Stochastic processes via the pathway model

STOCHASTIC PR OCESSES VIA THE P A THW A Y MODEL Arak M. Mathai Cen tre for Mathematical and Sta tistical Sciences, Pe ec hi Campus, KF RI P eec hi-680653 , Kerala, India directorcms458@gmail.com and Departmen t of Mathematics and Statistics, McGill Univ ersit y 805 Sherbro oke Street W est, Mon treal, Queb ec, Canada , H3A2K6 Hans J. Haub old Cen tre for Mathematical and Sta tistical Sciences, Pe ec hi Campus, KF RI P eec hi-680653 , Kerala, India hans.haub old@gmail.com and Office for Outer Space Affair s, United Nations P .O. Box 500, Vienna In t ernat io nal Center, A-1400 Vienna, Austria Abstract After collecting data from observ ations or exp erimen t s, the next step is to build an appropriate mathematical or s to c hastic mo del to describ e the data so that further studies can b e done with the help of the mo dels. In this article, the input-output type mec hanism is considered first, where reaction, diffus ion, reaction-diffusion, a nd pro duction-destruction ty p e phys ical situations can fit in. Then tec hniques ar e des crib ed to pro duce thic k er or thinner tails (p o w er law b eha vior) in sto c hastic mo dels. Then the pathw a y idea is describ ed where one can switc h to differen t functional forms of the probabilit y densit y function) through a parameter called the path wa y parameter. Keyw ords Data analys is, mo del building, input- output t yp e sto c hastic mo dels, thic ker or thinner-tailed mo dels, path w ay idea, pathw a y mo dels. 2000 Mathematics Sub ject Classific ation: 44A3 5 , 26A33, 47G10 1. In tro duction After collecting da ta from exp erimen ts or from observ ations, the next step is 1 to study the data and mak e inference out o f the da ta. This can b e ac hieve d b y using mathematical metho ds and mo dels if the ph ysical situation is deterministic in nature, otherwise create sto c hastic mo dels if the ph ysical situation is no n- deterministic in nature. If the underlying phenomenon, whic h created the data, is unkno wn, p ossibly deterministic but the underlying factors and the wa y in whic h these fa ctors act are unkno wn, thereb y t he situation b ecomes random in nature. Then w e go for sto c hastic mo dels o r non-deterministic type mo dels. One has to kno w or sp eculate ab out the underlying factors as w ell as the w ay in which these factors a ct so that one can decide whic h category of mo dels are appropriate. If the observ ations are a v ailable ov er time then a time series type of mo del ma y b e appropriate. If the time series sho ws p erio dicities then eac h cycle can b e analyzed b y using sp ecific types of sto c ha stic mo dels. Here w e will consider mo dels to describ e short-term b ehavior of data or b eha v- ior within one cycle if a cyclic b ehavior is noted. When monitoring solar neutrinos it is seen that there is lik ely to b e an elev en y ear cycle and within eac h cycle the b eha vior of the graph is something lik e slo w increase with sev eral lo cal p eaks to a maxim um p eak and slo w decrease with hum ps bac k to normal le v el. In such situ- ations, what is observ ed is not really what is actually pro duced. What is observ ed is the residual part of what is pro duced min us w hat is consumed or con v erted and th us the a ctual observ ation is made on the residual part only . Man y of na t ural phenomena belong