1. Introduction
Fractional calculus involving differential and integral operators of the non-integer order (see [
1,
2,
3,
4,
5,
6,
7]) has been actively used in recent decades to describe nonlocal systems and processes in physics (for example, see the handbooks [
8,
9] and books [
10,
11,
12,
13,
14,
15,
16,
17]). Fractional calculus has a rich history, which was first described in 1868 by Letnikov in his work [
18], and then continued by other authors in book [
1] and papers [
19,
20,
21,
22,
23,
24].
Nonlocal models are also studied in modern statistical physics, including the following areas (A) fractional physical kinetics and fractional anomalous diffusion; (B) statistical physics of lattices with long-range interactions; (C) fractional statistical mechanics. Let us note some reviews, books, and works on statistical physics, in which nonlocal models were considered.
(A) Fractional kinetics has been described in many reviews and books [
13,
17,
25,
26,
27], and [
28,
29,
30,
31,
32]. Regarding anomalous and fractional diffusion, there are many works [
13,
17,
33,
34,
35,
36,
37]. These works use fractional calculus to describe nonlocality in space and time.
(B) The lattice models of statistical physics, which take into account long-range interactions, began with the work of Dyson [
38,
39,
40] in 1969–1971, and other scientists in [
41,
42,
43,
44,
45], and then began to be actively developed. Remarkable properties of lattice models of systems with long-range interactions have been rigorously proven in works by [
46,
47]. These properties involve the continuous limits of lattice models of systems with power law types of long-range interactions that can be described by the equations with fractional derivatives of non-integer orders. The mapping of lattice models into continuous models is defined by a special transform operator [
10,
46,
47,
48,
49,
50,
51]. The models with long-range interactions are used in nonlinear dynamics to describe processes with spatial nonlocality [
52,
53,
54,
55,
56,
57,
58,
59].
(C) In statistical mechanics, the power law type of the nonlocality can be described using methods of integro-differential equations with non-integer order derivatives [
60,
61,
62,
63,
64,
65,
66]. In these works, fractional generalizations of some equations of statistical mechanics are suggested. To obtain these equations, the conservation of probability in the phase space is used [
60,
61]. The Liouville equations with fractional derivatives (with respect to coordinates and momenta) are derived in works [
60,
61,
62,
63,
64]. The use of fractional calculus to construct a non-local generalization of the normalization conditions for the density of the distribution and a generalization of the definition of the average value were first proposed in 2004 [
67,
68,
69].
Many models consider fractional differential equations that describe nonlocal statistical systems and processes. However, in many works, these equations are simply postulated and are not derived strictly from the basic laws and equations. This is largely due to the fact that these equations can be rigorously derived by introducing certain assumptions concerning the elements of probability theory for the nonlocal case. Because of this, it is important to understand the possibility of generalization of the standard probability theory to the nonlocal case.
As a mathematical tool for constructing basic concepts of the nonlocal probability theory, one could use fractional calculus instead of standard mathematical analysis, which uses integrals and derivatives of integer orders. Fractional differential equations of non-integer orders are powerful mathematical tools to describe nonlocal systems and processes. Unfortunately, in the theory of integrals and derivatives of non-integer orders, a narrow set of operator kernels is used, which is mainly of the power type.
The use of integral and integro-differential operators with the general form of operator kernels is very important to describe the widest possible types of nonlocalities in space. However, the use of integral and integro-differential operators of general forms has a significant drawback. The disadvantage is the lack of mutual consistency between integral and integro-differential operators, which leads to the fact that these operators do not form a general calculus.
As a generalization of FC, one can use the general fractional calculus (GFC) that is based on the concept of kernel pairs, which was proposed by Sonin [
70] in an 1884 article [
71] (see also [
72]). Recently, the GFC and its applications have been actively developed by Kochubei [
73,
74,
75], Samko and Cardoso [
76,
77], Luchko and Yamamoto [
78,
79], Kochubei and Kondratiev [
80,
81], Sin [
82], Kinash and Janno [
83,
84], Hanyga [
85], Giusti [
86], Luchko [
87,
88,
89,
90,
91,
92,
93], Tarasov [
94,
95,
96,
97,
98,
99], Al-Kandari, Hanna and Luchko [
100], and Al-Refai and Luchko [
101]. A very important form of the GFC was proposed by Luchko in 2021 [
87,
88,
89]. In articles [
87,
88], the fundamental theorems for the general fractional integrals (GFIs) and the general fractional derivatives (GFDs) were proved. The general fractional calculus with functions of several variables (the multivariable) GFC and the general fractional vector calculus (GFVC) was proposed in 2021 [
95].
The concept of the nonlocal probability theory (NPT) can be considered in a broad sense, namely, as a statistical theory of systems with nonlocality in space and time. Note that the theory of stochastic or random processes with nonlocality in time (with fading memory) is well known [
102,
103,
104,
105,
106,
107], and is actively used to describe economic processes.
To take into account nonlocality in the probability theory, it is necessary to generalize basic concepts of the standard (classical) probability theory. Such nonlocal generalizations must not only be correctly defined but also mutually consistent in order to form a self-consistent mathematical theory. At present, there is no mathematically correct and self-consistent “nonlocal probability theory”, if we do not include the so-called “quantum nonlocality” in the concept of nonlocality.
In this article, Luchko’s GFC and the GFVC are proposed to construct the nonlocal generalization of standard (classical) probability theory. The nonlocal probability theory is considered a theory in which nonlocalities in space can be described by the pairs of Sonin kernels from the Luchko sets. In the nonlocal probability theory, it is assumed that properties of the probability density and cumulative distribution function at any point P in the space depending on the properties of the probability density and cumulative distribution function in all other points in the space in addition to properties at point P. This paper proposes a generalization of integral and differential forms of the equations of the probability density and cumulative distribution functions to the nonlocal case. These equations have the form of fractional integral and differential equations with GFD and GFI with Sonin kernels from the Luchko set.
Let us note some mathematical features of the proposed approach. First of all, it should be emphasized that GF differential operators are nonlocal, since they are, in fact, integro-differential operators. In addition, the fractional and the general fractional derivatives of the Riemann–Liouville type of a constant function are not equal to zero. The standard Leibniz rule (the product rule) and the chain rule for fractional derivatives of the non-integer order and GF derivatives do not hold [
108,
109]. In addition, a violation of the chain rule leads to the fact that the operators defined in different orthogonal curvilinear coordinate (OCC) systems (Cartesian, cylindrical, and spherical) are not related to each other by coordinate transformations. At the same time, the equations defining these nonlocal operators in the OCC are correct for different coordinate systems [
95] since mutual consistency between integral and integro-differential operators in OCC is based on the fundamental theorems of GFC. The requirement of mutual consistency of the nonlocal generalizations of integral and differential operators, which is expressed as a generalization of fundamental theorems, imposes restrictions on the properties of operator kernels and their applications [
87,
95,
110,
111,
112]. A consistent formulation of the nonlocal probability theory should also be based on the basic theorems of some nonlocal calculus, which can be the general fractional calculus.
