Convergence of Fractional Brownian Motion with Reflections to Uniform Distribution

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1 Applied Mathematical Sciences, Vol. 11, 017, no. 43, HIKARI Ltd, Convergence of Fractional Brownian Motion with Reflections to Uniform Distribution G. Sh. Tsitsiashvili Institute of Applied Mathematics Far East Branch of the Russian Academy of Sciences Far Eastern Federal University, Vladivostok, Russia Copyright c 017 G. Sh. Tsitsiashvili. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A rate convergence of fractional Brownian motion with reflection on segment edges to uniform distribution is estimated. This problem plays an important role in the physical-technical models of mixing of impurities. In the present work a mathematical technique to construct such estimates is developed. In contrast to the anomalous or normal diffusion with independent increments it is obtained only power, not exponential, convergence to uniform distribution. Mathematics Subject Classification: 60J30, 60J60 Keywords: a partial Brownian motion, a reflecting edge, a convergence rate, a self-similarity property Introduction In [1] a mathematical model of an anomalous diffusion (defined by stable random process) in space was built. In [], [3] the model of anomalous diffusion on interval with reflecting edges was considered. The rate of convergence of the distributions of this process to a uniform distribution, that plays an important The author is supported by the Russian Foundation of Basic Research, project

2 104 G. Sh. Tsitsiashvili role in the physical-technical models of mixing impurities in the environment, was estimated. However, the latest physical research [4] - [7] require the consideration of models of diffusion, defined not only by stable random process, but also by fractional Brownian motion that occurs in environments with dependent time increments. In the present work a mathematical technique to construct such estimates is developed. In contrast to the anomalous or normal diffusion it is possible to obtain only power, not exponential, convergence to uniform distribution. This mathematical technique is based, firstly, on the property of selfsimilarity of fractional Brownian motion [8] - [11], second, on special assessments to the proximity of the sums of Darboux problems for integrals of the density function of a normal distribution multiplied by a linear function, with different shifts of argument. Assume that y(t), t 0 is fractional Brownian motion. The fractional Brownian motion with the Hurst parameter H is Gaussian process with zero mean and covariation function B H (t, s) = My(t)y(s) = σ [ t H + s H t s H ], t, s 0, y(0) = 0. The process y(t), t 0, satisfies the following self-similarity property: for any t 0, c > 0 random variables y(t), c H y(ct) coincide by a distribution. 1 Preliminaries In this section results of [] are represented shortly, without proofs. Construction of reflected Brownian motion on the segment [ 1, 1]. Each realization of the random process y = y(t), t 0, we associate a curve Γ on the plane (t, y). 3 y curve Γ 1 A B C D -3 E 3 t 3 y curve γ 1 A B C E D 3 t Figure 1. Construction of reflected curve γ. Put a curve Γ successive reflections from direct y = 1, y = 1. To do this, imagine that the plane (y t) is a transparent sheet of paper on which

3 Convergence of fractional Brownian motion with reflections 105 are plotted the curve Γ. Will consistently bend the sheet along the straight y = ±1, y = ±3,... in the transparent accordion in a band 1 y 1 coated with fragments of the original curve Γ. The resulting Γ is converted to a curve γ (see Fig. 1) of the form y = Y (t), t 0. Random process Y (t), t 0, in analogy with [1] can be seen as a model of anomalous diffusion with reflections on the boundaries of the segment [ 1, 1]. It is obvious that if a curve Γ coincides with some line, then corresponding curve γ is based on the law of reflection a ray of light from the borders the plane-parallel plate. It follows the rule equality of angle of incidence and angle of reflection. Analytical representation of reflected fractional Brownian motion. Along with the geometric method to build process Y (t), t 0, we give its analytical the idea with the functions s : R [, ), g : [, ) [ 1, 1] of the form s(u) = (u + )/mod 4, u, 1 u 1, g(u) = u, 1 < u <, u, u < 1 as follows: Y (t) = g(s(y(t))). Here u/mod A = A{u/A}, A > 0; {z} is fractional part of a real number z. s t Figure. Graphs of functions s, g. g 1 - t The derivation of the formula for fractional Brownian motion. Assume that p t = p t (u), P t = P t (u), f t = f t (u) are densities of random variables (r.v s) distributions y(t), s(y(t)), Y (t) accordingly. Using the graph of a function s (see Figure ) find: P t (u) = 0, u [, ), P t (u) = v: s(v)=u p t (v) = Define a helper function Q t (u) = p t (u 4k), u [, ). (1) p t (u 4k), < u <. The function Q t (u) coincides with P t (u) with u [, ), has period 4 and is even. Therefore, in according to Formula (1) and the graph of the function g

