**2. Mathematical models of Brownian motion**

The Brownian motion it is well known in physics, a random movement of a particle suspended in a liquid or a gas. The name of the movement is given after the botanist Robert Brown (1827), who was studying the movement of pollen grains suspended in water. There are many similar phenomena, where the time evolutions of the object depend on stochastic, microscopic contacts (collisions) with elements of the surrounded system. In mathematics, many models describing Brownian motion are well known and applied, e.g. the random walk stochastic process, Wiener stochastic process, Langevin stochastic differential equation, general diffusion equations and others.

Observations of the microscopic particle behavior show, that at any time step, the particle is changing its position in the space, according to collisions with liquid particles. Crashes of

Fuzzy Modelling Stochastic Processes Describing Brownian Motions 297

22 1 1 2

is a conditional probability density function, and 1 1 *f* (, ) *t x* is given by formula (1) with

For any 1 2 ... *<sup>n</sup> tt t* probability density function of the multidimensional random vector <sup>1</sup> ( ,..., ) *<sup>n</sup> X X t t* can be obtained, taking into account Markov features of the process and using

> <sup>1</sup> ( ) ( , ,..., , ) exp 2( ) 2( ) *i i n n i n*

Stochastic vector process [ ( ),..., ( )], 0 *Xt Xt t* <sup>1</sup> *<sup>n</sup>* is called the *nD stochastic Wiener process* if its every component, *Xt t <sup>i</sup>*( ), 0 , *i*=1,…,*n*, is the scalar stochastic Wiener

As an example of the 3*D* stochastic Wiener process we can show three coordinates of the

The Wiener process is also the special diffusion stochastic process, fulfilling the Fokker-

0, 0

2

 

( ,) ( ,) *xt xt t x*

where γ is a diffusion coefficient. The solution has the known exponential form. From the analytical form of the solution the second central moment of the displacement is expressed

> <sup>2</sup> ( )2 *EX t <sup>t</sup>*

2

where the solution is given by the normal probability density function, and the diffusion

In macroscopic scale, in physics and in industrial practice, the probability value *f*(x)*dx* that scalar variable *X* assumes its value from the interval [*x*, *x*+d*x*] is equivalent to the quotient *n*/*N* (concentration of particles), where *n* defines the power of subset of particles, whose feature *X* determines the value over the interval [*x*, *x*+d*x*], and *N* is the population size. This idea is consistent with Einstein's experiments who considered collective motion of Brownian

lim <sup>2</sup> *t x x*

*t*

*const*

(10)

(9)

(11)

( ,) *x t* at point

*x x ft x t x* 

1 1 1,..., 2

1 () (, /, ) exp 2 ( ) 2( ) *<sup>t</sup> x x ft x t X x* 

1

2 2 <sup>1</sup> ( ) *DX t <sup>t</sup>*

process and particular scalar stochastic processes are independent.

2

*f* ( ,) ( ,) *x t f x t t x* 

2

/ 2 (van Kampen, 1990).

particles. He assumed that the density (concentration) of Brownian particles

, <sup>2</sup>

.

where

parameters: 1 ( )0 *E Xt* , 1

(3), as follows (Sobczyk, 1991):

Brownian particle trajectory.

Planck diffusion equation

coefficient is equal to <sup>2</sup>

as

 

*x* at time *t*, met the following diffusion equation:

<sup>1</sup> 1 12 2 2 2 1 1 1 1 ( , ; , ) ( , / , )( , ) *<sup>t</sup> f txtx f tx tX x f t x* (6)

21 21

1 1 1,...,

 . (8)

*i i i n i i*

*t t t t*

*tt tt*

2 1 2

(7)

2 1

particles are frequent and irregular. It is usually assumed by mathematicians, that the displacements of the particle 1 2 , ,..., ,... *ZZ Zn* at particular time steps, are independent, identically distributed random variables. The stochastic process { , 1,2,..., .,...} *Zi n <sup>i</sup>* is named *random walk*.

In macroscopic scale, if the time between two observations of the particle, *t* , is larger than the time between successive crashes, then the increment of the particle positions, *X X <sup>t</sup>* , is a sum of many small displacements, 1,..., *t i i k XX Z* . Since the increments

*X X <sup>t</sup>* constitute sums of independent, identically distributed random variables, they are normal distributed random variables.

In mathematics, scalar stochastic process *X t <sup>t</sup>* , 0 , is the *Wiener process* if and only if


$$f(t, \mathbf{x}) = \frac{1}{\sqrt{2\pi t \sigma^2}} \exp\left(-\frac{\mathbf{x}^2}{2t\sigma^2}\right). \tag{1}$$

Wiener process is also known as the *Brownian motion process* (Fisz, 1967; van Kampen, 1990; Kushner, 1983; Sobczyk, 1991).