Let us briefly describe the content of this article.
In
Section 2, nonlocal generalizations of the probability density function, the cumulative distribution function, and probability are proposed. The properties of these functions are described and proved.
In
Section 3, relationships between local and nonlocal quantities are described to explain the concept of nonlocality. This section does not consider questions of constructing a nonlocal probability theory. It only describes what exactly is meant under nonlinearity in this paper.
In
Section 4, nonlocal analogs of uniform and degenerate distributions, which are called general fractional uniform and degenerate distributions, are described.
In
Section 5, general fractional distributions with some special functions are described. The GF distributions with the Mittag-Leffler function, the power law function, the Prabhakar function, and the Kilbas–Saigo function are suggested. Examples of GF distributions that can be represented as convolutions of the operator kernels, which describe nonlocality and standard probability density, are considered. A property of non-equivalence of equations with GFD and their solutions in different spaces is described.
In
Section 6, general fractional distributions of the exponential types are suggested as a generalization of the standard exponential distributions by using the solution of linear GF differential equations.
In
Section 7, truncated GF distributions and moment functions are defined. The truncated GF probability density function, truncated GF cumulative distribution function, and the truncated GF average values and moments, are considered. Two examples of the calculation of the truncated GF average value are given.
2. Toward Nonlocal Probability Theory
2.1. Remarks about the Concept of the Nonlocal Probability Theory
Let us give remarks about the concept of the nonlocal probability theory.
Requirements of the nonlocality Theory
To describe nonlocality in space, integral operators and integro-differential operators should be used. Moreover, the kernels of these operators must depend on at least two points in space. If these kernels are dependent on only one point, then they could be interpreted as densities of states that are often used in statistical physics. The density of states (DOS) describes the distribution of permitted states in space and the probability density function (PDF) describes the placement of particles by these permitted states. In this case, it is obvious that such kernels cannot describe nonlocality in space.
For example, let functions
and
be related by the equation
where
is a kernel, which can be interpreted as the density of states. Such a relationship of functions
and
cannot be interpreted as nonlocal since the equation can be represented in the form
Equation (
2) is a differential equation of an integer (first) order, in which the functions are determined by properties in an infinitesimal neighborhood of the considered point with coordinate
x. Because of this, we can formulate the requirement of nonlocality in the following form. The equations that describe nonlocality cannot be represented as an equation or a system of a finite number of differential equations of an integer order. It is possible to consider the kernel in the form
, instead of
; that is,
In the general case, Equation (
3) cannot be represented as a differential equation of the integer order for a wide class of operator kernels
.
Therefore, to take into account nonlocality, the operator kernels must depend on at least two points in space (, ). In this case, operator kernels can have physical interpretations of the nonlocal density of states in space. For simplicity, one can consider the dependence on the distance between points or the difference in the coordinates of these points. In the one-dimensional case, this leads, for example, to operator kernels of the form (, ).
Requirement of Self-Consistency of Theory
For the mathematical self-consistency of the theory, the integral and integro-differential operators must be mutually consistent and must form some calculus. In order to explain this requirement, consider the following.
Suppose that the generalized cumulative distribution function
F can be obtained from the generalized probability density function
f by the action of some nonlocal integral operator
, i.e.,
Suppose also that the generalized probability density function
f can be obtained from the generalized cumulative distribution function
F by the action of some nonlocal integro-differential operator
, i.e.,
If the operators are not mutually consistent, then the sequential action of these operators leads to the mathematical non-self-consistency of the theory. The non-self-consistency of the theory is expressed in the fact that the substitution of the first equation into the second, as well as the substitution of the second equation into the first, will not result in identities. This is expressed in the form of inequalities
In the standard probability theory, the requirement of self-consistency is satisfied by the virtue of the first and second fundamental theorems of the mathematical analysis.
The proposed two requirements can be satisfied by using the general fractional calculus (GFC) in the Luchko form.
(1) The requirement of the nonlocality in this case is realized as follows. In the GFC, the integral operators and integro-differential operators are represented as Laplace convolutions, in which operator kernels are differences in the coordinates of two different points (, ). The proposed approach to the nonlocal probability theory can be characterized as an approach, in which nonlocality is described by the pair of kernels from the Luchko sets, and equations with GFI and GFD. In GFC, operator kernels from the Luchko sets cannot be represented as the kernel pairs ,
(2) The requirement of self-consistency in this case is realized as follows. In the GFC, the integral operators and integro-differential operators are called the general fractional integrals (GFIs) and the general fractional derivatives (GFDs). These operators satisfy the first and second fundamental theorems of GFC.
In this paper, to construct a nonlocal generalization of standard probability theory (see books [
113,
114,
115,
116,
117] and handbook [
118,
119,
120]), the general fractional calculus (GFC) in the Luchko form [
87,
88] is proposed. Because of this, restrictions on function spaces, which can be used in nonlocal theory, are dictated by the restrictions that are used in the GFC. Therefore, one can consider a generalization of a special case of the standard probability theory, in which only continuous distributions on the positive semi-axis are considered. Obviously, in such a standard theory, a large number of probability distributions are left out of consideration, including, for example, the standard uniform distribution.
2.2. Standard Continuous Distributions on the Positive Semi-Axis
In this subsection, some equations of the standard probability theory will be written out to simplify further references. As an example, the one-dimensional continuous distributions on the positive semi-axis are used for these equations.
Let us consider a probability distribution of one random variable
X on the set
. Let the function
belong to the set
. For a function
to be a probability density function (PDF), it must satisfy [
113], p. 3, the non-negativity condition
and the normalization condition
Using the fact that continuous functions
are integrable, the distribution function
can be defined by the integration of the first-order. For each probability density function
, one can put in correspondence [
113], p. 3, its cumulative distribution function
defined by
Function (
9) satisfies all standard properties of the cumulative distribution function. It is a non-decreasing continuous function that takes values from 0 to 1. Since
belongs to the set
, Function (
9) is continuously differentiable, i.e.,
. Then, the probability density function
can be obtained by the action of the first-order derivative
This statement is proved by the substitution of (
9) into Equation (
10), which gives the identity
by the first fundamental theorem of calculus in the form
Substitution of (
10) into Equation (
9) should also give the identity
which is satisfied by the second fundamental theorem of calculus, if
. Substitution of (
10) into non-negativity and normalization conditions gives
The probability for the region
is described by the equation
where
is defined by Equation (
9).
2.3. Toward Generalizations of Standard PDF and CDF for Nonlocal Cases
Despite the fact that Equation (
9) uses an integral operator, the connection of functions
and
can be interpreted as local. This is due to the fact that the relationships of these functions, which are described by Equation (
9), can be represented as differential equations of the first-order (
10). Differential equations of integer powers for each point are given by the properties of functions in an infinitely small neighborhood of this point.
