Little Notes on stochastic processes

8/21/2019 Little Notes on stochastic processes

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C

k +

k −

B

A

C

BA

( ) ( )2

2

, , p x t p x t D

t x

∂ ∂=

∂ ∂



















A B















x x x + ∆ x x −∆

12

12



































A B







0 50 100 150 2000

0.5

1

1.5

time (ms)

Height

(nm)

0.8

0.60.4

0.2

50 100 150time (s)

0

B

B

δ

A B C





x

t

( ),t ξ ω

( )time t













( )time t

( )t ξ

( ) ( )1

t ξ ( ) ( )2

t ξ ( ) ( )3

t ξ

( )time t

0 T

( )time t

0 T

( )time t

0 T







Bandwidth

A) B)

ω

ω

τ

τ

( ) x B τ

( ) y B τ

( ) xS ω

( ) yS ω









R

C

T

( ) I t

( )Q t





















t

1 x

2 x

3 x















1n − 1n +nsite site

1ng

− ng

1nr +n

r





A)

ˆ x x

n=

1

2

( )ˆ x ρ

1

8n

B)

n

balls









C

k +

k −

B

A

C

BA





C ~ 10

C ~ 100

C ~ 1000

b)

time

a) c)

time

[ ]C [ ]C [ ]( )C t

































0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

i p i p i p

in

in

in

10n = 50n = 100n =

B)A) C)







b

a

Fixed Point isStable

Limit CycleAbout theFixed Point

1 1b a< +

1b a> +

2 X

1 X

1b a> + 2 X

1 X

1b a< +

a) b) c)





1. Initialize: t← t0, n← n0.

2. Pick τ according to the density functiona,b

3.Pick µ according to the density functiona

4.Advance the process:•ni← ni + Siµ

• t← t + τ.c

5. Record as required for sampling or plotting.

If the process is to continue, then return to 2;

otherwise, stop.

( ) ( ) ( )1| , exp . p t a aτ τ = − n n n

( ) ( )

( )2 | , p t

a

µ

µ

ν ∆ =

nn n

n

( ), for ,

, for ;

t t t t

t t

µ τ ′− − < <

′ = ′ =

n Sn

n







Number of X1 molecules (x103)

1 2 43 5

1

2

3

4

5

1 2 43 5

1

2

3

4

5

Number of X1 molecules (x103)

N u m b e r o f X 2

m o l e c u l e s ( x 1

0 3 )

N u m b e r o f X 2

m o l e c u l e s ( x 1

0 3 )

A B



Number of X1 molecules (x103)1 2 43 5

1

2

3

4

5

N u m b e r o f X 2 m o l e c u l e s ( x

1 0 3 )

B

1

2

3

4

5

N u m b e r o f X j m o l e c u l e s ( x 1 0 3 )

A

1 2 3 4 5 6 7 8 9 10

Time

Number of X1

molecules (x103)

6 8 1210 14

4

6

8

12

14

Number of X1

molecules (x103)

N u m b e r o f X 2

m o l e c u l e s ( x 1 0 3 )

N u m b e r o f X 2 m o l e c u l e s ( x 1 0 3 )A B

10

2

4

6

8

12

14

10

2

2 4 6 8 1210 142 4













Ω→∞

Large Volume( ) x t

( ), t α Π

a) b)





















1n

1n

2n

2n

a) b)

1000 2000 3000

1000

2000

3000

1000 2000 3000

1000

2000

3000

1n

2n

1000 2000 3000

1000

2000

3000





Transcription

Translation

p

m

mα

pα

p β m

β

A) B)





P r o t e i n n u m b

e r

0 100 200 300 400 5000

100

200

300

400

500

600

Time (min) Time (min)

P r o t e i n n u m b

e r

b 15

25

20

10

5

50 100 150

A) B)

























( ) ( ) ( ) ( )3 3 1 1 3 3 2 2 2 2 1 1 2 1 2 3, | , , | , , | , f x t x t f x t x t f x t x t dx t t t

∞

−∞

= < <∫

Chapman-Kolmogorov Equation - A (nonlinear) functional equationgoverning all conditional probability densities of a Markov process.

StationaryProcess

( ) ( ) ( )3 1 3 2 2 1 2| , | , | , p x x p x x p x x dxτ τ τ τ

∞

−∞

′ ′+ = ∫

Chapman-Kolmogorov Equation

( ) ( )2

212

p A y p B y pt y y

∂ ∂ ∂= − + ∂ ∂ ∂

Assumean=0, n>2

Kolmogorov Equation

Compute frommicroscopic theory:

( ) ( ) ( )

( ) ( )

0| , 1

|

p x z a x z

w x z o

τ τ δ

τ τ

′ ′= − −

′ ′+ +

( ) ( ) ( ) ( ) ( )3 1 3 2 2 1 2 3 3 1 2| , | | , | | , p x x w x x p x x w x x p x x dxτ τ τ

τ

∞

−∞

∂= −

∂ ∫

M ast er Equ at ion

Assume the jumps are “small”, and p( y,t ) is a slowly varying function of y.

