of water extracted from an exponential distribution with mean ␣. An example of daily rainfall values, measured at Duke Forest, NC 共USA兲, is shown in Fig. 4共a兲. For the sake of simplicity, canopy interception is ignored here 共see 关33,34兴 for details兲. Infiltration is modeled assuming that the soil accommodates all the incoming rainfall if the rainfall depth is smaller than the available storage capacity, 1 − x; otherwise, the rainfall in excess is instantaneously lost as runoff and drainage. Evapotranspiration E(x共t兲 , t) is assumed to be a function of both soil moisture x and potential evapotranspiration E p共t兲—i.e., the maximum rate of transpiration achievable by the extant vegetation under well-watered conditions 共x = 1兲. A simple way to model evapotranspiration is to assume E(x共t兲 , t) to decrease linearly from E p共t兲 at x = 1 to zero at x = 0 关25兴. However, as is apparent from Fig. 4共b兲, potential evapotranspiration fluctuates from day to day depending on climatic conditions 共mainly solar radiation, air temperature, and air humidity兲. A suitable way to model such fluctuations is to assume E p共t兲 to be the sum of a mean potential evapotranspiration ¯E p plus a white Gaussian noise E⬘共t兲 of intensity ␴⬘ 共we refer to 关15兴 for details兲. With these assumptions, Eq. 共27兲 can be normalized to become

FIG. 4. Time series in the year 2001 of 共a兲 precipitation and 共b兲 potential evapotranspiration from Duke Forest, NC 共USA兲.

dx˙ = I„x共t兲,t… − 关␩x共t兲 + x共t兲␰共t兲兴, dt

共28兲

where ␩x共t兲, with ␩ = ¯E p / w0, is the deterministic component of evapotranspiration and x共t兲␰共t兲 is the stochastic forcing of E, where the new noise intensity is ␴ = ␴⬘ / w0. Following 关33兴, the amount of water carried by each rainfall event is assumed to be extracted from an exponential PDF with mean 1 / ␥ = ␣ / w0. Although the problem is similar to the case presented in Sec. III, Eq. 共28兲 describes now a process that is bounded at x = 1 because of saturation. However, as discussed in 关33兴, the presence of the bound at x = 1 共which acts as a reflecting barrier兲 only rescales the PDF of x in the range 共0, 1兲. Thus the solution is the same as Eq. 共23兲, where the constant is now determined by the condition 兰10 p共x兲dx = 1. We note that the interpretation of the multiplicative noise in Eq. 共28兲 must be according to Stratonovich, since the approximation of external fluctuations in evapotranspiration as a white noise follows from an adiabatic approximation of the weakly colored 共differentiable兲 signal shown in Fig. 4共b兲 共see 关15兴 for details兲. Examples of two series of modeled soil moisture using different values of ␴ are shown in Fig. 5 along with their PDF’s. While in the time series the effect of the fluctuations in the potential evapotranspiration is hardly visible, unless the value of ␴ is chosen unreasonably high, the PDF’s show some interesting differences. Increasing the noise strength moves the mode of the PDF towards lower values, as is common in systems with multiplicative noise 关3兴, while the typical increase of the variance caused by increased noise is in part prevented by the presence of the bound in x = 1. Overall, the changes induced by evapotranspiration are relatively

026108-5

PHYSICAL REVIEW E 73, 026108 共2006兲

E. DALY AND A. PORPORATO

Solving the integrals with respect to x and reorganizing the terms gives ␭ ␭ − 共␩ − ␴2兲具x典 − ␥ ␥

冕

1

e−␥共1−z兲 p共z兲dz − ␴2关p共x兲兴x=1 = 0.

