In previous papers , the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.
In recent years, fractional Brownian motion has experienced significant growth in the applied problems of physics [
Algorithm of constructing of such estimates was described in [
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 y ( t ) , t ≥ 0 is a random process with a fixed initial value y ( 0 ) = 0 . Consider random process Y ( t ) comparable to y ( t ) but reflected at the ends of the segment [ 0,1 ] in the following way.
Construction of random process reflected on interval [0, 1]
The one-dimensional process y = y ( t ) , t ≥ 0 is mapped to the reflected (from the boundaries of the segment [ 0,1 ] random process Y ( t ) = g ( s ( y ( t ) ) ) , where the functions s : E 1 → [ 0,2 ) , g : [ 0,2 ) → [ 0,1 ] are defined by the equalities s ( u ) = ( u ) / m o d 2 , g ( u ) = u , 0 ≤ u ≤ 1 , g ( u ) = 2 − u , 1 < u < 2 , [
Reflection formula for random process with symmetric density
Let f t = f t ( u ) is a distribution density of a random variable (r.v.) Y ( t ) . Then by the formula Y ( t ) = g ( s ( y ( t ) ) ) we have:
f t ( u ) = ∑ k = − ∞ ∞ p t ( u − 2 k ) + ∑ k = − ∞ ∞ p t ( 2 − ( 1 + u ) − 2 k ) , u ∈ [ 0 , 1 ] ,
f t ( u ) = 0, u ∉ [ 0,1 ] . If for each t > 0 the density p t ( u ) is symmetric in u : p t ( u ) = p t ( − u ) , then for u ∈ [ 0,1 ]
f t ( u ) = ∑ k = − ∞ ∞ p t ( u − 2 k ) + ∑ k = − ∞ ∞ p t ( − u − ( 2 k + 1 ) ) = ∑ k = − ∞ ∞ p t ( u − k ) , (1)
f t ( u ) = 0 , u ∉ [ 0 , 1 ] .
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 ( [
Reflection formula for random process with periodic initial conditions
Define f t ( u − a ) the density of distribution of random process y ( t ) + a reflected from the ends of the segment [ 0,1 ] , 0 ≤ a ≤ 1 . Let S is a random variable, uniformly distributed on the set { 0, 1 / n , ⋯ , ( n − 1 ) / n } , and random variable S and random process y ( t ) are independent. We introduce the function F t ( u ) = f t ( u − S ) , 0 ≤ u ≤ 1 , then
F t ( u ) = 1 n ∑ s = 0 n − 1 f t ( u − s / n ) = 1 n ∑ s = 0 n − 1 ∑ k = − ∞ ∞ p t ( u − k − s / n ) = 1 n ∑ k = − ∞ ∞ p t ( u − k / n ) . (2)
Because of Formula (2) the function F t ( u ) possesses following properties:
F t ( u ) = F t ( u + 1 / n ) , 0 ≤ u ≤ 1 − 1 / n ,
F t ( u / n ) = 1 n ∑ k = − ∞ ∞ p t ( ( u − k ) / n ) , 0 ≤ u ≤ 1. (3)
The first equality in (3) means that the function F t ( u ) consists of n periods of length 1/n on the interval [ 0,1 ] . The second equality in (3) means that on the interval [ 0, 1 / n ] the function F t ( u ) characterizes the distribution density of a random process n y ( t ) reflected from the ends of the segment [ 0, 1 / n ] .
In turn, the function F t ( u − 1 / 2 n ) characterizes the distribution density of a random process y ( t ) + s ^ + 1 / 2 n with an initial condition s ^ + 1 / 2 n , which has a uniform distribution on the set of points 1 / 2 n , 3 / 2 n , ⋯ , ( 2 n − 1 ) / 2 n and is independent with random process y ( t ) .
Self-similar stochastic processes with reflection and periodic initial conditions
Let the random process y ( t ) is self-similar of order a [
y ( t / r − 1 / a ) = d r y ( t ) .
In terms of the density distribution this relation looks like
p t ( u r ) = p t / r − 1 / a ( u ) . (4)
We now turn to the calculation of the function F t ( u / n ) assuming that self-similar random process y ( t ) has a symmetric density p t ( u ) :
F t ( u / n ) = ∑ k = − ∞ ∞ 1 n p t ( ( u − k ) / n ) = ∑ k = − ∞ ∞ p t / n 1 / a ( u − k ) = f t / n 1 / a ( u ) , 0 ≤ u ≤ 1.
Hence in particular it follows the equality
F t ( u / n − 1 / 2 n ) = f t / n − 1 / a ( u − 1 / 2 ) , 0 ≤ u ≤ 1. (5)
Then from Formulas (3), (5) we get the equality
F t ( u − 1 / 2 n ) = f t n 1 / a ( u − 1 / 2 ) . (6)
Multidimensional random process with independent components
For simplicity of notation all future constructions without loss of generality, we spend for the flat case m = 2 . Consider a two-dimensional random process with independent components of y → ( t ) = ( y 1 ( t ) y 2 ( t ) ) , having symmetric and self-similar density distribution of order a. Construct a process Y → ( t ) with reflections from the boundaries of the square [ 0,1 ] 2 using the obvious equalities:
Y → ( t ) = ( Y 1 ( t ) , Y 2 ( t ) ) , Y 1 ( t ) = g ( s ( y 1 ( t ) ) ) , Y 2 ( t ) = g ( s ( y 2 ( t ) ) ) .
