Fractional Generalization of Kac Integral

Kac integral [1,2,3] appears as a path-wise presentation of Brownian motion and shortly becomes, with Feynman approach [4], a powerful tool to study different processes described by the wave-type or diffusion-type equations. In the basic papers [1,4], the paths distribution was based on averaging over the Wiener measure. It is worthwhile to mention the Kac comment that the Wiener measure can be replaced by the Lévy distribution that has infinite second and higher moments. There exists a fairly rich literature related to functional integrals with generalization of the Wiener measure (see for example [5,6]). Recently the Lévy measure was applied to derive a fractional generalization of the Schrödinger equation [7,8] using the Feynman-type approach and expressing the Lévy measure through the Fox function [9] In this paper, we derive the fractional generalization of the diffusion equation (FDE) from the path integral over the Lévy measure using the integral equation approach of Kac.

Let us consider the transition probability P (x, t|x ′ , t ′ ) that describes the evolution of the probability density ρ(x, t) by the equation

where

The function P (x, t|x ′ , t ′ ) can be considered as conditional distribution function. Then the normalization condition

holds. Assume that P (x, t|x ′ , t ′ ) satisfies the Markovian (semigroup) condition

known also as the Chapman-Kolmogorov equation.

In physical theories, the stability of a family of probability distributions is an important property which basically states that if one has a number of random variables that belong to some family, any linear combination of these variables will also be in this family. The importance of a stable family of probability distributions is that they serve as “attractors” for linear combinations of non-stable random variables. The most noted examples are the normal Gaussian distributions, which form one family of stable distributions. By the classical central limit theorem the linear sum of a set of random variables, each with a finite variance, tends to the normal distribution as the number of variables increases. All continuous stable distributions can be specified by the proper choice of parameters in the Lévy skew alpha-stable distribution [10] that is defined by

where

and

Here y is a shift parameter, β is a measure of asymmetry, with β = 0 yielding a distribution symmetric about y. In Eq. ( 6), parameter c is a scale factor, which is a measure of the width of the distribution and α is the exponent or index of the distribution.

Consider P (y, t ′ |x, t) as a symmetric homogeneous Lévy alpha-stable distribution

For α = 2, Eq. ( 8) gives the Gauss distribution

Eq. ( 8) gives the function

that can be presented as a Fourier transform

where

For α = 2, Eq. (11) gives

In the general case, the function K(x, t), given by Eq. ( 11), can be expressed in terms of the Fox H-function [7,8,9,11,12,13,14] (see Appendix).

Let us denote by C[t a , t b ] the set of trajectories starting at the point x a = x(t a ) at the time t a and having the endpoint

The Kac functional integral [2,3,15] is

where V (x) is some function, and

For ( 13), expression (15) gives

which is the Wiener measure of functional integration [15]. The integral ( 14) is also called the Feynman-Kac integral. Using (10) for α = 2, the path integral ( 14) can be written as

where the time interval [t a , t b ] is partitioned as

and

The functional integral ( 17) can be rewritten as

where

The Kac functional integral in the form ( 20) is a classical analog of the Feynman phase-space path integral, which is also called the path integral in Hamiltonian form.

For the fractional generalization of Wiener measure (15) and Kac integral ( 14), we consider K(x, t) given by (10). Substitution of (10) into

with

gives

Similarly to (20), (21) this expression can be written as

This expression is a fractional generalization of (20).

If we introduce formally imaginary time such that

transforms into the Feynman path integral with a generalized action [7,8]

as an action. Hamiltonian-type formal equations of motion are

where N α = αC α sign(p).

It is known that the Kac integral ( 14) can be considered as a solution of the diffusion equation [2,15]. Let us derive the corresponding diffusion equation for the fractional generalization of the Kac integral (25).

In (25) the integration is performed over a set C[t a , t b ] of trajectories that start at point (1) The set C f [0, t] consists of paths for which both the initial and final points are fixed.

The integration over this set obviously gives the transition probability

The conditional fractional Wiener measure corresponds to the integration over the set

of paths with fixed endpoints: x a = 0, x b = x.

(2) If we consider a set C a [0, t] of trajectories with arbitrary endpoint x b = x, the measure is called the unconditional fractional Wiener measure. This measure satisfies the normalization condition

since it is a probability

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