The increments, *X X <sup>t</sup>* , 0 *t* , are normal distributed random variables with the expected value and variance as follows:

$$E(X\_t - X\_\tau) = 0 \; \prime \; D^2(X\_t - X\_\tau) = (t - \tau)\sigma^2 \; \; . \tag{2}$$

Random variables 1 ,..., *<sup>n</sup> X X t t* , where

$$X\_{t\_n} = X\_{t\_1} + (X\_{t\_2} - X\_{t\_1}) + \dots + (X\_{t\_n} - X\_{t\_{n-1}}) \, \, \, \tag{3}$$

are also normal distributed with parameters:

$$E(X\_{t\_k}) = 0 \; \; \; \; D^2(X\_{t\_k}) = t\_k \sigma^2 \; \; \; k = 1, 2, \dots, n; \tag{4}$$

and with a non-zero covariance matrix.

If <sup>2</sup> 1 then *X t <sup>t</sup>* , 0 is the *standard Wiener process*. Probability, that a particle occurs in some interval [*a*, *b*], at the moment *t*, is given by the relationship

$$\Pr\{X\_t \in [a, b]\} = \int\_a^b f(t, \mathbf{x}) d\mathbf{x} = \frac{1}{\sqrt{2\pi t \sigma^2}} \int\_a^b \exp\left(-\frac{\mathbf{x}^2}{2t\sigma^2}\right) d\mathbf{x} \,. \tag{5}$$

For any 1 2 *t t* probability density function of the random vector variable 1 2 (,) *X X t t* can be obtained as follows:

$$f(t\_1, \mathbf{x}\_1; t\_2, \mathbf{x}\_2) = f(t\_2, \mathbf{x}\_2 \mid t\_1, X\_{t\_1} = \mathbf{x}\_1) f(t\_1, \mathbf{x}\_1) \tag{6}$$

where

296 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

particles are frequent and irregular. It is usually assumed by mathematicians, that the displacements of the particle 1 2 , ,..., ,... *ZZ Zn* at particular time steps, are independent, identically distributed random variables. The stochastic process { , 1,2,..., .,...} *Zi n <sup>i</sup>* is

than the time between successive crashes, then the increment of the particle positions,

In mathematics, scalar stochastic process *X t <sup>t</sup>* , 0 , is the *Wiener process* if and only if

iv. random variables *Xt* , are normal distributed, with the probability density function

<sup>1</sup> ( , ) exp

constitute sums of independent, identically distributed random variables, they are

. Since the increments

. (1)

. (2)

, *k*=1,2,…,*n*; (4)

2

*t*

2 2

. (5)

2 2

1,..., *t i i k XX Z* 

*t* are homogeneous (stationary) and independent for

2

*t* , are normal distributed random variables with the

 

1 21 <sup>1</sup> ( ) ... ( ) *n n <sup>n</sup> XX XX XX t t tt tt* , (3)

2 2

2 2 *<sup>x</sup> ftx <sup>t</sup> t*

Wiener process is also known as the *Brownian motion process* (Fisz, 1967; van Kampen, 1990;

( ) *<sup>k</sup> DX t t k*

Probability, that a particle occurs in some interval [*a*, *b*], at the moment *t*, is given by the

*b b*

*a a <sup>x</sup> X ab <sup>f</sup> t x dx dx*

For any 1 2 *t t* probability density function of the random vector variable 1 2 (,) *X X t t* can be

 *t*

<sup>1</sup> Pr{ [ , ]} ( , ) exp

 , 2 2 ( )( ) *DX X t <sup>t</sup>* 

, is larger

In macroscopic scale, if the time between two observations of the particle, *t*

iii. trajectories of the process *X t <sup>t</sup>* , 0 are continuous (almost surely),

, is a sum of many small displacements,

normal distributed random variables.

disjoint time intervals,

Kushner, 1983; Sobczyk, 1991).

The increments, *X X <sup>t</sup>*

If <sup>2</sup> 

relationship

obtained as follows:

 , 0 

ii. the initial condition, 0 *P X*( 0) 1 , is fulfilled,

 , 0 

> ( )0 *EX X <sup>t</sup>*

expected value and variance as follows:

Random variables 1 ,..., *<sup>n</sup> X X t t* , where

are also normal distributed with parameters:

*t*

and with a non-zero covariance matrix.

( )0 *<sup>k</sup> E Xt* , 2 2

1 then *X t <sup>t</sup>* , 0 is the *standard Wiener process*.

named *random walk*.

i. increments *X X <sup>t</sup>*

*X X <sup>t</sup>* 

*X X <sup>t</sup>* 

$$f(\mathbf{t}\_2, \mathbf{x}\_2 \mid \mathbf{t}\_1, \mathbf{X}\_{t\_1} = \mathbf{x}\_1) = \frac{1}{\sqrt{2\pi\sigma(t\_2 - t\_1)}} \exp\left(-\frac{\left(\mathbf{x}\_1 - \mathbf{x}\_2\right)^2}{2(t\_2 - t\_1)\sigma^2}\right) \tag{7}$$

is a conditional probability density function, and 1 1 *f* (, ) *t x* is given by formula (1) with parameters: 1 ( )0 *E Xt* , 1 2 2 <sup>1</sup> ( ) *DX t <sup>t</sup>* . For any 1 2 ... *<sup>n</sup> tt t* probability density function of the multidimensional random vector <sup>1</sup> ( ,..., ) *<sup>n</sup> X X t t* can be obtained, taking into account Markov features of the process and using (3), as follows (Sobczyk, 1991):

$$f(t\_n, \mathbf{x}\_n, \dots, \mathbf{t}\_1, \mathbf{x}\_1) = \prod\_{i=1,\dots,n} \frac{1}{\sqrt{2\pi\sigma(t\_i - t\_{i-1})}} \exp\left(-\sum\_{i=1,\dots,n} \frac{\left(\mathbf{x}\_i - \mathbf{x}\_{i-1}\right)^2}{\mathbf{2}(t\_i - t\_{i-1})\sigma^2}\right). \tag{8}$$