to this t ype of behavior of the form u = x − y where x is the input or pro duction v aria ble and y is the o ut put or consumption or destruction v ariable and u r epresen ts the residual part whic h is o bserv ed. A general analysis of input-output situation ma y b e s een from [21]. In man y situations one can assume that x and y are statistically indep enden tly distributed and that u ≥ 0 means pro duction dominates ov er destruction or input dominates o v er the output. In reaction- rate theory , when particles react with each other pro ducing new particles or pro ducing neutrinos w e ma y ha v e the f o llo wing t ype of situations. Certain particles may react with eac h other in short-span o r short-time p erio ds and pro duce small n um b er of particles, others ma y tak e medium time in t erv als and pro duce larger num b ers of particles and y et o thers ma y react ov er a long span and pro duce larger n um b er of part icles. F o r desc ribing suc h types of situations in the pro duction of solar neutrinos the presen t authors considered creating mathe- matical mo dels b y erecting triangles whose ares are prop ortiona l to the neutrinos pro duced, see [5],[6 ],[7 ],[8]. Another approach that w a s adopted w as to assume x and y as independen tly distributed ra ndom v ariables, then w ork out the densit y of the residual v ariable under the assumption that x − y ≥ 0. The simplest suc h situation is an exp onen tial 2 t ype input a nd an exp onen tial ty p e output. T hen w e can lo ok at sum of suc h indep enden tly distributed residual t ype v ariables. This is a reasonable type of assumption. Then input-output mo del has the Laplace dens it y , when x and y are iden tically and indep enden tly distributed and the densit y is g iv en b y , f 1 ( u ) = β 1 2 e − β 1 | u − α 1 | , 0 ≤ u < ∞ , β 1 > 0 , (1 . 1) and f 1 ( u ) = 0 elsewhere, where α 1 is a lo cation para meter. Note that β 1 can act as a scale para meter or as a dispersion or scatter parameter. Supp ose tha t this situation is rep eated at successiv e lo cations and with the scale parameter β = β 1 , β 2 , ... . Then the nature of t he graph will b e that of a sum o f Laplace densities. If the lo cation para meters are sufficien tly fa rther apart then the graph will lo ok lik e that in Figure 1(b). If suc h blips are o ccurring sufficien tly close together then we hav e a graph of the t ype in Figure 1(a). In these graphs w e ha v e tak en only fiv e to six lo cations f or simplic it y . But b y taking successiv e lo cations w e can generate many of the phenomena that are seen in nature, esp ecially in t ime series data. When the lo cations are sufficien tly closer w e get the graph with sev eral lo cal maxima/spike s and a con tin uous curv e. This is the t yp e of b eha vior seen in solar neutrino pro duction. Cyclic patterns can also a r ise dep ending up on t he lo cation and scale parameters. Here β 1 measures the in tensit y of t he blip and α 1 the lo cat io n where it happ ens, and eac h blip is the residual effect of an exp onen tial t ype input and an independen t exp onen tial t yp e output o f the same strength. If α 1 , α 2 , ... are farther apart then the con tributions coming from other blips will b e negligible and if α 1 , α 2 , ... a r e close tog ether then there will b e contributions from other blips. The function will b e of the follo wing form: f ( u ) = k X j =1 β j 2 e − β j | u − α j | , 0 ≤ u < ∞ , β j > 0 , j = 1 , ..., k < ∞ (1 . 