0

共31兲

FIG. 5. Top: realizations of soil moisture with different values of noise intensity 共␴ = 0.8, solid line, and ␴ = 0.4, dashed line兲. Bottom: soil moisture PDF’s for ␴ varying from 0.2 共solid line兲 to 0.8 共dotted line兲 with steps of 0.2. Common parameters: ␭ = 0.1 d−1, w0 = 9 cm, ␣ = 1.5 cm 共␥ = 6兲, and ¯E p = 0.4 cm d−1.

small, despite the fact that its potential variability is similar to the one of the rainfall process 关15兴, suggesting that rainfall variability is more effective than fluctuations in evapotranspiration, the action of which is also reduced by their multiplicative nature. Once the probability distribution is obtained, an equation for the dynamics of the mean water balance 共i.e., the mean of x兲 can be derived. The long-term water balance allows one to distinguish the long-term effect of each single component of the balance expressed by Eq. 共28兲, describing how the rainfall input is partitioned between the different soil water losses 关34兴. Multiplying Eq. 共3兲 by x and integrating between 0 and 1 leads to d具x典t = dt

冕

1

0

⳵ x 关p共x,t兲兴dx = − ⳵t

冕

1

x

0

⳵ J共x,t兲dx, ⳵x

共29兲

which, in stationary conditions, only depends on the probability current −

冕

1

0

x

冕 冋冕

d J共x兲dx = 关− xJ共x兲兴10 + dx +

冕 冕

1

0

+

␭e−␥x

1

J共x兲dx = −

0 x

1

0

␴2xp共x兲dx −

冕

1

0

V. CONCLUSIONS

Dynamical systems driven by Gaussian and Poisson noises 共jump-diffusion processes兲 have been studied here for both additive and multiplicative Gaussian noise. Formal solutions 共characteristic functions and cumulants兲 of the forward Chapman-Kolmogorov equation have been obtained for general jump distributions in case of constant and linear drift with additive Gaussian noise and jumps distributed as two- and one-sided exponential PDF’s. The case of linear Gaussian multiplicative noise with exponentially distributed jumps has been solved in stationary conditions. Such a solution is characterized by power-law tails, resulting from the interaction of the Poisson noise, which ensures repulsion of the system from the origin, with multiplicative Gaussian fluctuations. The same solution, with slight modifications, also provides a simplified description of the daily soil moisture dynamics in the presence of both rainfall variability and climatic fluctuations.

␩xp共x兲dx

0

ACKNOWLEDGMENTS

e␥z p共z兲dz dx

1

0

册

冕

The first term is the mean rainfall rate, which is the input of water into the system. The second represents the averaged losses due to evapotranspiration, and the last two terms are related to the presence of the bound at x = 1. In particular, the third term describes the averaged losses caused by leakage and runoff, while the last one, which is in general negligible, is a spurious term introduced by the interaction between the bound and negative tail of the potential evapotranspiration modeled with a Gaussian distribution. Interestingly, Eq. 共31兲 shows the physical effects of the spurious drift, ␴2具x典, resulting from the necessary Stratonovich interpretation of the fluctuations in potential evapotranspiration to account for their temporal autocorrelation. Here we only notice that the effect is one of a slight reduction of the transpiration losses, when compared to a similar case with same mean but no fluctuations in potential evapotranspiration 共e.g., constant E p = ¯E p = 0.5 cm d−1兲 关25兴, followed by a readjustment of the partitioning between the different soil water losses. A detailed discussion of the hydrologic implications of Eq. 共31兲 is presented elsewhere 关15兴.

d 2 2 关␴ x p共x兲兴dx = 0. dx 共30兲

This research was supported in part by the Office of Science 共BER兲, U.S. Department of Energy, Grant No. DEFG02-95ER62083, through its Southeast Regional Center 共SERC兲 of the National Institute for Global Environmental Change 共NIGEC兲 under Cooperative Agreement No. DEFC02-03ER63613.