To take into account a nonlocality in the probability theory, integral and integro-differential operators with kernels, which depend on at least two points, should be used. For simplicity, one can consider the dependence on the distance between points or the difference in the coordinates of these points, and .
As candidates for nonlocal analogs of Equations (
9) and (
10), one can consider the following equations
Instead of Equation (
17), one can also consider the equation
where
.
For the kernels
where
is the Heaviside step function and
is the Dirac delta function, Equations (
16) and (
17) (or (
18)) give standard Equations (
9) and (
10), respectively.
The properties of operator kernels and integrands must be such that Equations (
16) and (
17) exist. In this case, these equations must be mutually consistent, such that the substitution (substituting one equation into another) should give the identities. These restrictions must also be imposed on the operator kernels
and
. These restrictions on operator kernels and function spaces are described by fundamental theorems of the general fractional calculus. The main restrictions, which can be used for these purposes, are proposed by Luchko [
87,
88] in the form of the following conditions.
The main restrictions are Sonin’s condition and Luchko’s conditions.
Definition 1 (Sonin’s condition).
Let functions and satisfy the conditionfor all , where ∗ denoted the Laplace convolution (see [4], p. 19, and [113], p. 6-7). In condition (20), denotes the function that is identically equal to 1 on [87], p. 3. Then, the Sonin condition is satisfied.
In order for Functions (
16), (
17) to exist, condition (
20) to be satisfied, and the nonlocal analog of the fundamental theorem of calculus be proved, one can use Luchko’s conditions.
Definition 2 (Luchko’s first condition).
Let functions , be represented in the form the functions , can be represented in the form , and for all , where , and .
Then, the functions belong to the set , and the first Luchko condition is satisfied.
Definition 3 (Luchko’s second condition).
Let function be represented as for all , where , and . Then, the set of such functions is denoted as ,
Let a function satisfy the condition . Then, the set of such functions is denoted as .
Let functions , satisfy the condition , .
Then, the second Luchko condition is satisfied.
In the proposed approach to nonlocal probability theory, it will be assumed that nonlocality is described by the kernel pairs that belong to the Luchko set.
Definition 4 (Luchko set).
A pair of kernels belongs to the Luchko set, if the Sonin condition and Luchko’s first condition are satisfied.
Note the following inclusions
It should be also noted that kernels (
19) do not belong to the Luchko set.
Equations (
16) and (
17), in which the pair of kernels
belong to the Luchko set and functions
and
satisfy Luchko’s second condition, can be written by using the general fractional integral (GFI) and general fractional derivatives (GFDs) [
87,
88].
Definition 5 (General fractional integral).
Let and , . If and satisfy the Sonin condition (20), then the general fractional integral is defined by the equation Definition 6 (General fractional derivatives).
Let and , . If and satisfy the Sonin condition (20), then the general fractional derivative of the Riemann–Liouville (RL) type is defined by the equationand the general fractional derivative of the Caputo type is defined by the equationwhere .
Using the definitions of GFI and GFD, Equations (
16)–(
18) can be represented in the forms
Substitution of expression (
25) into expression (
26) gives the identity
if the pair of kernels
belongs to the Luchko set and
belongs to the space
. This identity follows from the first fundamental theorem of GFC for the GFD of the Riemann–Liouville type [
87,
88].
Definition 7 (Luchko’s third condition).
Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function be represented as a GFI with the kernel , such thatwhere . Then, the function belongs to the set , and the third Luchko condition is satisfied.
It should be noted that using the GFD of the Caputo type
one can consider an action of the GFD (
30) on Function (
25), which gives the identity
if
belongs to the space
and the kernel pair
belongs to the Luchko set. This identity follows from the first fundamental theorem of GFC for the GFD of the Caputo type [
87,
88].
A function
that belongs to the set
is a continuous function on the positive semi-axis, for which the following inclusions are satisfied
Theorem 1 (First fundamental theorem for the GFC).
Let a kernel pair belong to the Luchko set. If belongs to the space , thenfor all . If belongs to the space , thenfor all . Proof. Theorem 1 is proved in [
87,
88] (see Theorem 3 in [
87], p. 9, and Theorem 1 in [
88], p. 6). □
Substitution of expression (
26) into expression (
25) gives the identity
if the pair of kernels
belongs to the Luchko set and
belongs to the space
. This identity follows from the second fundamental theorem of GFC for the GFD of the RL-type [
87,
88].
Theorem 2 (Second fundamental theorem for the GFC).
Let a kernel pair belong to the Luchko set. If belongs to the space , thenfor all . Proof. Theorem 2 is proved in [
87,
88] (see Theorem 4 in [
87], p. 11, and Theorem 2 in [
88], p. 7). □
Note that the equation
holds for all
, if
. Using Equation (
38), one can see that the functions
coincide, if
.
2.4. General Fractional (GF) Probability Density Function
The Luchko conditions ensure the existence of GFI, GFD, and the fulfillment of the fundamental theorems of GFC for these operators, but do not guarantee the fulfillment of the probabilities density properties for . In order for a function to be a probability density, it is necessary to impose additional conditions on the function.
Definition 8 (General fractional (GF) probability density).
Let a pair of kernels belong to the Luchko set.
Let be a function that satisfies the following conditions.
- (1)
The function is a continuous function on the positive semi-axis , such that - (2)
The function is a non-negative function () for all .
- (3)
The function satisfies the normalization conditionwhere
Then, the functionis called the GF probability density. The set of such functions is denoted by the symbol . One can define the standard probability density function in the following form. In Definition (8), one can consider that the function
belongs to the set
instead of condition (
40), and the function
satisfies the standard normalization condition instead of condition (
41).
Definition 9 (Standard probability density function).
Let be a function that satisfies the following conditions.
- (1)
The function is a continuous function on the positive semi-axis , such that - (2)
The function is a non-negative function () for all .
- (3)
The function satisfies the condition
Then, such a function is called the standard probability density, and the set of such functions is denoted as .
Note that the kernels
and
, a pair of which belongs to the Luchko set, are non-negative and non-increasing functions [
87,
88].
Remark 1. Note that cannot be considered as a subset of since the kernel for all cannot be a kernel of a pair from the Luchko set.
2.5. General Fractional (GF) Cumulative Distribution Function
Let us formulate some restrictions on the nonlocal generalization of the standard cumulative distribution function. In this case, the definitions will not be formulated in maximum generality, and will consider only a simplified case of continuity and differentiability at all points of an open interval .
Definition 10 (General fractional (GF) cumulative distribution function).
Let a pair of kernels belong to the Luchko set.
If , then the function that is defined by the equationis called the GF cumulative distribution function. The set of such functions is denoted as . If , then the function that is defined by the equationis called the standard cumulative distribution function. The set of such functions is denoted as . The following theorem is important for describing the properties of the GF cumulative distribution functions (
46).
