Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

Developing Holtzmark’s idea, the distribution of nonstationary ﬂuctuations of local interaction of moving objects of the system with gravitational inﬂuence, which is characterized by the Riesz potential, is constructed. A pseudodiﬀerential equation with the Riesz fractional diﬀerentiation operator is found, which corresponds to this process. The general nature of symmetric stable random L´evy processes is determined.


Introduction
One of the main problems of celestial mechanics is the analysis of the nature of the force of interaction between objects in one or another star system. Force F acting on a specific star of the system has two components: the first one, K, is the influence of the entire system and the second, F, is the local influence of the immediate environment: e influence of the entire system can be described by the gravitational potential R(r; t) [1] the average spatial distribution of stars of different masses m at time t. Such force, K, related to the unit of mass acting on the star Z 0 under consideration by the system (as a whole), is determined by the equality: K(r; t) � − gradR(r; t).
(1) e force K(r; t) is a function with a slow change in space and time since the corresponding potential R(r; t) characterizes the "smoothed" distribution of matter in the stellar system, whereas the other force F(t), related to the unit of mass, has relatively fast, sharp changes caused by instantaneous changes in the local distribution of stars from the environment Z 0 at time t. e value of F(t) is subject to fluctuations, so we can only talk about its probabilistic value. e famous Danish astronomer Holtzmark studied the stochastic properties of F(t) [1,2]. He used Newton's gravitational law, according to which where G is the gravitational constant, m j is the mass of a typical star Z j , r j is the radius-vector of Z j , r ∘ j ≔ r j /|r j |, |r j | 2 � (r j , r j ), scalar square in the Euclidean space R 3 , and N(t) is the number of stars that at the moment t form the local environment Z 0 . He assumed the constancy of the average density n(r; m; t) ≡ n of the spatial distribution of stars as well as the fulfillment of the equality: Using classical means of probability theory in combination with integral calculus, Holtzmark found a stationary distribution W(F) of the quantity F in the form where a ≔ (4/15)(2πG) 3/2 n〈m 3/2 〉 is the fluctuation coefficient, in which 〈m 3/2 〉 is the average value of m 3/2 , which corresponds to the observed law of stars distribution in the stellar system, and F is the Fourier transform operator. is Holtzmark distribution belongs to the class of distributions of symmetric stable random processes described by the French researcher Zolotarev et al. [3,4] at the beginning of the last century. e appearance of the class of distributions L α λ (·), in addition to these Holtzmark studies, was preceded by the studies of Poisson and Cauchy. About a hundred years before the publication of [4], first Poisson and then Cauchy paid attention to the distribution with density: which has the property and the parameter λ is uniquely determined by the values λ 1 and λ 2 . Within his statistical studies, being interested in the cases, when the error of an individual observation might be comparable to the mean error in a series of large numbers of independent observations, Cauchy found that such a case would occur if the observation errors were subject to a probability distribution with a density P λ (·). It should be noted that Cauchy [5] knew that each of the functions L α λ (·) for α > 0 possesses property (7), but he could not select the nonnegative functions from them (which were, obviously, the densities of some probability distributions) except for (8).
e famous Hungarian mathematician Polya [6] managed to prove that L α λ ≥ 0 for α ∈ (0; 1) only at the beginning of the last century. e fact that the functions L α λ (·) are densities of probability distributions only for α ∈ (0; 2] was fully determined by Le � vy in 1924 [4].
e Gauss-Wiener distribution of Brownian motion in the space R 3 , according to which heat and diffusion propagate in 3D environment, is also a good representative of the class L α λ (·). In the scientific literature [7][8][9][10][11][12], there are many examples of real applications of the Holtzmark, Cauchy, and Gauss distributions in astronomy, nuclear physics, economics, in the industrial and military sectors, etc. Each of these applications characterizes the stochastic features of Le � vy distributions with one or another value of α. However, the symmetric stable random processes L α λ (·), besides their individual characteristics, have a general nature. In this paper, it is determined that each of such Lévy processes at α ∈ (0; 2) can be regarded as a process of local influence of moving objects in the gravitational field of M. Rees, i.e., in a system, in which the interaction between masses occurs according to a certain power law (·) − β . In particular, the classical Holzmark process (α � 3/2) corresponds to the interaction with β � 2 (the case of Newtonian gravity), while the Cauchy process corresponds to the interaction with β � 3. Here, we also consider the Holtzmark problem in a general formulation and obtain a pseudodifferential equation (PDE) with the Riesz operator of fractional differentiation, which corresponds to the Holtzmark random nonstationary processes. e presence of this equation allows one to study Holtzmark stochastic processes in domains with boundary by means of the theory of boundary value problems for the PDE.
e main content of the work is as follows. In Section 2, the stochastic problem of the local influence of moving objects in the evolutionary gravitational field is studies. e connection of the Holtzmark fluctuations' distributions of nonstationary gravitational fields with one classical PDE is determined in Section 3. A brief historical overview of this equation study and its further application is also presented here. Section 4 is the conclusions.

