Stochastic process consisting both coding and proof, and some computation for constructing a transition matrix

RECALL : Sym =. -2-2

RW- wi zd .

Want to classify as recurrent or transient -

case I :d= ( Recurrent ? =-

-

caseIn.de#.LDenieusion - mi 3). it plays. . ft. any

.

,

,

,

,

.aY o o.w .

⇐ . - - - - - - ÷i÷ §( L ÷, y . y .

Pla , ne) =p . Recall : p Cn , n - D. =L -P

'

-

* * i ,

→ .

W . - Bailar , p-Z).P

-

- K .

pyw.=n } = Cnn ) -

Again for n - odd; Pulo, o) . = o -

W .

We need to estimate -

Z Penlop). '

h=0 -

The walk must take an equal number of hips in each of the 2 dissections associated with

any dimension to return to the starting point of 0 .

let the number of steps along each dimension be givin by i, j, k -

ng - Then itjtk . - n .

R

i steps → - y

← jour Then , - the number of steps the RW takes can

be represented as following a multinomial distribution :

-Mnttimnid . t.sn ., p . - ÷) .

=

(Imposing restriction that iii.k. are such that itjtkn?

i . Pzn .

10,0 ). = turn over rpnob - associated with each ii. j, k) combination: such that

itjtk = n .

T 2n = L

. in.im .) (f.) ""sites

.

is jsk 30-

i.+sixteen . -⑧

Aside : 2h - objects - - -are ÷÷ei÷÷÷-

t.I.i.i.i.ie) -- .ii. j ! k ! ii. silk ! as. 2litg.tk) - 2n .

Continuing from ⑧ s

kn) !Pan 10,0) .

= [ - "sik # o. Li ! j ! my

(f) " its'the?

itjtk .-n -

- En) ! h ! Ml.=L .

- -

i. sin >o . n ! h ! iii. j ! my

.

467 "

in. - .

itjtk --n - -

- til .

li.%a5.FI:5 "

is.nu.is. -- EH .

"

E .

I :*.5155 "

i , gish 70.

oitjtk = n. '①

Where do me go next ? hun *⑦ over all n .

& - determine if I Paco,O ? → .

⇒Riamont

-_ { i b- ⇒ Transient The smoothed out vermin of the MN pmf in a - 3 dim . space

. :

The pouf takes on its highest value on this

. "mound "

j i#

where i-- y k = m .

K .

Lets consider -the expression . I Bencao). under the 3 diff . ciuaumstances :

EI:3:L "e. iii. m]

.

letn-3m.IQ Then -

( i. "

j . .. )

.

⇐ (m? mm;m ) .

& CA .

can he

expressed as :

Pan 190). = pin) ( t) "

-2 .

-

÷÷: . EE⇒ ..

'?

⇐ HIEI "

:÷÷±:c:".). Hmm? Est

←

= t.tt: Eliz. ~:*:*.

"melts "

we are left # with this - =L . as- f 's

. .

] .

-MN -Ln, pets n objects

-

⇐ 'sis '¥n .

- II¥ .

Pznlo,D.IC(dn. by application 'appofme flirting's'

c. ÷⇐* , q b ( This converges ).

i .

A Random walk is transit . One can also show that if n•=3m

- l .

, 3.m

.

- ?

{ q"" ÷÷n " n⇒n⇒.-7.Pan 10,0) = Pgm 10,0). 3 ftp.Pjmy .

(907 .

In 10,0). if n=3m - l .

Oh.=3m- 2-

theorem A swipe RW on Ed .

is recurrent - if . D= I or 2.gg

& - transient if d 73 . .

1-mce.JAj.mu Fin . state z. ; for each . stater ; define

Pf.Xn=z. for some n 30- IX. = a}.

We see that a Cz) .

= p4Xn=z - for some riot Xo -- z} .

= I .

Suppose the chain is recurrent; then we world expect

Acn). ⇐ I .

t a-

Sps . the chain is transient. . there must be stator

with acne) 8*1 .

Infact , then should be states a . further & further among .

from Z with does. being arbitrarily small . =

Also ; for Z Fa ;

din) = Pfxn -- Z for some n ⇒ o - l Xo

.

-

7.} .

=p { Xu ' z. for some. n> i . IX. a-

o - a}

.

= ytfg.PL Xn . - Z- for some n> i. IX ,

-

- y } Pdx, -- y 1×0=23. --

dee) .

= E Ny) . play ). YES .

For a transient chain ; doe) satisfies :

(D .

OE din) E L .

7-(2) Nz). = 1. , inf . {din). : see 5} = O .

' 7¥ HE is '

③ .

deal .

= TL . Ny). plays. if a #Z-

- YEE

For a recurrent chain ;

Alu) = I . t NE Ii & . Nz) = I .

:c The criterion is :

Xu Frauen.int ⇒ F ! solution . to us, 14,133.

Xn .

recount L⇒ no solutions exists for 43,123,133.

trample .

: RW with partially reflecting boundary : p → Pms Is - - . . .

A -P '

←2 3. ←

i -p I-P

p la , set D= p t a30 .

p Ck , a - D. =L - p f a 31.

P ( O, O)

.

= I -p .

Questions : suppose 7=0 . Can we find a solution to 11161,133 ?

Recalls : Nas = PL Xu -Ez .

for some n>of Xo -

- a}

hyppose - Z -- O , se -0-# d.(z)

.

= 210?

= PL Xu -- O for some h2o ( Xo - o }.

= I .

i. Criterion ② satisfied .

Jnppose - -2=0. , N > o- : -