PRINCIPAL EIGENVALUE CHARACTERIZATION CONNECTED WITH STOCHASTIC PARTICLE MOTION IN A FINITE INTERVAL 1 FETHI

describing stochastic dynamics of a particle moving under the influence of an indefinite potential , a drift , and a constant diffusion . We treat 7ÐBß >Ñ ,ÐBß >Ñ H the periodic-parabolic eigenvalue problem (1) for finite domains having absorbing barriers. We show that under conditions required by the maximum principle, the positive principal eigenvalue (and the negative principal eigenvalue is connected to the probability eigendensity function by a -‡Ñ :ÐBß >Ñ Raleigh-Ritz like formulation. In the process, we establish the manner of effect of the drift and any inducing potential on the size of the principal eigenvalue. We show that the degree of convexity of the potential plays a major role in this regard. Fokker-Planck Equation, Indefinite Potential, Principal


Introduction
In the absence of randomness, the directed motion of a particle on the real line due to an existing differentiable potential (see [17]) is given by FÐBß >Ñ .B `F .>`B oe oe ,ÐBß >ÑÞ Ð"Þ"Ñ The authors wish to acknowledge the support of the Kuwait University Research Administration (KURA) through grant: SM 04/01.
The drift function is then the velocity with which the particle travels, and in the event ,ÐBß >Ñ that does not vary with time, the motion of the particle is steady and satisfies the equation Assuming an additional random component for the motion of the particle without violating its continuity, equation (1.1) can be expanded to the Langevin equation: .BÐ>Ñ oe ,ÐBß >Ñ  =ÐBß >Ñ Ð>Ñ.>Þ Ð"Þ$Ñ F Here, the function is rapidly fluctuating with time, having null statistical mean and Dirac FÐ>Ñ $ F variance function.The quantity is commonly known to be closely related to the Ð>Ñ.>Wiener process (see [13]).The function indicates the scale of randomness in the AÐ>Ñ =ÐBß >Ñ particle motion and is connected with the phenomenon of diffusion in the prescribed medium (see [9]).In fact, the diffusion coefficient of the medium is known to be given by the HÐBß >Ñ relation: Clearly, is a nonnegative function, and in this regard the Langevin equation may HÐBß >Ñ technically be called the stochastic drift-diffusion equation, and be written: .BÐ>Ñ oe ,ÐBß >Ñ.>  #HÐBß >Ñ.AÐ>ÑÞ Ð"Þ&Ñ È The problem at hand is to estimate the influence of the drift on the behavior of the ,ÐBß >Ñ particle motion described by the Langevin equation in a finite interval having Ò= ß = Ó " # absorbing ends.Assuming that the particle is subjected to a reactive potential that 7ÐBß >Ñ changes sign in the interval , we may ask how is the long term behavior of the particle Ò= ß = Ó " # affected if the drift is induced by .The standing conjecture (see [2,5]) is that for the 7ÐBß >Ñ reflecting barrier case a drift along should settle the particle mostly in areas where the 7 B potential is optimally positive.This is treated in [6].Here, we only consider the absorbing 7 barrier case.

Stratonovich Versus Ito Integration s
Techniques for solving stochastic equations like (1.5) depart greatly from those for solving ODEs.In this regard, the authors of [11] state that: The stochastic nature of the Langevin equation allows an infinite number of possible trajectories for any given initial conditions.
The probability of any one trajectory is governed by the statistical properties of the randomly fluctuating force.The value of equation (1.5) comes from our ability to integrate it when possible.Integrating (1.5) in the interval yields: The first integral in the previous equation is a Riemann-Stieltjes one, and it may be calculated by rules of ordinary calculus, while the second integral is stochastic in nature.Given a continuous, nonanticipating function (see [13]), and a partition of the interval , we choose 's such that 7 7 and compute the sequence of partial sums:

= =
The value of depends greatly on how the 's are chosen and on the value of .If we oe >  Ð"  Ñ> !Ÿ Ÿ " W oe W Ð ß 0ß = ß = Ñ 0 may be defined in the following manner: where means the limit in the mean square norm).The Ito stochastic integral is given 7= ÐP s # by: The Stratonovich stochastic integral, on the other hand, is given by: Note that the Stratonovich integral looks more like the regular Riemann integral and the term that occurs in the Ito formulation does not show up in the Stratonovich model.While the Ito integral may be mathematically and technically more satisfactory, the s Stratonovich quadrature may naturally be a better choice physically.Despite these discrepancies, it can be easily shown (see [1,2]) that both approaches yield if Ð. Ð>ÑÑ oe != $ $ =  # Ð. Ð>ÑÑ oe .>, and in the mean square. #

Stratonovich Versus Ito Dynamical Model Formulation s
For developmental and historical reasons, the stochastic equation (1.5) describing a Markov process (See [14,18]) has become to be known as the Ito Differential Equation (IDE): s .BÐ>Ñ oe ,.>  #H.Ð>ÑÞ Ð$Þ"Ñ È = Another rival model to the IDE also describing particle motion due to drift and diffusion is given by the Stratonovich Differential Equation (SDE): Ito and Stratonovich formulations are identical if the diffusion is independent of the s H location.It is shown by the authors of [19] that both, the IDE (3.1) and the SDE (3.2), can lead to natural solutions.Here among other results, we develop their ideas further.The Ito s change of variable formula is given by (see [1] for a proof): .0 ÐBÐ>ÑÑ oe Ò,0 ÐBÐ>ÑÑ  H0 ÐBÐ>ÑÑÓ.> #H0 ÐBÐ>ÑÑ.Ð>Ñ On the other hand, the Stratonovich chain rule is that of ordinary calculus as can be seen from the next result.
Theorem equalities, and manipulations of the IDE, we get: Now, we convert this IDE back to an SDE and obtain:

=
Using the previous defining relations, we recover the theorem conclusion.It was shown (see [1,6]) that a Langevin particle subject to diffusion and to drift HÐBß >Ñ ,ÐBß >Ñ :ÐBß >Ñ , hence governed by the IDE (3.1), resulting in the probability density evaluating the chances of the particle being at location at every time , and a probability B > flux: . It must satisfy the Fokker-Planck Equation (FPE): The next result establishes the correct form of the FPE due to a Stratonovich drift..12

Theorem 3.2: Assuming that the Langevin particle moves under the influence of a
Exchanging for we get: Substituting for in equation (3.13) we get the Stratonovich FPE (3.11).
Clearly, both formulations coincide if diffusion is isotropic; constant.HÐBÑ oe H oe Moreover, both equations (3.10) and (3.11) are valid in the absence of reactions.In the next section, we treat reactive versions of the FPEs with constant diffusion.In the absence of drift, equations (3.1) and (3.5) lead to the heat equation versions of (3.10) and (3.11) (see [10,12,18]).Introduction of drift into the FPEs helped explain many cellular dynamic phenomena (see, for instance, [3,4,9,21]).

Indefinite Reactions Effects on the Dynamics of a Langevin Particle
In this section, we treat the isotropic case of a Langevin particle subject to a -periodic drift X ,ÐBß >Ñ X 7ÐBß >Ñ and a -periodic indefinite reactive potential .We seek a probability density function to the -periodic parabolic eigenvalue problem :ÐBß >Ñ X `: `: `> `B `B `" #  H  ,: oe 7: Ð= ß = Ñ ‚ Ò!ß X Óß Ð%Þ"Ñ ' "in with initial density , such that at all times, the given data at the boundary points and is .This means that we have Dirichlet conditions at the endpoints.The driftless problem with reflecting barriers at the endpoints was Š ‹ `: `B oe !treated in [8].Further developments including drift can be found in [3,5,6,16].Here, however, we focus on the Dirichlet case.
Using Dirichlet boundary conditions, the first integral on the right-hand side vanishes, and the second integral yields:  N : .B oe H Ð: Ñ .Integrating equation (4.5) on interval, we obtain: Ò Equations (4.11)  and (4.12) are obtained from (4.2) and (4.9) respectively, by the obvious substitution, for the Dirichlet eigenvalue problem in :