2) If one requires f ( u ) to b e a densit y within a n um b er of spik es then divide the sum b y k so that w e ha v e a conv ex com bination of Laplace densities, whic h will again b e a densit y . The mo del do es not require that w e create a densit y out of the pattern. I f the arriv al of the lo cation p oints ( α j ) is go v erned b y a P oisson pro cess then we will ha v e a P oisson mixture of Laplace densities. 3 (a) (b) Figure 1 (a) (b) A symmetric Laplace densit y will b e of the following form: f 2 ( u ) = 1 2 β e − | u | β , − ∞ < u < ∞ (1 . 3) and the graph is of the follo wing form: Figure 2 Symmetric Laplace densit y This is the symmetric case where u < 0 b ehav es t he same w a y as u ≥ 0. If the b eha vior of u is differen t fo r u < 0 and u ≥ 0 then we get the asymmetric La pla ce case whic h can b e written a s g ( u ) = ( 1 ( β 1 + β 2 ) e u β 1 , − ∞ < u < 0 1 ( β 1 + β 2 ) e − u β 2 , 0 ≤ u < ∞ (1 . 4) 4 Figure 3 Asymmetric Laplace case When α 1 > 1 , α 2 > 1 , α 1 = α 2 = α , β 1 = β 2 = β w e hav e indep endently and iden tically distributed gamma random v ariables for x and y and u = x − y is the difference b etw een them. Then g 1 ( u ) can b e seen t o be the follo wing: g 1 ( u ) = u 2 α − 1 e − u β β 2 α Γ 2 ( α ) Z ∞ z =0 (1 + z ) α − 1 z α − 1 e − 1 β (2 uz ) d z (1 . 5) for u ≥ 0, α > 0 , β > 0. This b eha v es lik e a gamma density and prov ides a symmetric mo del for u ≥ 0 and u < 0. The following is the nature of the gra ph. Figure 4 g 1 ( u ) in the symmetric gamma t ype input-output v aria bles 2. Mo dels with Thic k er and Thinner T ails 5 F or a large num b er of situations a gamma type mo del ma y be appropriate. A t w o parameter gamma densit y is of the type f ( x ) = 1 β α Γ( α ) x α − 1 e − x β , x ≥ 0 , α > 0 , β > 0 . (2 . 1) Sometimes a member from this parametric family of functions ma y b e a ppro priate to de scrib e a data set. Sometimes the data require a sligh tly thic ke r-tailed mo del due to chances of higher probabilities or more area under the curv e in the tail. Tw o of suc h mo dels dev eloped b y t he authors’ groups will be describ ed here. One t ype is where the mo del in (2.1 ) is app ended with a Mittag-Leffler series and another t ype is where (2.1) is app ended with a Bessel series, see also [25]. 2.1. Gamma mo del with app ended Mittag-Leffler function Consider a gamma densit y o f t he t yp e g 3 ( x ) = c 1 x γ − 1 e − x δ , δ > 0 , γ > 0 , x ≥ 0 . Supp ose that w e a pp end this g 3 ( x ) with Mittag- L effler function E β α,γ ( − x α ) where E β α,γ ( − ax α ) = ∞ X k =0 ( β ) k k ! ( − a ) k x α k Γ( γ + αk ) , α > 0 , γ > 0 . Consider the function f ∗ ( x ) = c ∞ X k =0 ( β ) k k ! ( − a ) k x αk + γ − 1 e − x δ Γ( γ + αk ) , x ≥ 0 where c is the normalizing constan t. Let us ev aluate c . Since the total in tegra l is 1, 1 = Z ∞ 0 f ∗ ( x )d x = c ∞ X k =0 ( β ) k k ! ( − a ) k Z ∞ 0 x αk + γ − 1 e − x δ Γ( γ + αk ) d x = c ∞ X k =0 ( β ) k k ! ( − a ) k δ αk + γ = c δ γ (1 + aδ α ) − β , | aδ α | < 1 for β > 0 , α > 0 , δ > 0 , aβ δ α < 1 , | aδ α | < 1 . Therefore the densit y is f ∗ ( x ) = (1 + aδ α ) β δ γ x γ − 1 e − x δ ∞ X k =0 ( β ) k k ! ( − a ) k δ αk Γ( γ + αk ) 6 for 0 ≤ x < ∞ , α > 0 , γ > 0 , δ > 0 , β > 0, | aδ α | < 1 , aβ δ α < 1. That