The Problem of the Local Influence of Stars
Let us consider a star system, in which the interaction between masses is subordinate to M. Riesz potential [13]. It means that the gravity between two arbitrary stars of the corresponding masses M and m is described by the law: where G is a certain gravity constant and r is a vector of distance between these stars. Let the star Z 0 under consideration be at the origin of the coordinate system and F(t) be the force acting on the unit of mass of the star Z 0 from its closest environment at time t. Developing Holtsmark's idea, we will find the nonstationary distribution W β (F(t)) for the force F(t). For this, we also assume that the distribution of stars in the neighborhood Z 0 is subject to fluctuations according to some empirically established law and that, at each moment of time t, the fluctuations of the density of stars are subject to the condition of the constancy of their average density per unit volume: If we now assume that the star Z 0 under study is at the origin of the system and its spherical circle of radius R at time t contains N(t) stars, then, according to the above, First, let us fix t and consider the distribution of W β,N(t) (F(t)) in the center of the spherical vicinity of the radius R, which contains N(t) of stars of the system. Let us find the probability W β,N(t) (F°(t))dF°(t) that the quantity F(t) will be cubed [ F°( t); F°(t) + dF°(t)] ⊂ R 3 . Using the well-known method of characteristic functions, we obtain (14) where Here, K R (0) is a sphere of radius R with the center at the origin and τ j (r j ; m j ; t) is the distribution of probability that, at time t, the star Z j has mass m j and is in position r j .
Since it is believed that there are only fluctuations of stars compatible with the constancy of their spatial average density, then where τ(m; t) is the frequency, with which stars of different masses meet at time t.
Taking it into consideration, we will receive the equality: , (17) in which Directing R ⟶ + ∞ and N(t) ⟶ + ∞, according to condition (13), we obtain where For all values of t, the equality is true, so Furthermore, the absolute convergence in (22) of the integral with the integration variable r in the entire space R 3 at β > (3/2) allows us to write equation (22) in the form From here, we get the image in which e last equality can be written in the form For this, it is necessary to replace the integration variable r with the variable η in the inner integral of this equality according to rule (18) and then to turn to a spherical coordinate system, in which the z-axis is directed along the ξ vector.
It should be noted that the integral of the last equation coincides only at β > 3/2. Integrating it by parts, we arrive at the equality

Journal of Mathematics
where where Γ(·) is Euler's gamma function.

Theorem 1.
With the above assumptions, for every β > 3/2, the function is the probabilities distribution of the force F(t) of the local influence of moving objects in the system with the interaction that takes place according to the power law (10).
Remark 1. For β ≤ 3/2, the integral I(β) diverges, so the coefficient a β (·) becomes immeasurable. erefore, we conclude that there is no corresponding random Holtzmark process in the sense of the problem of local interaction of moving objects. Let us introduce the notation H c (F; t) ≔ W β (F(t)), where c ≔ 2/β. e function H c (·; ·) is called the Holtzmark distribution of the order c of fluctuations of nonstationary gravitational fields. e classical situation considered by Holtzmark corresponds to β � 2; in this case, the distribution order c � 1. Taking the ratio β > 3/2 into account, this is the only case of the integer order c, and the rest of the possible values of c have a nonzero fractional part: c ∈ (0; 4/3). In Section 3, we find the corresponding differential equation, the fundamental solution of which is the Holtzmark distribution H c (·; ·).

Connection with the PDE
It would be more desirable to study the fluctuations in the local interaction of moving objects, especially in a limited environment with certain conditions at the boundary, by reducing to solve the corresponding boundary value problems for differential or pseudodifferential equations.
is would allow us to use the advanced computing apparatus of the theory of boundary value problems and use its known results. In this regard, there is a need to obtain an appropriate differential equation that adequately reflects the process under study. Under certain conditions, we will try to derive this equation "starting" from the distribution function H c . To do this, let us first find out the properties of this function.
It will be assumed here that the coefficient a β (·) is a positive, continuous-differential function on the interval [0; T]. It follows directly from [14,15] that, for all c ∈ (0; 4/3), the function H c (x; t) on the set R n × (0; T] is differentiable by t and infinitely differentiable by the variable x; for its derivatives, the following estimates are fulfilled: with some positive constants c 1 and c 2 . Estimate (32) ensures that H c (·; t) belongs to the Lebesgue class L 1 (R 3 ) for each fixed t ∈ (0; T]. erefore, this guarantees the existence of the Fourier transform of the function H c (·; t) and satisfies the equality We shall arbitrarily fix t ∈ [0; T], and for △t ≠ 0, we consider According to Lagrange's theorem on finite increments, we have a β (t + △t) − a β (t) � a β ′ (t + θ△t)△t, θ ∈ (0; 1). (35) Hence, given the continuity of a β ′ (·), we obtain that In addition, using Lagrange's theorem again, we find en, for all 0 < |△t| ≤ a β (t)/(2a) and ξ ∈ R 3 , the following estimates are true: Relations (36) and (38) ensure the correctness of the equality according to which, from (34), we obtain Now, taking (33) into account, we finally arrive at us, the Holtzmark distribution H c is a classical solution of the equation with Riesz operator A ] fractional differentiation of the ] ≔ (3c/2) order [16,17].
Furthermore, we clear up the question of existence of the limit value of distribution H c (·; t) at the point t � 0.
is value α � 2 corresponds to c � 4/3, which is also the boundary value of the c-Holzmark spectrum. However, for such c, there is no random Holzmark process. is means that the process of classical diffusion occurs according to the laws that differ in nature from the laws of random vortices of local gravity of moving objects.
Taking into account the results of the research in Section 3, equation (42) can be called the pseudodifferential equation of local fluctuation of nonstationary Riesz gravitational fields.

Conflicts of Interest
e authors declare that they have no conflicts of interest.