Theorem 3 (The Luchko theorem about set
)
Let a pair belong to the Luchko set.
If , then The inverse statement is also satisfied: If conditions (48) are satisfied, then . Proof. The statements of this theorem are proven by Luchko in [
87], (see comments on p. 9, and Remark 1 on p. 10 of [
87]). □
Using the Luchko theorem (Theorem 3) and the properties of functions
, one can prove the following properties of functions (
46); the following properties the GF cumulative distribution functions eqrefDEF-FM can be proved.
Theorem 4 (Property of GF cumulative distribution functions).
Let a pair belong to the Luchko set and a function belong to the set .
Then, the function , which is defined by Equation (46), satisfies the following properties. - (A)
The function belongs to the set i.e., - (B)
The behavior of the function at zero is described as - (C)
The behavior of the function at infinity is described as - (D)
The GF derivatives of the Caputo type of is a non-negative function - (E)
The GF derivatives of the Riemann–Liouville type of is a non-negative function
Proof. (A + B) By Definition 8 of a GF probability density function
, the function
belongs to the set
. This means (see Definition 7) that the function
can be represented as
for all
, where
. According to the Luchko theorem (Theorem 3), such functions have two following important properties
Using Definition 10, the GF cumulative distribution function
is defined by the equation
where
. Using Equation (
57), the properties (
55) and (
56) can be rewritten as
These properties coincide with properties A and B.
(C) By Definition 8, the GF probability density function
satisfies the normalization condition
Using Equation (
57), which defines a GF cumulative distribution function
(see Definition 10), Equation (
60) can be rewritten in the form
This property coincides with property C.
(D) Using (
58), one case see that
. For such functions there exists a GF derivative of the Caputo type (
24) (see Definition 6).
Using Definition 10, the GF cumulative distribution function
is defined by Equation (
57). Then, the GFD of the Caputo type of Equation (
57) has the form
where
.
Using the first fundamental theorem for the GFC (Theorem 1) for the GFD of the Caputo type, the equality
is satisfied, if
belongs to the set
. Therefore, Equations (
62) and (
63) give
By Definition 8, the GF probability density function
satisfies the property
Using Equation (
64), the inequality (
65) can be represented in the form
This property coincides with property D.
(E) Similar to the proof of property D for the GFD of the Caputo type, the proof for the GFD of the Riemann–Liouville type can be realized. Using the first fundamental theorem of GFC (Theorem 1) for the GFD of the Riemann–Liouville type, the equality
is satisfied, if
belongs to the set
. Taking into account the inclusion
one can state that the first fundamental theorem for the GFC for the GFD of the Riemann–Liouville type is satisfied for
. Therefore, for the GFD of the Riemann–Liouville type, one can obtain
This property coincides with property E.
This ends the proof. □
Remark 2. It should be emphasized that the properties described in Theorem 4 hold for any pair of operator kernels from the Luchko set. It should also be noted that the fact that the GF probability density function belongs to the set is important to prove these properties. Note that the condition also guarantees the fulfillment of the first fundamental theorem of GFC.
Corollary 1 (GF probability density through GF distribution function).
Let a pair belong to the Luchko set and the function belongs to the set .
Then, the functions, which are defined by the equationswhere is the GFD of the Caputo type and is the GFD of the Riemann–Liouville type, and are the samefor , and belong . Then, one can use the notation or for functions (70) and (71) describe the GF PDF of the random variable X on a positive semi-axis. Proof. The proof is based on the identity connecting the GF derivatives of two types in the form
that is satisfied if
[
87,
88] (see Equation (
49) in [
87], p. 8, and Equation (
29) in [
88], p. 6).
Using Equations (
70) and (
71), Equality (
73) leads to the equation
Using that the property of the function
, in the form (
50), Equation (
74) gives Equality (
72), if
.
This ends the proof. □
2.6. General Fractional (GF) Probability for Region
The GF probability for the region
can be described by an expression similar to Equation (
15) in the form
where
is defined by Equation (
25). Equation (
75) can be represented in the form
where
and
is the GFI that is defined in [
95] by the equation
if
, and, for
, by the equation
As a result, one can propose the following definitions.
Definition 11 (GF probability).
Let a pair belong to the Luchko set, a function belong to the set , and function belong to the set .
Then, the real value is defined by the equationwhere , is called the GF probability of a random variable X being in the interval . Remark 3. It should be emphasized that the GF cumulative distribution function is not non-decreasing (in the standard sense) for all , in the general case. Only the general fractional derivative of this function is non-negative. The first-order derivative of this function must not be nonnegative for all . This means that the function can be decreased at some intervals. For , the non-decreasing function in the standard sense is only the GF integral for all since Therefore, there may exist such an interval that the first-order derivative of the function is negative. Then, on this interval, the function decreases in the standard sense, and. This means thatwhere , and the GF probability (79) can be negative At the same time, the non-decreasing conditionshould be satisfied andfor every , since Therefore, it is important to consider not only the general case, in which the GF probability on the interval can be negative, but also the special case, when the GF probability on the interval is non-negative.
2.7. Condition for the GF probability Density Function to be Non-Negative
The GF probability density functions (
) are non-negative functions (
) for all
that satisfy the GF normalization conditions. The GF probability density functions belong to the set
. This property means that the function
can be represented as
where
.
The non-negativity of the function
means the non-negativity of the convolution
where the GFD kernel
is the non-negative function
for all
.
The properties of the non-negativity of the kernel
from the Luchko set and the non-negativity of the convolution (
86) do not guarantee the non-negativity of the function
. In the general case, the function
need not be non-negative in all points of the positive semi-axis. The function
can also take negative values at some intervals.
Therefore, it is important to consider two following cases for the GF probability densities and GF cumulative distributions:
- (A)
The function
is non-negative on the positive semi-axis, i.e., the condition
holds. Further, it will be proved that condition (
87) will be satisfied in this case.
- (B)
The function
can be negative on some intervals of the positive semi-axis, i.e., condition (
88) is violated. In this case, the condition of non-negativity of the convolution (
87) must be satisfied for all
.
Theorem 5 (Non-negativity of GFI).
Let a pair belong to the Luchko set.
Let belong to the set , i.e., the function can be represented aswhere . Then, if the function is non-negative for all , then the function is also non-negative for all ; that is Proof. Using that the kernels
, which belong to the Luchko set, are non-negative functions for all
and the assumption that
is the non-negative function for all
, the convolution
is the non-negative function for all
by the definition of the integral. Therefore, Function (
89) is also non-negative for all
.
This ends the proof. □
Note that the statement, which is opposite to the statement of Theorem 5, is not true; that is, the statement that for all , if for all is not a true statement.
Using Theorem 5, the following property can be proved.
Theorem 6 (Non-negativity of the GF probability density function).
Let a pair belong to the Luchko set.