is, f ∗ ( x ) = (1 + aδ α ) β δ γ x γ − 1 e − x δ [ 1 Γ( γ ) + ∞ X k =1 ( β ) k k ! ( − 1) k δ αk Γ( γ + α k ) ] . Note that a = 0 corresp onds to the original gamma densit y . The follow ing are some gra phs of the app ended Mittag-Leffler-gamma densit y . When a < 0 w e hav e thinner tail and when a > 0 we ha v e thic k er tails compared to the gamma tail. Figure 5 Gamma densit y with Mittag-Leffler function app ended 2.2. Bessel app ended gamma densit y Consider the mo del o f the t yp e of a basic g a mma densit y app ended with a Bessel function, see also [25]. ˜ f ( x ) = c x γ − 1 e − x δ ∞ X k =0 x k ( − a ) k k !Γ( γ + k ) , δ > 0 , γ > 0 , x ≥ 0 , where c is the normalizing constan t. The app ended function is o f the form 1 Γ( γ ) 0 F 1 ( ; γ : − ax ) whic h is a Bessel function. Let us ev aluate c . 1 = c ∞ X k =0 ( − a ) k k ! Z ∞ 0 x γ + k − 1 Γ( γ + k ) e − x δ d x = c δ γ ∞ X k =0 ( − a ) k δ k k ! = c δ γ e − aδ . 7 Hence the densit y is of the f orm ˜ f ( x ) = e aδ δ γ x γ − 1 e − x δ ∞ X k =0 x k ( − a ) k k !Γ( γ + k ) , x ≥ 0 , γ > 0 , δ > 0 . Figure 6 Gamma app ended with Besse l function Note : Instead of app ending with Bessel function one could hav e app ended with a general h ypergeometric series. But a general h yp ergeometric series do es not simplify in t o a con v enien t form. W e ha v e chose n sp ecialized parameters as w ell as suitable functions so that the normalizing constan ts simplify to con venie n t fo r ms thereb y general computations will b e m uc h easier and simpler. 3. Pa th w ay Idea Here w e cons ider a mo del whic h can switc h to three functional for ms co v ering almost all statistical densities in curren t use, see [23]. Let f ∗ 1 ( x ) = c ∗ 1 | x | γ [1 − a (1 − α ) | x | δ ) η 1 − α , α < 1 , η > 0 , a > 0 , δ > 0 , (3 . 1) and 1 − a (1 − α | x | δ > 0, where c ∗ 1 is the normalizing constan t. When α < 1 the mo del in (3.1) sta ys as the generalized t ype-1 b eta family , extended o v er the real line. When α > 1 write 1 − α = − ( α − 1) with α > 1. Then the functional form in (3.1) changes to f ∗ 2 ( x ) = c ∗ 2 | x | γ [1 + a ( α − 1) | x | δ ] − η α − 1 (3 . 2) for α > 1 , a > 0 , η > 0 , −∞ < x < ∞ . Note that (3.2) is the extended generalized t ype-2 b eta family of functions. When α → 1 then b o th (3.1) and (3.2 ) g o to 8 f ∗ 3 ( x ) = c ∗ 3 | x | γ e − aη | x | δ , a > 0 , η > 0 , δ > 0 , −∞ < x < ∞ . (3 . 3) Eq. (3.3) is the extended generalized gamma family of functions. Th us (3.1) is capable of switc hing to three families of functions. This is the pathw a y idea and α is the path w ay parameter. Through this parameter α one can reac h the three families of functions in (3 .1 ),(3.2), and (3 .3 ). The pathw a y idea was in tro duced b y Mathai [23]. The normalizing constan t s can b e seen to b e the following: c ∗ 1 = δ 2 [ a (1 − α )] γ +1 δ Γ( γ +1 δ + η 1 − α + 1) Γ( γ +1 δ )Γ( η 1 − α + 1) (3 . 4) for a > 0 , α < 1 , δ > 0 , γ > − 1 , η > 0, c ∗ 2 = δ 2 [ a ( α − 1)] γ +1 δ Γ( η α − 1 ) Γ( γ +1 δ )Γ( η α − 1 − γ +1 δ ) (3 . 5) for α > 1 , a > 0 , δ > 0 , η > 0 , δ > 0 , η α − 1 − γ +1 δ > 0, c ∗ 3 = δ 2 ( aη ) γ +1 δ Γ( γ +1 δ ) , a > 0 , δ > 0 , η > 0 , γ > − 1 . (3 . 