4 106 G. Sh. Tsitsiashvili (see Figure ) have f t (u) = 0, u [ 1, 1], f t (u) = Q t (u) + Q t (4 u) = p t (u 4k), u [ 1, 1]. () It should be noted that Formula () is true for any even the density p t (u). It is interesting to note that Formula (), giving the distribution of the reflected diffusion process, very similar to its structure formula obtained by reflection [1, Ch. III, 13, paragraphs 5, 6] and gives the solution of the wave equation for a finite string with fixed ends. Fractional Brownian motion with periodic initial conditions. The diffusion process on the interval with periodic initial conditions occurs for example, when the study of mixing of fuel in the ramjet engine [3]. For its modelling we take a natural n and we define a Markov process Z n (t) = g(s(y(t) 1 + z n )), t 0, where r.v. z n has a uniform distribution on a finite set I n = ( k + 1 n : k = 0, 1,..., n 1 and z n, y(t) are independent. Then the random process Z n (t) t 0, can be viewed as the anomalous diffusion on segment [ 1, 1], but with periodic initial conditions P (Z n (0) = 1 + q) = 1 n, q I n. Denote F t, n = F t, n (u), P t, n = = P t, n (u) densities of distributions of r.v. s Z n (t), s(y(t) 1 + z n ), t > 0. Theorem 1.1 The following formula is true F t, n (u) = 1 n n 1 k=0 ) ( f t, 1/n u + 1 k + 1 ). (3) n It should be noted that the equality (3) true for any process y(t) with independent and symmetrically distributed increments., The convergence of fractional Brownian motion to the uniform distribution Fractional Brownian motion with reflections from the cut ends [ 1, 1]. Next consideration closely connecting with a concept of a self-similar random process (see for example [4]]) has a large application in modern physics [5] -

5 Convergence of fractional Brownian motion with reflections 107 [10]. Put f(u) = 1/, u [ 1, 1], f(u) = 0, u / [ 1, 1]. For function ϕ defined on (, ), we define the norm ϕ = sup{ ϕ(u), < u < } and denote v k = 4k u σt H, h = v k+1 v k = 4 σt H, ψ(v) = v exp( v /) π. Lemma.1 When h < 1/ the following inequality holds: sup 1 u 1 f t (u) u Ch3 = ε, C = π. (4) Proof. Compute the derivative of ψ(v) : ψ (v) = (1 v ) exp( v /)/ π, ψ (v) = ψ(v)(v 3), sup ψ (v) = 1/ π (5) <v< Using formula () and the theorem on the differentiability of a number of functions we calculate for a fixed u, 1 u 1, derivatives f t (u) u = p t (u 4k) u = Ih 16, I = h 1 ψ(v k )h. (6) Differentiability of a number of functions standing in the right part of the equality follows from the absolute convergence of the series I. We now construct an upper bound of I for fixed u, 1 u 1, assuming k (v) = vk+1 ψ(v k ) ψ(v), B k = k (v)dv : I = B k. Highlight on the real v k h axis (, ) the following segments: A 1 = (, 3], A = [ 3, 1], A 3 = = [ 1, 0], A 4 = [0, 1], A 5 = [1, 3], A 6 = [ 3, ) so as to satisfy the inequalities: ψ (v) 0, ψ (v) 0, v A 1 ; ψ (v) 0, ψ (v) 0, v A ; ψ (v) 0, ψ (v) 0, v A 3 ; ψ (v) 0, ψ (v) 0, v A 4 ; ψ (v) 0, ψ (v) 0, v A 5 ; ψ (v) 0, ψ (v) 0, v A 6. Define auxiliary signs of summation: =, = 6 3+h<vk 5 ==, =, =, 4 h<v k <1 h 3 1+h<v k < h 3+h<v k < 1 h 1+h<v k < 3 h,

6 108 G. Sh. Tsitsiashvili = and put S i = B k, i = 1,..., 6. In accordance with the 1 v k < h 3 h i issued for the segment of A k, k = 1,..., 6, inequalities we get: h 6 ψ (v k+1 ) S 6 h 6 ψ (v k ), h 5 ψ (v k ) S 5 h 5 ψ (v k+1 ), h 4 ψ (v k ) S 4 h 4 ψ (v k+1 ), h 3 ψ (v k+1 ) S 3 h 3 ψ (v k ), h ψ (v k+1 ) S h ψ (v k ), h 1 ψ (v k ) S 1 h 1 ψ (v k+1 ). Denote C k = vk+ vk ψ (v)dv, D k = ψ (v)dv and we deduce from the last v k+1 v k 1 inequalities following relationships: C k S D k, 5 D k S 5 5 C k, 4 D k S 4 4 C k, 3 C k S 3 3 D k,, C k S D k, 1 D k S 1 1 C k. Of them it is easy to obtain: ψ (v)dv S 6 3+3h ψ (v)dv, 3 3 3h 1+h 1 h ψ 3 (v)dv S 5 1+h 1 h ψ (v)dv S 4 3h h ψ (v)dv S 3 1 h ψ (v)dv S 3+3h 3 3h 1+h 1 h 3 ψ (v)dv, ψ (v)dv, ψ (v)dv, ψ (v)dv, 3 ψ (v)dv S 1 ψ (v)dv. Therefore, we have: ψ( 3 + 3h) S 6 ψ( 3), ψ(1 + h) ψ( 3 3h) S 5 ψ(1 + h) ψ( 3), ψ(1 h) S 4 ψ(3h) ψ(1 h), ψ( 1) S 3 ψ( 1 + h) ψ( h), ψ( 1 h) + ψ( 3 + 3h) S ψ( 1 h) + ψ( 3),