Let belong to the set , i.e., the function is a standard probability density function in the sense of Definition 9, and the following conditions are satisfied Then, function , which can be represented asbelongs to the set , i.e., the function is the GF probability density function in the sense of Definition 8. Proof. (1) If
, then
. Therefore, Function (
95) satisfies the condition
that follows directly from the definition of the set
.
- (2)
The statement that the assumption of the non-negativity of a function
for all
leads to the non-negativity of Function (
95) is proven as Theorem 5.
- (3)
The GF integration of Equation (
94) gives
Using the associativity of the Laplace convolution and the Sonin condition (
for all
), one can obtain
If
belongs to the set
, then the standard normalization condition
is satisfied.
Using Equation (
98), condition (
99) can be written as
This ends the proof. □
Remark 4. Note that the statement opposite to the statement of Theorem 6 is not true, since the non-negativity of the function for all does not lead to the non-negativity of the function for all .
2.8. Condition for the GF Probability to Be Non-Negative:
Complete the GF Probability
Let us describe conditions for the GF probability density functions, for which the GF probability is non-negative.
Definition 12 (Set of functions
).
Let functions and be kernels of GFI and GFD, respectively, and let the pair of these kernels belong to the Luchko set.
Let a function satisfy the following condition Then, the set of such functions is denoted as .
The set of functions, for which condition (102) is violated, is denoted as . Theorem 7 (Property of set
)
Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function belong to the set .
Then, the function can be represented as a GFI with the kernel , such thatwhere andfor all . Proof. If a function
belongs to the set
, then
satisfies the condition
for all
. Substitution of Equation (
103) into Equation (
105) gives
Using the first fundamental theorem of GFC, Equation (
106) takes the form
for all
. □
Definition 13. [Complete the GF probability density function, and ].
Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function be the GF probability density function (i.e., ) that satisfies the condition Then, the function is called the complete GF probability density function, and the set of such functions is denoted as or .
The function , for which the condition is not satisfied (i.e., ) will be called the non-complete GF probability density function, and the set of such functions is denoted as or .
Note that the set
is the subset of
and
Definition 14 (Complete the GF cumulative distribution function and
).
Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function be complete GF probability density function (i.e., ).
Then, the function , which is defined by the equationis called the complete GF cumulative distribution function and the set of such functions is denoted as . If ), then function (109) is called the non-complete GF cumulative distribution function, and the set of such functions is denoted as . Definition 15 (Complete GF probability).
Let a pair of belong to the Luchko set, function belongs to the set , and function belongs to the set .
Then, the real value , which is defined by the equationwhere , is called the complete GF probability of a random variable X being in the interval . If belongs to the set , then the value (110) is called the non-complete GF probability of the interval . Let us prove that the complete GF cumulative distribution function is non-decreasing and the complete GF probability of all intervals is non-negative.
Theorem 8 (Non-decreasing GF cumulative distribution function).
Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function be a complete GF probability density function, i.e., .
Then, the GF cumulative distribution function, which is defined by the equationsatisfies the standard non-decreasing condition in the formfor all , the GF probability is non-negativefor all , where . Proof. If
, then the function
can be represented as
where
, and
for all
. Substitution of Equation (
114) into Equation (
111) gives
Using the associativity of the Laplace convolution
Then, inequality (
112) takes the form
for all
.
Using the first fundamental theorem of the standard calculus, the inequality (
118) is written as
for all
.
Inequality (
119) means that the GF cumulative distribution Function (
111) is a non-decreasing function on the interval
. Then,
if
.
Using Definition 15, inequality (
120) shows that inequality (
113) holds for all
, where
.
□
Corollary 2. Let functions and be kernels of the GFI and GFD, respectively, and let this pair of kernels belong to the Luchko set.
Let a function be a complete GF probability density function, i.e., , and be a complete GF cumulative distribution function, i.e., , defined by the equation Then, the GF probabilitywhere satisfies the standard properties of the standard probability theory. Let , be intervals, such that , where . Then, the following properties of the complete GF probability density are satisfied.
- (1)
The non-negativity,for every . - (2)
- (3)
- (4)
- (5)
- (6)
Proof. The proof of these properties follows directly from the properties of the GF cumulative distribution function and Equation (
122) that defines the GF probability. □
The conditional GF probability is defined by the equation
where
.
Remark 5. It should be noted that for the GF probability density functions from a set , the GF probability on the interval can be negative for some intervals. However, the GF probabilityfor all . This property is true because it is described as the Laplace convolution of two non-negative functionswhere the GFI kernel for all , and the GF probability density function for all . This statement does not depend on which of the two subsets or is considered. The negativity of the GF probability on the interval is due to the fact that nonlocality affects the change in the probability density. This influence leads to the fact that the distribution function may decrease in some regions. Such influence of the nonlocality is in some sense similar to the behavior of the Wigner distribution function in quantum statistical mechanics [
121,
122] and some non-Kolmogorov probability models [
123,
124,
125,
126]. This property of the nonlocality in the proposed generalization of the standard probability theory should not be excluded from consideration. Because of this, it is proposed in the theory of probability not to be limited only to sets
and
. It is useful to study and consider wider sets
and
.
It should be emphasized that the proposed non-local probability theory cannot be reduced to a standard theory that uses classical probability densities and distribution functions. This impossibility is analogous to the fact that fractional calculus and general fractional calculus cannot be reduced to standard calculus, which uses standard integrals and derivatives.
2.9. Operator Kernels in Nonlocal Probability Theory
In the standard probability theory, the dimension of the probability density is always the inverse of the dimension of the random variable
The standard cumulative distribution function and probability are dimensionless quantities
For the correct use of the general fractional calculus in the construction of the nonlocal generalization of probability theory, it is necessary to specify the physical dimensions of the GF integral and GF derivative.
For reasons of convenience, it is proposed to use the following requirement. To preserve the standard physical dimension of quantities, the dimension of the DFD and DFI should coincide with the dimension of the derivative and integral of the first order, respectively. Then, the dimensions of the kernels
,
,
of the GF integrals and dimensions of the kernels
,
,
of the GF derivatives are the following
where
denotes a dimensionless quantity.
The mathematical property that a pair of kernels
belongs to the Luchko set, then the kernel pair
also belongs to the Luchko set, is violated, if the assumption (
134) is used. However, this property of interchangeability of the operator kernels cannot be applied to the physical dimensions of these kernels, since GFI-kernel
is
, and GFD-kernel
has
.
Therefore, the mathematical property of interchangeability should be somewhat reformed by using the following property of the variability of the kernel dimension.
The Sonin condition for the kernels
and
that belong to the Luchko set has the form
One can see the following property: If the kernel pair belongs to the Luchko set, then the kernel pair with also belongs to the Luchko set.
As a result, the following proposition is proved.
Theorem 9 (Interchangeability of Operator Kernels).
Let a kernel pair belong to the Luchko set.
Then the kernel pair with and also belongs to the Luchko set.