6) Note that (3.1) is a finite rang e mo del, suitable to describe situations where the tails are cut off. When α comes closer and closer to 1 then the cut-off p oint mov es a w ay from the origin and ev entually go es to ±∞ . When α → 1 then mo del (3.1) go es to mo del (3.3) whic h is an extended generalized gamma mo del. The mo del in (3.2) is type-2 b eta form, spreads out ov er t he whole real line and the shap e will b e closer to that of a gamma t ype mo del when α approache s 1. Th us the pat hw ay mo dels in (3.1),(3.2), and (3.3) co v er all t ypes of situations where the tails are cut off, tails are made thinner or thic k er compared to a gamma t ype mo del. The extended ga mma type mo del in (3.3) also contains the Gaussian mo del, Brow nian motion, Maxw ell-Bo ltzmann density etc. If Gaussian or Maxw ell-Boltzmann is the stable or ideal f o rm in a ph ysical situation then the unstable neigh b orho o ds are co v ered by (3 .1) and (3.2) or the paths leading to t his s table form is describ ed b y (3.1) and (3.2). It is w orth noting that (3.1) for x > 0 , γ = 0 , a = 1 , δ = 1 , η = 1 is the Tsallis statistics for non-extensiv e statistical mec hanics. Also note that (3.2) for a = 1 , δ = 1 , η = 1 is sup erstatistics. This sup erstatistics can also b e deriv ed as the unconditional densit y when both the conditional dens it y of x giv en a parameter θ 9 and the margina l density of θ are gamma densities or exponential t yp e densities, the details may b e seen from Mathai and Haubo ld ([15], [26], [27 ], [29], [30]). V arious t yp es of mo dels whic h are applicable in a v ariety of situatio ns ma y b e seen from Mathai [2 5]. 4. Reaction Rate P robabilit y I n t egral Mo del Starting from 1980’s the presen t authors had pursued mathematical mo dels for reaction-rate theory in v arious situations suc h as non-resonant reactions and resonan t reactions under v arious cases suc h as depletion, high energy tail cut off etc, see [2],[3 ],[4], [5], [9], [11]. Th e basic mo del is a n inte gral o f the following form: I (1) = Z ∞ 0 x γ − 1 e − ax δ − z x − ρ , a > 0 , z > 0 , ρ > 0 , δ > 0 . (4 . 1) F or ρ = 1 2 , δ = 1 one has the basic probabilit y in tegral in the non-resonan t case, see [4]. F or γ = 0 , ρ = 1 one has Kr¨ atzel in t egra l [24]. F or γ = 0 , δ = 1 , ρ = 1 one has inv erse Gaussian densit y . Computational asp ect of (4.1) is discuss ed in [1] and related materia l ma y b e seen f rom [20]. Since the integral in (4.1) is a pro duct of integrable functions one can ev aluate the integral in (4.1) with the help of Mellin conv olution of a pro duct b ecause the in tegra nd can b e written as Z ∞ 0 1 v f 1 ( v ) f 2 ( u v )d v , f 1 ( x ) = x γ e − ax δ , f 2 ( y ) = e − y ρ (4 . 2) for u = z 1 ρ , u = xy . Then the Mellin conv olution of the integral in (4.1), denoting the Mellin transform of a function f with Mellin para meter s as M f ( s ), w e ha v e from (4.1) M I (1) ( s ) = M f 1 ( s ) M f 2 ( s ) , (4 . 3) where M f 1 ( s ) = Z ∞ 0 x s − 1 f 1 ( x )d x = Z ∞ 0 x γ + s − 1 e − ax δ d x = 1 δ Γ( s + γ δ ) a s + γ δ , ℜ ( s + γ ) > 0 and M f 2 ( s ) = Z ∞ 0 y s − 1 e − y ρ d y = 1 ρ Γ( s ρ ) , ℜ ( s ) > 0 . 10 Hence M I (1) ( s ) = M f 1 ( s ) M f 2 ( s ) = 1 ρδ a γ δ Γ( s + γ δ )Γ( s ρ ) a s δ . Therefore the in tegral in (4.1 ) is the in v erse Mellin transform of (4.3). That is, I (1) = 1 2 π i Z c + i ∞ c − i ∞ 1 ρδ a γ δ Γ( s + γ δ )Γ( s ρ )( ua 1 δ ) − s d s, u = z 1 ρ , i = √ − 1 = 1 ρδ a γ δ H 2 , 0 0 , 2 [ z 1 ρ a 1 δ   (0 , 1 ρ ) , ( γ δ , 1 δ ) ] (4 . 