7 Convergence of fractional Brownian motion with reflections 109 and so the sum S = ψ( 3 3h) S 1 ψ( 3) 6 S k satisfies the inequalities: k=1 ψ( 3 + 3h) ψ( 3 3h) + ψ(1 + h) ψ(1 h) + ψ( 1) ψ( 1 h)+ +ψ( 3 + 3h) ψ( 3 3h) S ψ(1 + h) ψ(1 h) + ψ(3h) how do we get that S ψ( h) + ψ( 1 + h) ψ( 1 h) max(3h + 5h + 3h, 6h + 3h + h + 6h) π = 16h π. Because of the formula (5), the resultant inequality B k h / π and the definition of the sums, k = 1,..., 6, we find: k I S h 3 h vk 3+h h v k 3+h + 1 h k 1+h h v k h + B k 0h/ π. 1 h v k 1+h Therefore, the following inequality holds I 36h/ π, which leads to the assertion of the Lemma.1. Theorem. The following inequality holds f t (u) f(u) 3Ct 3H 0, t. Proof. Because of Formula () we have f t (u) = f(u) = 0, u / [ 1, 1]. Let δ t (u) = f t (u) f t ( 1), u [ 1, 1]. In Lemma.1 the following inequality holds δ t (u) (u + 1)ε, u [ 1, 1], and therefore 1 = 1 1 f t (u)du == f t ( 1) δ t (u)du and hence f t ( 1) 1/ ε. Here we come to the statement of Theorem.. Self-similarity of Brownian motion with reflections from the ends of the segment [ r, r]. Consider the random process Y r (t), t 0 for r > 0 : Y r (t) = rg(s(y(t)/r)). Process Y r (t), t 0, obtained from a random process y(t), t 0, by previous reflections, but not from the boundaries of the segment [ 1, 1], and from the +

8 110 G. Sh. Tsitsiashvili borders of the segment [ r, r]. Denote f t, r = f t, r (u) the distribution density of r.v. Y r (t). It is obvious that f t, 1 = f t. We introduce the normalized r.v. W t, r = Y r(t) = g(s(y(t)/r)), t 0, r with the density f t, r (ru). Then from Theorem. if τ r = r 1/H should the inequality f t, r (u) f(u/r)/r C1 3C(t/τ r ) 3H 0, t. (7) Fractional Brownian motion with periodic initial conditions. Using theorem 1.1 and the formula (7), it is easy when T n = τ r, r = /n to obtain the inequality F t, n (u) f(u) C1 3C(t/T n ) 3H 0, t. (8) Inequality (8) can be interpreted as a reduction in n 1/H times the characteristic time of convergence of the distribution of fractional Brownian motion to a uniform distribution on the interval [ 1, 1] with periodic initial conditions. This the result is easily extended to the general case when r.v. Z n (0) has a continuously differentiable density distribution ρ n (u) that satisfy the conditions of periodicity ρ n (u) = ρ n u + ) (, 1 u 1 n n, symmetry ρ n ( n v) = ρ n ( ) n + v, 0 v 1, and boundary condition n dρ n (v) dv = 0 for v = ±1. 3 Conclusion It should be noted that fractional Brownian motion models the processes with chaotic behaviour. It is easy to consider the multidimensional version of fractional Brownian motion with independent components and reflection at the boundary of the cube. However, in general case, such consideration needs more detailed analysis and is planned for later using this work approaches. Acknowledgements. The author is supported by Russian fund of Basic Researches, project References [1] V.V. Uchaikin, Multidimensional symmetric anomalous diffusion, Chemical Physics, 84 (00),

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10 11 G. Sh. Tsitsiashvili [11] P.-O. Amblard, J.F. Coeurjolly, F. Lavancier, A. Philippe, Basic properties of the multivariate fractional brownian motion, (010). [1] V.S. Vladimirov, Equations of Mathematical Physics, Moscow, Nauka, (In Russian) Received: August 4, 017; Published: August 1, 017