In recent years, fractional Brownian motion has experienced significant growth in the applied problems of physics     in connection with the necessity of modelling chaotic behaviour of the diffusing impurity in a variety of environments and alloys. Therefore there is a need to analyse the speed of mixing of impurities (convergence to the uniform distribution) in areas with reflecting boundaries, which cannot be obtained by the method of Fourier series.
Algorithm of constructing of such estimates was described in  for one-dimensional case and was based on the method of analysis of anomalous diffusion  , simulating stable random process with independent increments. It is based on reflection formula for the density of the anomalous diffusion process. However, in recent years, there have been a lot of physical researches, in which models of fractional Brownian motions are used.
Therefore, in the present work the algorithm of the corresponding estimates for the fractional Brownian motion on interval with reflecting edges is constructed. This algorithm is based on a calculation of a derivative of series which is describing density of fractional Brownian motion distribution  .
Let is a random process with a fixed initial value . Consider random process comparable to but reflected at the ends of the segment in the following way.
Construction of random process reflected on interval [0, 1]
The one-dimensional process is mapped to the reflected (from the boundaries of the segment random process , where the functions , are defined by the equalities , , ,  . Here ; ―the fractional part of a real number z.
Reflection formula for random process with symmetric density
Let is a distribution density of a random variable (r.v.) . Then by the formula we have:
. If for each the density is symmetric in , then for
It is interesting to note that the formula (1), giving the distribution of the reflected diffusion process, is very similar by its structure to the formula obtained by reflection (  , Chapter III, 13, paragraphs 5, 6) and gives the solution of the wave equation for a finite string with fixed ends.
Reflection formula for random process with periodic initial conditions
Define the density of distribution of random process reflected from the ends of the segment , . Let S is a random variable, uniformly distributed on the set , and random variable S and random process are independent. We introduce the function , then
Because of Formula (2) the function possesses following properties:
The first equality in (3) means that the function consists of n periods of length 1/n on the interval . The second equality in (3) means that on the interval the function characterizes the distribution density of a random process reflected from the ends of the segment .
In turn, the function characterizes the distribution density of a random process with an initial condition , which has a uniform distribution on the set of points and is independent with random process .
Self-similar stochastic processes with reflection and periodic initial conditions
Let the random process is self-similar of order a  , i.e. for every the random variables coincides in distribution:
In terms of the density distribution this relation looks like
We now turn to the calculation of the function assuming that self-similar random process has a symmetric density :
Hence in particular it follows the equality
Then from Formulas (3), (5) we get the equality
Multidimensional random process with independent components
For simplicity of notation all future constructions without loss of generality, we spend for the flat case . Consider a two-dimensional random process with independent components of , having symmetric and self-similar density distribution of order a. Construct a process with reflections from the boundaries of the square using the obvious equalities:
In this case equalities are true and so
Let is a random vector, uniformly distributed on the set of numbers , and independent random vector and a random process . We introduce the function , then
Because of the equality (7) the function has the following properties:
Let , then from Formula (7) it is easy to obtain the equality
For a function defined on the interval , we introduce the norm . A similar norm is introduced for a function , defined on the square . Let , then the following inequality holds
3. Examples of Random Processes Anomalous Diffusion
Let―homogeneous random process with independent increments,. The difference is symmetric on stable distribution with parameter and the characteristic function.
Process is describing in  anomalous diffusion on an infinite straight line. From this definition it follows that this process satisfing equality (4) is self-similar and therefore satisfies the equality (6).
Introduce on the binary operation “”, the unary operation of taking inverse element “”:, ,. It is easy to check that so defined on operations generate commutative group C in addition with the identity element 0 and the inverse u element of. It is obvious that the mapping is a homomorphism of the additive group of real numbers on the group C, i.e. the fair equalities,.
Therefore, when we have . Consequently, the random process with independent and homogeneous increments and the distribution density is a homogeneous Markov with values in the group C, and with the density of the conditional distribution
, hence the equality
is true. Denote, then
From equality (9), it follows that
Let densities of uniform distributions on the intervals, respectively.
Lemma 1. For an arbitrary
Proof. Assuming and using Formula (10) we obtain for:
that is for
With the help of Formulas (10), (12) it is easy to obtain that
From Formulas (1), (13) follows the inequality (11). Lemma 2 is proved.
So we obtain geometric by t convergence rate of the density to the density of the uniform distribution with. Moreover, due to (6) with
Hence when we have n-periodic initial conditions, the characteristic time mixing with anomalous diffusion is reduced to times.
Fractional Brownian motion
Let is a fractional Brownian motion  . Fractional Brownian motion with Hurst parameter a is Gaussian process with zero mean and covariance function
The process of fractional Brownian motion satisfies the condition of self-similarity (4) and has a symmetric density distributions of. Using the process we define reflected from the cut ends, the process by the equation.
Lemma 2. When the following relation is valid:
Proof. Fix and denote
Using Formula (1) and the theorem on the differentiability of a series of the functions compute the derivative
Differentiability of a series of functions standing in the right part of the equality follows from the absolute convergence of the series I.
Put in virtue of Formula (16) the number I can be represented in the form.
Compute the derivative of:
Highlight on the real axis the following segments:
In virtue of (18) run inequalities:
In accordance with the issued for the segment of inequalities we get:
Designate and we deduce from the last inequalities following relationships:
From them it is easy to obtain:
Therefore, we have:
and so the sum satisfies the inequalities:
how do we get that
Because of Formula (18), the resultant inequality and the definition of the sums, we find:
Therefore, the following inequality holds, which leads to the relation (11). Lemma 2 is proved.
Theorem 3. If the following inequality holds
Proof. Because of Formula (1) equalities take place. Let. In Lemma 2 the following inequality holds, therefore
and hence. Here we come to the statement of Theorem 3.
Because of Formula (6) from inequality (19) we have
Hence for n-periodic initial conditions the characteristic time mixing under fractional Brownian motion is reduced to times.
Obtained in the present work, an upper estimate of the convergence rate of the density function of a multidimensional fractional Brownian motion with reflection at the boundaries of a square is not exponential as in the usual Brownian motion or a stable process with independent increments. Apparently this is due to the fact that the multidimensional fractional Brownian motion models the processes with chaotic behavior  .
In the work  , it is considered a model of fractional Brownian motion with dependent components. However, this model fails to obtain the corresponding results in the case of periodic initial conditions. But in the case of independent components multidimensional Brownian motion satisfies the statement of Theorem 3.
The paper is supported by RFBR, project 17-07-00177.
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