4) where H ( · ) is the H-function, see [28],[32]. F rom the basic result in (4.4) w e can ev aluate the reaction-rate probabilit y integrals in the other cases of non-r elativistic reactions. 4.1. Generalization of reaction-rate mo dels A companion integral corresp o nding to ( 4.1) is the integral I (2) = Z ∞ 0 x γ e − ax δ − z x ρ d x, a > 0 , δ > 0 , ρ > 0 , z > 0 . (4 . 5) In ( 4.1) we had x − ρ with ρ > 0 whereas in (4.5) w e hav e x ρ with ρ > 0. F or δ = 1, (4.5) corresp onds to the L a place transfor m or momen t generating function o f a generalized gamma de nsit y in statistical distribution theory . The in tegral in (4.5) can b e ev aluated. Note that (4.5) can b e written in the form of an in tegral of the form Z ∞ 0 v f 1 ( v ) f 2 ( uv )d v , f 1 ( x ) = x γ − 1 e − ax δ , f 2 ( y ) = e − y ρ (4 . 6) for u = z 1 ρ . The inte gral in (4.6) is in the structure of a Mellin transform of a ratio u = y x , so that the Mellin transform of I (2) is then M I (2) ( s ) = M f 2 ( s ) M f 1 (2 − s ) . (4 . 7) The in v erse Mellin transform in (4.7) giv es the in tegral I (2) . The pair of integrals I (1) and I (2) b elong to a par t icular case of a general vers atile mo del considered b y the authors earlier [33]. A generalization of I (1) and I (2) is the pathw a y generalized mo del, whic h results in the v ersatile in tegral. The pathw a y generalization is done b y r eplacing the t w o exp o nen tia l functions b y the corresp onding path w ay form. Consider the in tegrals of the following t yp es: 11 I p = Z ∞ 0 x γ [1 + a ( q 1 − 1) x δ ] − 1 q 1 − 1 [1 + b ( q 2 − 1) x ρ ] − 1 q 2 − 1 d x, (4 . 8) where q 1 > 1 , q 2 > 1 , a > 0 , b > 0 , . W e will ke ep ρ free, could b e negat iv e or p ositiv e. Note that lim q 1 → 1 [1 + a ( q 1 − 1) x δ ] − 1 q 1 − 1 = e − ax δ and lim q 2 → 1 [1 + b ( q 2 − 1) x ρ ] − 1 q 2 − 1 = e − bx ρ . Hence lim q 1 → 1 ,q 2 → 1 I p = Z ∞ 0 x γ e − ax δ − bx ρ d x whic h is t he integral in (4.5 ) a nd if ρ < 0 then it is the in tegra l in (4.1). The general in tegral in (4.8) b elongs to the general family of v ersatile inte grals. The factors in the integrand in (4.8) ar e of the generalize d t yp e-2 b eta form. W e could hav e taken eac h factor in type-1 b eta o r t yp e-2 b eta form, th us pro viding 6 differen t com binations. F or eac h case, w e could ha v e the situation of ρ > 0 or ρ < 0. The whole collection of suc h mo dels is kno wn as the v ersatile integrals. In t egr a l transforms, k no wn as P -transforms, a re also asso ciated with the integrals in (4.8), see for example [18],[19]. 4.3. F ractional c alculus mo dels In a series of pa p ers the authors ([12], [13], [14], [15], [17 ], [22], [28]) hav e show n recen tly that fra ctional in tegrals can b e classified into the forms in (4.2) and (4.6) or fr actional in tegral op erators o f the second kind or righ t-sided fractional integral op erators can b e considered as Mellin con volution of a pro duct as in (4.2 ) and left-sided or fractional integral op erators of the first kind can b e considered as Mellin con v olution of a ratio where the functions f 1 and f 2 are of the follow ing forms: f 1 ( x ) = φ 1 ( x )(1 − x ) α − 1 , 0 ≤ x ≤ 1 , f 2 ( y ) = φ 2 ( y ) f ( y ) (4 . 