Let us give examples of kernel pairs that belong to the Luchko set and have physical dimensions and . In these examples, , , , and .
Example 1. The power law nonlocality:
Example 2. The Gamma distribution nonlocality:
Example 3. The two-parameter Mittag-Leffler nonlocality:
Example 4. The Bessel nonlocality:
Example 5. The hypergeometric Kummer nonlocality:
Example 6. The cosine nonlocality:
Remark 6. Note that this list of examples can be expanded by using kernel pairs of the form for each pair of examples. For example, using the kernel pair (138), one can consider the following new pair In these examples, the following special functions are used:
is the incomplete gamma function (see Section 9 in [
127], pp. 134–142);
is the two-parameter Mittag-Leffler function (see Section 3 in [
128], pp. 17–54, [
129] and Section 1.8 in [
4], pp. 40–45);
is the Bessel function (see Section 7.2.1 in [
127], pp. 3–5, and Section 1.7 in [
4], pp. 32–39);
is the modified Bessel function (see Section 7.2.2 in [
127], p. 5, and Section 1.7 in [
4], pp. 32–39);
is the confluent hypergeometric Kummer function (Section 1.6 in [
4], pp. 29–30).
Remark 7. Note that GFD and GFI, in which kernels are standard probability density functions up to the numerical factors, can be interpreted as integer-order derivatives and integrals with continuously distributed lag [130]. For example, the GFI with the kernels and the GFD with kernels , which are defined in Equations (137) and (138), can be used to take into account a continuously distributed lag as a special form of nonlocality. Remark 8. It should be emphasized that kernels (19) do not belong to the Luchko set. At the same time, it should be noted that general fractional calculus is a generalization of fractional calculus of the Riemann–Liouville fractional integrals, the Riemann–Liouville and Caputo fractional derivatives of the order α. These operators are defined by kernels (136). In the GFC, the kernel pair (136) does not belong to the Luchko set, if . The GFI with the kernel from the pair (136) is the Riemann–Liouville fractional integral of the order α:for . Then, the GFD of the RL type is the Riemann–Liouville fractional derivative of the order α: For the Riemann–Liouville fractional integral, Equation (143) is also used for and , where the relationshold true. For the Riemann–Liouville fractional derivative, Equation (144) is also used for and , where the relationsalso hold true. Therefore, the function can be interpreted as the Heaviside step function.
Therefore, the function can be interpreted as a kind of Dirac delta function that plays the role of unity with respect to multiplication in form of the Laplace convolution [87], p. 7. As a result, using power law kernels (136), we obtain the consideration of a nonlocal probability theory in the framework of a fractional calculus approach, which uses the fractional integral and derivatives of an arbitrary order . In the framework of this calculus, the standard probability theory can be considered a special case, when the order α of operators is equal to integer values. It should also be noted that fractional integrals and derivatives of generalized functions and distributions were described in Section 8 of Chapter 2 in book [1], pp. 145–160, including generalized functions on the test function space in the framework of the Schwartz approach. 2.10. Multivariate Probability Distribution
The proposed approach to the consideration of the univariate probability distribution can be extended to multivariate probability distributions.
In the two-dimensional space
, one can consider a multivariate probability distribution consisting of random variables
X and
Y on the set
The probability density
is non-negative (
) and is normalized
If
, then the cumulative distribution function
is defined by the integration
If
, then the density
is defined by the differentiation
Expressions (
25) and (
26) can be generalized for the region
such that
For region (
151), a generalization of Equations (
25) and (
26) for
has the form
where
and
.
Remark 9. It should be emphasized that the sequence of actions of the GF derivatives must be the reverse of the action of GF integrals in Equations (152) and (153), i.e., the -sequence of GFI and -sequence of GFD. In general fractional calculus, this requirement is due to the need to fulfill the identity after substituting expression (152) into expression (153) in the formwhere the first fundamental theorem of GFC is used twice (first on x, and then on y). Identity (154) holds if the pairs of kernels and belong to the Luchko set and belongs to . Remark 10. The action of the GFD of the Caputo type with respect to x on GF distribution function , which depends on y, and vice versa, gives zerosince the action of the GFD of the Caputo type on a constant function is equal to zero. The GFD of the Riemann–Liouville type of a constant function is not equal to zero For the GFD of the Riemann–Liouville type, the following equation is satisfiedsince Therefore, the action of the GFD of the Riemann–Liouville type with respect to x on the GF distribution function , and vice versa cannot give zero A consequence of this property is the following non-standard equality. If the function has the formthen These facts should be taken into account for multivariate GF probability distributions.
It should be noted that the standard product (Leibniz) rule is violated for GFD. Therefore, the following inequalities exist
The GF derivative of the Caputo type satisfies a similar inequality.
Note that the GF differential equations can describe nonlocality in the space due to the fact that these equations are actually integro-differential, which depends on the region.
2.11. General Fractional Average (Mean) Values
In this subsection, nonlocal generalizations of the standard (local) average value are proposed for continuous distributions on the semi-axis.
First, let us briefly write out the standard formulas that define the average values of the function of a random variable X, which is distributed with a density on the semi-axis.
Let
be a standard probability density function,
be a function of a random variable
X, such that
, and the function
be the standard cumulative distribution function. Then, the standard average value is described as
In constructing definitions of the nonlocal generalizations of the standard expression (
165), one should take into account the need to satisfy the following properties in addition to linearity. For the GF average values of the function
of the random variable
X on the semi-axis
, the following properties should be satisfied.
The first property is the normalization condition for the unit function of a random variable
that should be satisfied for all types of the average GF values. Equation (
166) can be interpreted as a normalization condition of the GF probability density.
The second property is the principle of correspondence with the definition of the standard (local) average value with the GFI kernel
equal to unit for all
, i.e.,
for all
,
that should be satisfied for all types of GF average values. Note that the operator kernel
does not belong to the Luchko set. Therefore, the correspondence principle is verified by substituting the power kernel
, which belongs to the Luchko set together with the kernel
, and considering the limit
.
Let us define three types of GF average values of function
, for which the first property (
166) and the second property (
167) are satisfied. These properties can be easily proven (verified) by direct substitution of the identity function
for the function
and by the described limit passage
for the operator kernel
.
Definition 16 (GF average values of function
).
Let a pair belong to the Luchko set.
Let be a GF probability density, be a function of a random variable X, and the functionis the GF cumulative distribution function. Let . Then, the valueis called the GF average (mean) value of the first type for the function of the random variable X. Let . Then, the valueis called the GF average (mean) value of the second type for the function of the random variable X. Let . Then, the valueis called the GF average (mean) value of the third type for the function of the random variable X. The proposed GF average values can be represented by using the notations of the general fractional calculus in the following forms.