( ) ( )

( ) ( )

2

1 22

1

2

,n

n

p

a y p a y pt y y

a y z w y z dz

∞

−∞

∂ ∂ ∂

= − + ∂ ∂ ∂

= ∫

Fokker-PlanckEquation



0 y ( ) y t eq

y

0t >

( ), p y t ( )eq p y

















t t t

y y y

( ),P y t ( ),P y t ( ),P y t

( )0 y yδ −

( )

( )

where y t

dy A y

dt = ( ) 0

y t y A t = + ⋅

Width

2 D t ⋅

( ) 0

k t y t y e

− ⋅

=

( ) ( )2

Width

2 / 2 1 k t

D k e− ⋅

⋅ −

A) B) C)

















A) B)

xA

B

C

( )U x

0 x

W

A

Aω B

exp W

kT

−









θ











































1. Initialize: t← t0, y← y0.

2. Choose a suitably small ∆t>0.a

3. Draw a sample value n of the unitnormal random variable N(0,1).b

4.Advance the process:

•y← y + n· c(y,t) [∆t]1/2 + A(y,t)∆t• t← t + ∆t.c

5. Record y(t)=y as required forsampling or plotting. If the process isto continue, then return to 2 or 3;d

otherwise, stop.































t<<τcατc>>1Static Averaging

t>>τcατc<<1Kubo Renormalization

ατc<<1Bourret Convolution

αt<<1ατc>1

α2τct<<1ατc<1Born Iteration

Condition on t Condition onKubo number

Approximation

or



















( ) ( ) ( ) ( )0,1 x t t x t x t t + ∆ = − ⋅ ∆ ⋅N

( ) ( ) ( ) ( )

( ) ( ) ( )2 11 0,1

2

exp

c

c

x t t x t x t t t

t t t

t

η

η ρ η ρ τ

ρ τ

+ ∆ = − ⋅ ∆

+ ∆ = ⋅ + −

∆≡ −

N

1.

2.1 2 3 4 5

2

4

6

8

10

121.

2.

( ) x t

time

B.A.





























A) B)











Noise – Induced

Oscillations

( )110 Ab

−∆ ×

R

A

δ

δ

0.25 0.5 0.75 1

0.02

0.04

0.06

0.08

0.10

0.12

0.14Oscillations

Monostable

X

1 2 3 4 5 6 7

500

1000

1500

( ) R t

days

A) B)

0λ ′ =

0λ =







A

A

γ δ [ ]( )ln g A

[ ]ln A

AK

AK

f

A B C









λ N

µ N

Transport between subvolumes:

( ) ( )1, , 1Tr

iw N

i i i i N N N N

λ µλ λ µ λ µ

+ → +

Each subvolume is a

spatially-homogeneous

reaction vessel













0 0 1 0

1 1 1 1 0 0 1 0

0 0 0 1 1 1 1 1 0 0 1 0

0 0 0 1 1 1 1 1

0 0 0 1

− − −

− − −

− − −

1 2 3 4

λ λ λ λ ν ν ν ν

1 1 1 1

1 2 3 4


+ + + +1 1 1 1

1 2 3 4


− − − −

1

1

n

n

n

λ

λ

λ

−

+



1 1 1 1 0 0 1 0 0 0 0 1

0 0 0 1 1 1 1 1 0 0 1 0

0 0 1 0 0 0 0 1 1 1 1 1

− − −

− − −

− − −

1

2

3

n

n

n

Transport from n1 to n3

Transport from n3 to n

1

S =

1 1 0 1 0 0 1 0 0 0 0 0

0 0 0 1 1 1 1 1 0 0 1 0

0 0 0 0 0 0 0 1 1 1 1 0

− −

− − −

− −

1

2

3

n

n

n

No transport to the left

S =

No transport to the right





position, x

50

15.02

=

=

c

Dt

0 0.2 0.4 0.6 0.8 10

50

100

150

200

c o n c

e n t r a t i o n

B

0 0.2 0.4 0.6 0.8 10

50

100

150

200

position, x

c o n c

e n t r a t i o n

A

50

1.02

=

=

c

Dt

0 0.2 0.4 0.6 0.8 10

50

100

150

200

position, x

c o n c e

n t r a t i o n

C

50

2.02

=

=

c

Dt

0 0.2 0.4 0.6 0.8 10

50

100

150

200

position, x

c o n c e

n t r a t i o n

D

50

12

=

=

c

Dt



0 0.2 0.4 0.6 0.8 10

50

100

150

200

0 0.2 0.4 0.6 0.8 10

100

200

300

400

0 0.2 0.4 0.6 0.8 10

10

20

30

40

position, x

c o n c e n t r a t i o n

position, x position, x

A B C

2 1

50

Dt

c

=

=

2 1

100

Dt

c

=

=

2 1

10

Dt

c

=

=





0 0.2 0.4 0.6 0.8 10

50

100

150

200

250


position, x

A

0 0.2 0.4 0.6 0.8 10

1020

30

40

50

60


position, x

B

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

Uncertainty inthe interface

position is lessthan 4%


position, x

C Uncertainty inthe high stateis about 25%

Target molecule:

AverageStandarddeviation

Simulation















0.5 1 1.5 2

2

4

6

8

10

Stable

Turing Unstable

NoiseNoise--inducedinducedTuringTuring--likelike

instabilityinstability

H D

H ρ Inhibitor degradation rate,

Ratio of

diffusivity Unstable

a

b

d

c



Temporal instabili ty Turing instabili tyNoise-induced

spatial patterningSpatial patterning in

activator aloneA B C D

30

20

10

40 80 120 160 200

p o s i t i o n

time

00

200 600 1000 1400

30

20

10

40 80 120 160 200

p o s i t i o n

time

00

100 200 300 500

30

20

10

40 80 120 160 200

p o s i t i o n

time

00

100 200 300 400

30

20

10

40 80 120 160 200

p o s i t i o n

time

00

100 200 300400





0 0.2 0.4 0.6 0.8 10

50

100

Wavenumber, k

( ) HH S k

inhibitor

0 0.2 0.4 0.6 0.8 10

50

100

Wavenumber, k

( ) HH S k

inhibitor

0 0.2 0.4 0.6 0.8 10

50

100

Wavenumber, k

( ) AAS k activator

A

0 0.2 0.4 0.6 10

20

40

0.8

( ) AAS k activator

B

Wavenumber, k







E

EE

E

E

E P P

P

1 volt

1

1

1

1

.824 .785

.876 .503 .317 0

00

1

1



Macroscopiclimit cycle

1n

2

n s

r

φ



a)

2000 4000 6000 8000 10000

2000

4000

6000

8000

10000

1n

2n

2000 4000 6000 8000 10000

2000

4000

6000

8000

10000b)

1n

2n





c− c+

0U

W −

W +

( )U x

x

Variance of white noise forcing

Signal-to-noise ratio



Kalman

Filter

LeastSquares

StochasticProcesses

ProbabilityTheory

DynamicalSystems

LinearAlgebra





















( )nF x

( )nF x

−

( )F x

xn

x



( ) xθ

x0

1

( )

d x

dx

θ δ =

x0











A) B)

C) D)

( )baU X ,=

x

( ) f x

ab −

1

x

( ) X E a=

( ) f x

a1

a e

a

µ−σ µ µ+σ

2σ( )2, X N µ σ =

x

( ) f x

22

1

πσ

µ−σ µ µ+σ

2σ

( ), X C µ σ =

x

( ) f x

1πσ

( ) ( )

1

0 otherwise

b a a x b f x

− − ≤ ≤=

( ) [ ]exp 0 f x a ax x= − ≥

( ) ( )

2

22

1exp

22

x f x

µ

σ πσ

−= −

( ) ( )

( )2 2

f x x

σ π

µ σ

=− +

a b



A) B)

C) D)

0.2

0.4

0.6

0.8

1

x

( ) xF

( )baU X ,=

µ-σ µ

0.2

0.4

0.6

0.8

1

µ+σ x

( ) xF

( )2, X N µ σ =

0.2

0.4

0.6

0.8

µ-σ µ µ+σ x

( ) xF

( ), X C µ σ =

0.2

0.4

0.6

0.8

1

x

( ) xF

( ) X E a=

( ) a x

F xa b

−=

−

( ) 1 ax

F x e−

= −

( )2

11

2 2

xF x erf

µ

σ

−= −

( ) 1 1

arctan2

xF x

µ

π σ

− = −

a b

















0.0625001/24

0.062450107

0.062643106

0.062140105

0.063300104

0.061000103

Estimate ofprobability to

flip 4 heads

Number oftrials, N

N=input('How many trials?');

number_of_4_heads=0;for i=1:N

f1=rand;f2=rand;

f3=rand;f4=rand;if f1>0.5 && f2>0.5 && f3>0.5 && f4>0.5

number_of_4_heads=number_of_4_heads+1;

endend

sprintf('Average number of 4 heads tossed:%0.6f',number_of_4_heads/N)

A) B)



0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1 y

x



0.785398π /4

0.785627107

0.785315106

0.783310105

0.786300104

0.805000103

Estimate ofarea of quarter

circle

Number of

points, NN=input('How many points?');

pts_inside=0;for i=1:N

p1=rand;p2=rand;if p1^2+p2^2 <= 1

pts_inside=pts_inside+1;end

end

sprintf('Area of the quarter circle: %0.6f',pts_inside/N)

A) B)



P









a

λ

d

r

θ x

























































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Little Notes on stochastic processes