9) where φ 1 and φ 2 are pre-fixed functions, f ( y ) is arbitrary and f 1 ( x ) = 0 outside the in terv al 0 ≤ x ≤ 1. Th us, essen tially , all fractional in tegr a l op erator s b elong to the categories o f Mellin con v olution of a pro duct or ratio where one function is a m ultiple of type-1 b eta form and the other is arbitrary . The righ t-sided or t ype-2 fractional in tegral of order α is denoted b y D − α 2 ,u f and defined as D − α 2 ,u f = Z v 1 v f 1 ( u v ) f 2 ( v )d v (4 . 10) 12 and the left-sided o r type-1 fractional in tegral of order α is giv en b y D − α 1 ,u f = Z v v u 2 f 1 ( v u ) f 2 ( v )d v (4 . 11) where f 1 and f 2 are a s giv en in (4.9). Let n be a p ositive in teger suc h that ℜ ( n − α ) > 0. The smallest suc h n is [ ℜ ( α )] + 1 = m where [ ℜ ( α )] denotes the in teger part of ℜ ( α ). Here D − α 2 ,u f and D − α 1 ,u f are defined as in (4.10) and (4.11) resp ectiv ely . Let D = d d u the ordinar y deriv ativ e with resp ect to u and D n b e the n -th order deriv ativ e. Then the fractional deriv ativ e of order α is defined as D α f = D n [ D − ( n − α ) i,u f ] in the Riemann-Liouville sense and D α f = [ D − ( n − α ) i,u D n f ] in the Caputo sense (4 . 12 ) for i = 1 , 2, see also [31]. The input-o utput mo del t hat we started with, when applied to reaction- diffusion problems can result in fractional order r eactio n- diffusion differen tial equations. Suc h fractio na l o rder differen tial equations are seen t o pro vide so- lutions whic h are more relev ant to practical situations compared to the solutions coming from differen tial equations in the con v entional sense o r inv olving in teger- order deriv at iv es. Some of the relev ant papers in this direction ma y b e seen from [12], [13], [16], [17]. Ac kno wledgemen t The authors w ould lik e to thank the D epart ment of Science and T ec hnology , Go v ernmen t of India, for the financial assis tance for this w ork under Pro j ect Num b er SR/S4/MS:287 /05 and t he Cen tre for Mathematical Sciences India for facilities. References [1] W.J. Anderson, H.J. Haub old and A.M. Mathai (1994): Astroph ysical ther- mon uclear functions, Astr ophysics and Sp ac e Scienc e , 214(1-2) , 4 9 -70. [2] H.J. 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Mathai (2012) : Sto c hastic mo dels under p o w er transfor ma t io ns and exp o nen tia t ion, Journal of the Indian So ciety fo r Pr ob ability and Statistics , 13 , 1-19. [26] A.M. Mathai a nd H.J. Haub o ld (2 0 07): On generalized en tro p y measure and path w ays , Physic a A , 385 , 493-500 . [27] A.M. Mathai and H.J. Haub old (20 08): Path w a y parameter and t hermonu- clear functions, Physic a A , 387 , 2462-2470 . [28] A.M. Mathai a nd H.J. Haub old (2 008): Sp e cial F unctions for Applie d Sci- entists , Springer, New Y o rk. [29] A.M. Mathai a nd H.J. Haubold (2011): Mittag-Leffler functions to path wa y mo del to Tsallis statistics, Inte gr al T r ansforms and Sp e cial F unctions , 21(11) , 867-875 . [30] A.M. Mathai and H.J. Haub old (201 1 ): A path w ay for Ba y esian statistical analysis to sup erstatistics, Appli e d Mathema tics and Computations , 218 , 799-804 . [31] A.M. Mathai and H.J. Haub old (20 13): Erdelyi-Kob er fractional in tegral op erators from a statistical p ersp ectiv e I-IV, arXiv:13 03.3978-3981 . [32] A.M. Mathai, R.K. Saxe na and H.J. Haub o ld (2010) : The H-function: The- ory and Applic ations , Springer, New Y or k. [33] A.M. Mathai and H.J. Haub old (201 1): A v ersatile in tegral in ph ysics and astronom y , . 15