- (1)
For the GF average value of the first type, one can use the fact that the condition
leads to
. Then, the equation
leads to the equality
Therefore, Equation (
169) can be written as
- (2)
For the GF average value of the second type, one can use the fact that the condition
leads to
. Then, Equation (
168) gives the equality
where
is the FG derivative RL type with the kernel
instead of the kernel
. Therefore, Equation (
170) can be written as
- (3)
For the GF average value of the third type, similarly to the second type, one can use Equation (
175). Therefore, Equation (
171) can be written as
Let us make some remarks about the proposed three types of GF average values.
Using the GF average value of the first type (
174), in fact, in addition to the “old density function”
, “new density function”
should be also finite at
. The following conditions should satisfy at the same time
and
for which it is necessary to find the conditions of the parameters. For most GF probability density functions and operator kernels, for which the analytical expressions are known, the average value (
174) gives a finite value at
only. Because of this, the GF mean value that is derived by a simple replacement of the first-order integral with a general fractional integral
leads to a not-very useful characteristic of the nonlocal distribution. Such a definition of the GF average value can be used for truncated GF average values over finite intervals
of truncated GF distributions. Such distributions and their corresponding to truncated GF average values are discussed in
Section 7.
Using the notation of the GF mean value through integration with the GF cumulative distribution functions (see equations (
169), (
170) and, (
171)), it becomes clearer that the second and third types of GF mean value are more adequate generalizations of the standard average values.
Due to the fact that Equations (
170) and (
171) contain the differentials of the GF cumulative distribution function
, the Riemann–Liouville type of GF derivative should be used in Equations (
176) and (
177).
It should also be emphasized that the GF derivative, which is used in Equations (
170) and (
171), contains the kernel
, instead of the GFD kernel
. Because of this, in the limit case
, which is described in the correspondence principle, the GF derivative does not give the standard derivative of the first order, but the function itself
The proposed three types of average values can be combined into one generalized form with two different operator kernels.
Definition 17 (GF average values with two kernels).
Let two kernel pairs and belong to the Luchko set.
Let be a GF probability density, be a function of a random variable X, such thatand the functionis the GF cumulative distribution function. Then, the valueis called the GF average (mean) value with two kernels for the function of the random variable X. Equation (
185) can be written as
All three types of GF average values are particular cases of their proposed generalization, if we include in the considerations the operator kernel as the limiting case of the power law kernels.
- (1)
If
, Equation (
185) gives the standard average value (
165)
- (2)
If
and
, Equation (
185) gives the GF average value of the first type
- (3)
If
and
, Equation (
185) gives the GF average value of the second type
- (4)
If
, Equation (
185) gives the GF average value of the third type
- (5)
If
,
and
, Equation (
185) does not coincide with the three types of average GF values.
The use of two operator kernels and in Definition 17 can be interpreted as follows. The first kernel describes the influence of the nonlocality on the function of random variables (on “classical observable” in the physical interpretation). The second kernel describes the influence of the nonlocality on the probability density (on the distribution of states in the physical interpretation).
For the GF average (mean) value with two kernels for the case , there are problems in finding examples, for which these GF average values are non-zero finite values. Therefore, these GF average values with can be used to consider truncated GF distributions on finite intervals .
Because of this, it seems that the most interesting for use in applications are the non-truncated GF average values of the second type.
Let us give two examples of average values of the second type.
Example 1. Using the operator kernels
and the GF probability density function
that describes the uniform GF distributions (for details see
Section 4), the GF average value of the second type has the form
Example 2. Using the operator kernels
and the GF probability density function
the GF average value of the second type has the form
Remark 11. It should be noted that a generalization of the normalization condition and average value by using fractional integration of non-integer order was first proposed in [67,68,69] and then it was used in papers [67,68,69] to describe complex physical systems in fractional statistical mechanics [10,131,132,133]. These generalizations are proposed for the case of the power law nonlocality only. The GF characteristic function of the real-valued random variable is defined by the GF probability distribution.
Definition 18 (GF characteristic function).
Let a pair belong to the Luchko set.
Let be a GF probability density, and the functionis the GF cumulative distribution function. Let the following conditions be satisfiedfor all . Then, the valueis called the GF characteristic function of the second type for the random variable X. Using the GF probability density function, Equation (
200) can be defined by the equation
As a result, the GF characteristic Function (
201) is the Fourier transform of the GF derivative with the kernel
of the GF probability density function.
3. Relationship between Local and Nonlocal Quantities
This section will not consider the questions of constructing a nonlocal probability theory. Here, an explanation of the nonlocality will be given.
The purpose of this section is to describe the relationship between nonlocal and local concepts, but, first of all, to point out the differences between nonlocal theory and the standard (local) theory.
In this section, the following relationships are described.
A relationship between the functions
and
where
.
A relationship between the functions
where
and
.
A relationship between functions (
202) and
.
A relationship between the functions
and
where
.
For convenience, the description begins with well-known mathematical facts and theorems.
3.1. Mean-Value Theorems for Integrals of the First Order
Let us describe the sets of functions, which are used in this section, and well-known theorems, including the mean-value theorem. The following choice of sets of functions and operator kernels is determined by the general fractional calculus, which will be used to construct a nonlocal probability theory. Let functions and belong to the set .
The set
is the space of functions that are continuous on the positive real semi-axis and can have an integrable singularity of a power function type at the point zero. The condition
means that
and it can be represented as
, where
,
. Note that there are the following inclusions
The kernels of integral and integro-differential operators will be assumed to belong to the subset of the set . The condition means that and it can be represented as , where and .
In standard calculus, the Weierstrass extreme value theorem states that if a real-valued function
is continuous on the closed interval
, then
is bounded on that interval. This means that there exist real numbers
and
, such that
for all
(see Theorem 3 in [
134], p. 161). In addition, there is a point on the interval at which the function takes its maximum value and a point where it assumes its minimal value.
To describe connections of the nonlocal quantities with standard (local) quantities one can use the first mean-value theorem for the integral (see Theorem 5 in [
134], p. 352). In [
134], this name of the theorem is used for the somewhat more general proposition that can be useful for the general fractional integral. Note that the kernels from the Luchko set are nonnegative. The first mean-value theorem for the integral can be formulated in the following form.
Theorem 10. (First mean-value theorem for integrals)
Let be integrable functions on , If is nonnegative (or non-positive) on , thenwhere . If in addition, it is known that , then there exists a point , such that This theorem for the integral is proved in [
134] (see Theorem 5 in [
134], p. 352) for the case
. The above statement of Theorem 10 is given for the positive semi-axis for use in general fractional calculus.
The standard mean-value theorem can be considered as a corollary of Theorem 10 (see, also Corollary 3 in [
134], p. 352),
Corollary 3. Let be an integrable function on . If , then there exists a point , such that Proof. Let us consider the function
for all
. One can see that this function
is a nonnegative function and
for
. Using Theorem 10, Equation (
212) with
gives
This ends the proof. □
3.2. Expression of in Terms of
Let a function
belong to the set
, and a kernel
belong to the set
, such that
for all
. Then, the following functions are defined as
Let us consider a relationship between function
and function
, which are defined by Equations (
216) and (
215), respectively. Using the mean value theorems and the additivity property of the first-order integral, one can write an equation relating these functions.