026108-6

PHYSICAL REVIEW E 73, 026108 共2006兲

PROBABILISTIC DYNAMICS OF SOME JUMP-… 关1兴 G. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. 共Springer-Verlag, Berlin, 1990兲. 关2兴 W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology 共Springer-Verlag, Berlin, 1984兲. 关3兴 A. Schenzle and H. Brand, Phys. Rev. A 20, 1628 共1979兲. 关4兴 C. R. Doering, Phys. Lett. A 122, 133 共1987兲. 关5兴 D. Perry and W. Stadje, Probab. Eng. Mech. 16, 19, 共2002兲. 关6兴 O. Kella, D. Perry, and W. Stadje, Probab. Eng. Mech. 17, 1 共2003兲. 关7兴 J. Łuczka, R. Bartussek, and P. Hänggi, Europhys. Lett. 31, 431 共1995兲. 关8兴 D. R. Cox and H. D. Miller, The Theory of Stochastic Processes 共Methuen, London, 1965兲. 关9兴 T. Czernick, J. Kula, J. Łuczka, and P. Hänggi, Phys. Rev. E 55, 4057 共1997兲. 关10兴 J. Łuczka, T. Czernick, and P. Hänggi, Phys. Rev. E 56, 3968 共1997兲. 关11兴 G. L. Gerstein and B. Mandelbrot, Biophys. J. 4, 41 共1964兲. 关12兴 P. Lánský and J. P. Rospars, Biol. Cybern. 72, 397 共1995兲. 关13兴 N. Hohn and A. N. Burkitt, Phys. Rev. E 63, 031902 共2001兲. 关14兴 M. Abundo, Open Syst. Inf. Dyn. 11, 105 共2004兲. 关15兴 E. Daly and A. Porporato 共unpublished兲. 关16兴 H. C. Tuckwell and F. Y. M. Wan, J. Appl. Probab. 21, 695 共1984兲. 关17兴 Ph. Blanchard and M.-O. Hongler, Phys. Lett. A 180, 225 共1993兲. 关18兴 H. Takayasu, A.-H. Sato, and M. Takayasu, Phys. Rev. Lett. 79, 966 共1997兲.

关19兴 Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共Dover, New York, 1965兲. 关20兴 C. Van Den Broeck, J. Stat. Phys. 31, 467 共1983兲. 关21兴 P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation 共Birkhäuser Verlag Basel, 1971兲, Vol. I. 关22兴 D. Sornette, Critical Phenomena in Natural Sciences, 2nd ed. 共Springer-Verlag, Berlin, 2004兲. 关23兴 R. Zygadło, Phys. Lett. A 329, 459 共2004兲. 关24兴 D. R. Cox and V. Isham, Adv. Appl. Probab. 18, 558 共1986兲. 关25兴 A. Porporato, E. Daly, and I. Rodriguez-Iturbe, Am. Nat. 164, 625 共2004兲. 关26兴 D. Sornette and R. Cont, J. Phys. I France 7, 431 共1997兲. 关27兴 D. Sornette, Physica A 250, 295 共1998兲. 关28兴 S. Solomon and M. Levy, Int. J. Mod. Phys. C 7, 754 共1996兲. 关29兴 A. Porporato and P. D’Odorico, Phys. Rev. Lett. 92, 110601 共2004兲. 关30兴 A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations 共CRC Press, Boca Raton, 1995兲. 关31兴 A. P. Prudnikov, Y. A. Brychov, and O. I. Marichev, Integrals and Series 共Gordon and Breach, New York, 1986兲, Vol. 5. 关32兴 J. M. Sancho, M. San Miguel, S. L. Katz, and J. D. Gunton, Phys. Rev. A 26, 1589, 共1982兲. 关33兴 I. Rodriguez-Iturbe, A. Porporato, L. Ridolfi, V. Isham, and D. R. Cox, Proc. R. Soc. London, Ser. A 455, 3789 共1995兲. 关34兴 F. Laio, A. Porporato, L. Ridolfi, and I. Rodriguez-Iturbe, Adv. Water Resour. 24, 707 共2001兲. 关35兴 F. Laio, A. Porporato, L. Ridolfi, and I. Rodriguez-Iturbe, Phys. Rev. E 63, 036105 共2001兲.

026108-7