Stochastic vector process [ ( ),..., ( )], 0 *Xt Xt t* <sup>1</sup> *<sup>n</sup>* is called the *nD stochastic Wiener process* if its every component, *Xt t <sup>i</sup>*( ), 0 , *i*=1,…,*n*, is the scalar stochastic Wiener process and particular scalar stochastic processes are independent.

As an example of the 3*D* stochastic Wiener process we can show three coordinates of the Brownian particle trajectory.

The Wiener process is also the special diffusion stochastic process, fulfilling the Fokker-Planck diffusion equation

$$\frac{\partial f(\mathbf{x},t)}{\partial t} = \gamma \frac{\partial^2 f(\mathbf{x},t)}{\partial \mathbf{x}^2}, \quad \lim\_{\Delta t \to 0, \Delta \mathbf{x} \to 0} \frac{\left(\Delta \mathbf{x}\right)^2}{\Delta t} = \text{const} = \mathbf{2}\gamma \tag{9}$$

where the solution is given by the normal probability density function, and the diffusion coefficient is equal to <sup>2</sup> / 2 (van Kampen, 1990).

In macroscopic scale, in physics and in industrial practice, the probability value *f*(x)*dx* that scalar variable *X* assumes its value from the interval [*x*, *x*+d*x*] is equivalent to the quotient *n*/*N* (concentration of particles), where *n* defines the power of subset of particles, whose feature *X* determines the value over the interval [*x*, *x*+d*x*], and *N* is the population size. This idea is consistent with Einstein's experiments who considered collective motion of Brownian particles. He assumed that the density (concentration) of Brownian particles ( ,) *x t* at point *x* at time *t*, met the following diffusion equation:

$$\frac{\partial \rho(\mathbf{x},t)}{\partial t} = \gamma \frac{\partial^2 \rho(\mathbf{x},t)}{\partial \mathbf{x}^2} \tag{10}$$

where γ is a diffusion coefficient. The solution has the known exponential form. From the analytical form of the solution the second central moment of the displacement is expressed as

$$E\left(\left.X\_t\right|^2\right) = 2\gamma t\tag{11}$$

Fuzzy Modelling Stochastic Processes Describing Brownian Motions 299

*x x* , *i,j=1,2,…,I*. Membership functions for joint fuzzy events *Ai Aj* should

*xx xx*

The joint 2D probability distribution of linguistic values (fuzzy states) of the stochastic process *X*(*t*) is determined by the joint probability distribution of the linguistic random

2

*PA A x x dp* 

(Walaszek-Babiszewska, 2008, 2011). From the joint probability distribution (17), the

1 2 (, ) ( ) (,) *i j t t i j AA t t*

*x x*

*PX X A*

[( /( )] (,)

*PX A X A*

2 1

 2 1 2 1

1

The following relationships should be fulfilled for the conditional distributions of fuzzy

The proposed model of the stochastic process, formulated into fuzzy categories, for two moments 1 2 *t t*, , 2 1 *t t* , is a collection of file rules, in the following form (Walaszek-

( ) *Ai L X* , *i*=1,…,*I*

*PX A*

( )

[( /( )] 1; . *t jt i*

*P X A X A i const*

*t ti j I t jt i j I t i*

*i j* 

0 ( ) 1, , 1,..., *PA A ij I i j* and

conditional probability distribution of the fuzzy transition

1,...,

*j I*

1 2 1 2 <sup>2</sup> ( , ) 1, ( , ) *AA t t t t i j*

1 2 , 1,2,..., ( , ) { ( )}, *PX X PA A tt i <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>I</sup>* (17)

(18)

( )1

(19)

. (21)

*i j*

*PA A*

1 2

1 1

2 1 [( ) /( )] *PX A X A <sup>t</sup> <sup>j</sup> t i* , *j*=1,2,…,*I; i*=*const* (20)

1,...,

1,...,

(22)

*i j*

*I I*

. (16)

1 2 (,) *Ai j At t*

vector 1 2 (,) *X X t t*

and fulfilling

calculated according to

can be determined according to

states (probability of the transitions)

**3.2 Rule based fuzzy model** 

Babiszewska, 2008, 2011):

fulfill

Diffusion coefficient, γ, has been expressed by Einstein as a function of macro- and microscopic parameters of the fluid and particles, respectively. Einstein confirmed statistical character of the diffusion law (cited by van Kampen, 1990).