Theorem 11 (Function
in terms of
).
Let and be nonnegative functions for all .
Then, the function , which is defined by Equation (216), can be described aswhere and , and , where is defined by Equation (215).
If for all , then Equation (217) gives Proof. Using the additivity property of the integral in Equation (
216) of the function
, one can obtain
where
.
Then, using Theorem 10 for
and the non-negativity of the GFI kernel
for all
, integrals (
219) can be represented by the equations
for all
, where
. Using Equation (
215), the integral in Equation (
220) can be written in the form
Substitution of Equation (
221) into Equation (
220), and then the resulting expression into Equation (
219) gives
Therefore, using the limit
gives (
217).
If
for all
, then Equation (
222) gives
Therefore, Equation (
223) gives (
218).
This ends the proof. □
Remark 12. Equation (217) allows to state that the functionwhere , can be represented in the formif , where and and . Equation (
225)
means that depends on the “trajectory” of changes in the function in space, and not only on the initial and final points as in the standard (local) casein which for all . 3.3. Expression of in Terms of
Let a function
belong to the set
, and a kernel
belong to the set
, such that
for all
. Then, the following functions can be defined.
and
Equations (
228) and (
229) can be represented as a sum that is described by the following Theorem.
Theorem 12. (Function in terms of )
Let , , and be nonnegative functions for all .
Then, functions (228) and (229) can be represented in the formwhere , and . Proof. Let us consider the function
where
. Then, one can use the equality
that holds if
and
[
87], where
.
If
, the additivity property of the integral in Equation (
232) can be used to obtain
where
. Then, using Theorem 10 and the non-negativity of the kernel
, integral (
233) is represented by the equation
where
. Using Equation (
227), the integral in Equation (
234) can be written in the form
Substituting of Equation (
235) into Equation (
234), and then the resulting expression into Equation (
233) gives
Using that
, one can obtain (
230).
This ends the proof. □
Remark 13. In the general case, one can use the instead of in Equation (230) of Theorem 12. 3.4. Expression of through
Using the mean value theorems and the additivity property of the first-order integral, in addition to Theorem 11, one can prove an equation relating the functions through .
Theorem 13 (Expression of
through
).
Let and be nonnegative functions for all .
Then, the function , which is defined by Equation (216), is described by the equationwhereand with . If , then the functioncan also be described by Equation (237) in the form Proof. Using the additivity property of the first-order integral, Equation (
216) can be represented in the form
where
.
Then, using Corollary 3 for the function
with
and
, one can obtain
where
and
.
Substitution of (
243) into Equation (
241) gives Equation (
240).
This ends the proof. □
3.5. Expression of Function Via
Let us consider the relationship between the functions
and
where
.
Theorem 14. (Expression of function
via
)
Let , , and be nonnegative functions on .
Then, the function , which is defined by Equation (244), is described by the equation , and .
Proof. Using Equations (
245) and (
232), the conditions
,
, and the function Equation (
231) gives
As a result, it is proved the equation
where
is defined by Equation (
245).
Then, the additivity property of the integral in Equation (
248) can be used to obtain
where
.
Then, the mean value theorem is used to obtain
where
.
This ends the proof. □
8. Conclusions
In this paper, a nonlocal generalization of the standard probability theory of the continuous distribution of the semi-axis is formulated by using general fractional calculus (GFC) in the Luchko form as a mathematical tool.
Let us briefly list the most important results proposed in this paper.
- (1)
Basic concepts of the nonlocal probability theory, nonlocality, described by the pairs of Sonin kernels that belong to the Luchko set, are suggested. Nonlocal (GF) generalizations of the probability density function, the cumulative distribution function, probability, average values, and characteristic functions are proposed. The properties of these functions are described and proved.
- (2)
Nonlocal (general fractional) distributions are suggested and their properties are proved. Among these distributions, the following distributions are described:
- (a)
Nonlocal analogs of uniform and degenerate distributions;
- (b)
Distributions with special functions, namely with the Mittag-Leffler function, the power law function, the Prabhakar function, the Kilbas–Saigo function;
- (c)
Convolutional distributions that can be represented as a convolution of the operator kernels and standard probability density;
- (d)
Distributions of the exponential types are suggested as generalizations of the standard exponential distributions by using solutions of linear general fractional differential equations.
- (3)
The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are considered. Examples of the calculation of the truncated GF average value are given.
It should be emphasized that the proposed nonlocal probability theory cannot be reduced to a standard theory that uses classical probability densities and distribution functions. This impossibility is analogous to the fact that fractional calculus and the general fractional calculus cannot be reduced to standard calculus, which uses standard integrals and integral derivatives.
Obviously, all aspects and questions of the nonlocal probability theory could not be considered in one article. Generalizations of all the concepts and methods of the standard theory of probabilities for a nonlocal case could not be proposed here. Moreover, it is obvious that only one type of nonlocality is considered in this work. Nonlocality is actually described by the Laplace convolution only. Many important and interesting questions and problems have not been resolved in this work and require further study and research in the future.
As a further development of the nonlocal probability theory, the following directions of its expansion seem important.
First, it is important to expand the types of nonlocalities for which a mathematically correct NPT can be constructed. For example, in addition to the nonlocalities described by the Laplace convolution, it is important to study the nonlocalities described by the Mellin convolution. However, for this type of nonlocality, unfortunately, a general fractional calculus similar to Luchko’s GF calculus has not yet been created.
Secondly, it is important to describe discrete theories of non-local probabilities, which make it possible to correctly describe nonlocal discrete distributions. Unfortunately, a discrete analog of the general fractional calculus in the Luchko form has not yet been created.
Thirdly, it is interesting to further develop the approach proposed in this paper, including a description of the properties of the proposed probability distributions. For example, it is important to write (mathematically accurately) the descriptions of the nonlocal probabilities for piecewise continuous distributions and probability distributions on the entire real axis, and not just on the positive semi-axis. Such formulations clearly go beyond the function spaces used in the GFC in the Luchko form. One can assume that the piecewise continuous case can be described by using some of the tools used in [
95]. Note that the general fractional calculus of many variables, which is partially described in [
95], can be used for the detailed study of GF distributions in multidimensional spaces.
The proposed mathematical theory can be used primarily to describe the nonlocal models of statistical mechanics [
147,
148,
149,
150,
151], physical kinetics of plasma-like media [
65,
152,
153,
154,
155], non-Markovian quantum physics of open systems [
66,
97], statistical optics [
156,
157], and statistical radiophysics [
158]. This may be due to the role of non-standard spatial and frequency dispersions. A nonlocal theory of probability can be an important tool for describing complex processes in the economy, in technical and computer sciences, where nonlocality can make a significant contribution to the studied processes and phenomena.