Stochastic Meaning Theory 5

Language as Brown Motion

For ZHANG Taiyan 2

TANAKA Akio

[A]

1

Abstractive space Ω

σ additive family that consists of subset of Ω F

Measure that is defined over F P

P satisfies P (Ω ) = 1.

P probability measure over ( Ω, F )

Ω sample space

( Ω, F , P ) Probability space

Element of Ω sample ω

Element of F event A

Probability that event A occurs probability P ( A )

Real number valued Borel measurable function over Ω random variable X = X ( ω )

Random variable is integrable.

Mean (Expectation) of X E[X] = ∫_Ω X ( ω ) P ( dω )

2

Measurable space ( S, S )

X : ( Ω, F ) → ( S, S )

X is measurable.

X S-value random variable.

Random variable X₁, …, X_d

X : = (X₁, …, X_d ) R^d-value random variable

3

R^d-value random variable X

E[X _i ²] < ∞

E[(X - E[X])²] variance

4

S-value random variable. X

P_X : = P ( X ∈A ), A∈S distribution

5

Real number space R

Borel set family over R B ( R )

Probability measure over ( R, B ( R ) ) μ

6

R^d-value random variable X

ψ_X (ξ ) : = E[e^iξ・X], ξ∈R^dcharacteristic function

7

Lebesgue measure dx

Mean m∈R

Variance v >0

Measure over R μ ( dx ) = e ^{-(x-m)2 / 2v} dx / Gauss distribution ( normal distribution)

8

(2p – 1 ) !! : = (2p – 1 ) ・(2p – 3 ) … 3・1

E[X^2p] = (2p – 1 ) !! v ^p moment of X

9

Event A, B∈F

When (A∩B) = P(A) P(B), A and B are independent each other.

10

Integrable and independent random variable X, Y

Product XY integrable

E[XY] = E[X]E[Y]

11

Time t

t ∈[0, ∞)

Family of R^d-value random variable ≥ X = ( X_t ) _{t ≥ 0} d-dimensional stochastic process

∀ω∈Ω

When X_t (ω) is continuous as function of t., d-dimensional stochastic process is called to be continuous.

12

σ additive family F_t

F_t ⊂F

0 ≤ s ≤ t

F _s ⊂F_t

(F_t ) = (F_t ) _{t ≥ 0} increase information system

13

d-dimensional stochastic process X = ( X_t ) _{t ≥ 0}

∀t ≥ 0

X_t : Ω → R^d is F_t – measurable.

X = ( X_t ) _{t ≥ 0} is (F_t ) – adapted.

14

Mapping ( t, ω) ∈([0, ∞)×Ω, B([0, ∞)]×F) ↦ Xt ( ω) ∈( R^d, B ( R^d ) )

When the mapping is measurable, X = ( X_t ) _{t ≥ 0} is called to be measurable.

15

X = ( X_t )

F_t⁰ = F_t^0,X : = σ ( X_S ; s≤t )

16

Probability space ( Ω, F , P )

Stochastic process defined over ( Ω, F , P ) (Bt)_{t ≥ 0} = (Bt(ω)) _{t ≥ 0}

(B_t)_{t ≥ 0} that satisfies the next, it is called Brownian motion.

(i) P ( B₀ = 0 ) = 1

(ii) For ∀ω∈Ω, B_t (ω) is continuous on t.

(iii) For 0 = t₀<∀t₁<…<t_n, ∀n∈N, {Bt_i-Bt_i-1} satisfies the next.

a) {Bt_i-Bt_i-1} are independent each other.

b) {Bt_i-Bt_i-1} are followed by mean 0 and variance t_i-t_i-1 of Gauss distribution.

17

(Existence theorem)

Over adequate probability space, there exists Brownian motion.

18

Ω = W₀

F = B ( W₀ )

Brownian motion has the next.

(i)B_t ( w ) = W_t

(ii) w = ( wt ) _{t ≥0} ∈W ₀

Measure over ( W0, B ( W₀ ) ) P

P is called Wiener measure.

19

d-dimensional Brownian motion B = ( B_t ) _{t ≥ 0}

d×d orthogonal matrix A

AB_t d-dimensional Brownian motion

Sphere S : = δ B (0, r), B (0, r) = {|x| ≤ r }

Hitting time σS (ω) : = inf{t >0; B_t ∈S }

Hitting place B_{σS (ω)}

Distribution of B_{σS (ω)} uniform stochastic measure

20

d-dimensional Brownian motion B = ( B_t ) _{t ≥ 0}

x∈R^d

Brownian motion from x ( x + B_t ) _{t ≥ 0}

W ^d = B ( W ^d )

Space (W ^d, W ^d )

Distribution over (W ^d, W ^d ) P_x

Mean on P_x E_x [ ・ ]

Probability space (W ^d, W ^d , P_x )

Stochastic process over (W ^d, W ^d , P_x ) B_t ( w ), w∈W ^d ; B_t ( w ) = w_t

Sub σ additive family of W ^d F_t⁰ =σ (B_s ; s≤t ) , F_t = F_t⁰ ⋁ N, t≥0 ; N : = {N∈W ^d ; P_x (N) = 0, ∀x ∈R^d }

F_t^* = F_t+ : = ∩s>t F_s

Shift operator over W ^d θ_s : W ^d → W ^d , s≥0 ; (θ_s (w) ) _t : = w_t+s

B_t ∘ θ_s = B_t+s

21

(Markov property)

∀x∈R^d

∀s≥0

∀Y = Y (w) : W ^d –measurable bounded function over W ^d

E_x[Y∘θ_s ・1_A] = E_x[E_Bs(w)[Y]∘θ_s ・1_A] , ∀A∈F _s

By conditional mean

E_x[Y∘θ_s | F_s] (w) = E_Bs(w)[Y

P_x-a.s.w

22

(Blumenthal’s 0-1 law)

When A∈F₀ ( = F₀^* ), P_x (A) = 0 or 1

23

Random variant of 1-dimensional Brownian motion starting from the origin B

σ _(0,∞) : = inf {t >0; B_t∈(0,∞) }

A = {σ_(0,∞) = 0 }

A ∈F₀^*

P (σ_(0,∞) = 0 ) = 0 or 1

t↓0

P (σ_(0,∞) = 0 ) = 1

From symmetry of Brownian motion B_t = -B_t

[B]

Language that has Brownian motion L_B

L_B has actual language and imaginary language.

[References]

Mirror Theory For the Structure of Prayer / Dedicated to the Memory of CHINO Eiichi / Tokyo June 5, 2004

Mirror Language / Tokyo June 10, 2004

Guarantee of Language / For LÉVI-STRAUSS Claude / Tokyo June 12, 2004

Actual Language and Imaginary Language / To LÉVI-STRAUSS Claude / Tokyo September 23, 2004

To be continued

Tokyo August 12, 2008

Sekinan Research Field of Language

www.sekinan.org