In this case equalities p t ( u 1 , u 2 ) = p t ( u 1 ) p t ( u 2 ) are true and so
f t ( u 1 , u 2 ) = ∑ k 1 , k 2 = − ∞ ∞ [ ( p t ( u 1 − 2 k 1 , u 2 − 2 k 2 ) + p t ( 2 − ( 1 + u 1 ) − 2 k 1 , u 2 − 2 k 2 ) ) + p t ( u 1 − 2 k 1 , 2 − ( 1 + u 2 ) − 2 k 2 ) + p t ( 2 − ( 1 + u 1 ) − 2 k 1 , 2 − ( 1 + u 2 ) − 2 k 2 ) ] = f t ( u 1 ) f t ( u 2 ) .
Let s = ( s 1 , s 2 ) is a random vector, uniformly distributed on the set of numbers I = { ( p 1 , p 2 / n ) , p 1 , p 2 = 0, ⋯ , n − 1 } , and independent random vector s and a random process y → ( t ) . We introduce the function F t ( u → ) = f t ( u → − s → ) , u → ∈ [ 0,1 ] 2 , then
F t ( u → ) = 1 n 2 ∑ i → ∈ I f t ( u → − s → ) = 1 n 2 ∑ i → ∈ I ∑ k 1 = − ∞ ∞ ∑ k 2 = − ∞ ∞ p t ( u → − k → − i → ) = 1 n 2 ∑ k 1 , k 2 = − ∞ ∞ p t ( u → − k → / n ) = 1 n ∑ k 1 = − ∞ ∞ p t ( u 1 − k 1 / n ) 1 n ∑ k 2 = − ∞ ∞ p t ( u 2 − k 2 / n ) = F t ( u 1 ) F t ( u 2 ) . (7)
Because of the equality (7) the function F t ( u → ) has the following properties:
F t ( u → ) = F t ( u → + i → ) , u → ∈ [ 0 , 1 / n ] 2 , i → ∈ I ,
F t ( u → / n ) = 1 n 2 ∑ k 1 , k 2 = − ∞ ∞ p t ( ( u → − k → ) / n ) , u → ∈ [ 0 , 1 ] 2 .
Let f ( u → ) = ( f ( u 1 ) , f ( u 2 ) ) , then from Formula (7) it is easy to obtain the equality
F t ( u → ) − f ( u → ) = ( F t ( u 1 ) − f ( u 1 ) ) ( F t ( u 2 ) − f ( u 2 ) ) + f ( u 1 ) ( F t ( u 2 ) − f ( u 2 ) ) + f ( u 2 ) ( F t ( u 1 ) − f ( u 1 ) ) .
For a function φ ( u ) defined on the interval [ 0,1 ] , we introduce the norm ‖ φ ‖ = sup { | φ ( u ) | , u ∈ [ 0 , 1 ] } . A similar norm is introduced for a function φ ( u → ) , defined on the square [ 0,1 ] 2 . Let Δ n ( t ) = ‖ F t ( u ) − f ( u ) ‖ , then the following inequality holds
‖ F t ( u → ) − f ( u → ) ‖ ≤ ‖ F t ( u 1 ) − f ( u 1 ) ‖ ‖ F t ( u 2 ) − f ( u 2 ) ‖ + f ( u 2 ) ‖ F t ( u 1 ) − f ( u 1 ) ‖ + f ( u 1 ) ‖ F t ( u 2 ) − f ( u 2 ) ‖ ≤ Δ n 2 ( t ) + ( f ( u 1 ) + f ( u 2 ) ) Δ n ( t ) = Δ n 2 ( t ) + 2 Δ n ( t ) → 0 , t → ∞ . (8)
Anomalous diffusion
Let
Process
Introduce on
Therefore, when
is true. Denote
and so
From equality (9), it follows that
Let
Lemma 1. For an arbitrary
Proof. Assuming
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
Hence when we have n-periodic initial conditions, the characteristic time mixing with anomalous diffusion is reduced to
Fractional Brownian motion
Let
The process of fractional Brownian motion
Lemma 2. When
Proof. Fix
remark that
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
Compute the derivative of
Highlight on the real axis
In virtue of (18) run inequalities:
In accordance with the issued for the segment of
Designate
From them it is easy to obtain:
Therefore, we have:
and so the sum
how do we get that
Because of Formula (18), the resultant inequality
Therefore, the following inequality holds
Theorem 3. If
Proof. Because of Formula (1) equalities
and hence
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
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 [
The paper is supported by RFBR, project 17-07-00177.
Tsitsiashvili, G. (2018) One Dimensional Random Motion on Segment with Reflecting Edges and Dependent Increments. Journal of Applied Mathematics and Physics, 6, 488-497. https://doi.org/10.4236/jamp.2018.63045