19 one dimensional wiener process and exit from an interval

Question 1

The standard Brownian motion in 1D — recap

in a finite domain [0, L] with absorbing boundary
conditions,

how do we write B_n?

Answer 1

p(x,t) = Σ_[n=1,..∞]
Bₙ sin( (nπ/L) x)*exp (-D((nπ)/L)²t)

The coefficients are found from
p(x, 0)

Bₙ= (2/L) ∫₀ ᴸ p(x,0)sin((nπ/L) x) .dx

Question 2

The standard Brownian motion in 1D —a recap

in a finite domain [0, L] with absorbing boundary
conditions,

how do we re-write p(x,t)?

Answer 2

p(x,t) = Σ_[n=1,..∞]
Bₙ sin( (nπ/L) x)*exp (-D((nπ)/L)²t)

Question 3

The standard Brownian motion in 1D — recap

with reflecting boundary conditions:
pdf

b_n?

Answer 3

then we have
p(x,t) = (b₀/2) +
Σ_[n=1,..∞]
bₙ cos( (nπ/L) x)*exp (-D((nπ)/L)²t

bₙ =
(2/L) ∫₀ ᴸ p(x,0)cos((nπ/L) x) .dx

Question 4

The standard Brownian motion in 1D — recap

0———x——-L
equally likely L or R
time plot position against time zig zag up down
against time W_t

exit time?

Answer 4

0—-a—–x—–b–L

P(x) = P[τₐ < τᵦ]
probability that i hit boundary a before b

T(x) =E(τ )
mean hitting time to any boundary (either of them)

Question 5

The standard Brownian motion in 1D
Exit times what conditions do they impose on
[0,L]

important to remember

Answer 5

d²P/dx² = 0 with P(0)=1 P(L)=0
(ie if i start at 0 the prob that i hit 0 is 1, hit L is 0)

D d²T/dx² = -1 with T(0) = 0 T(L) = 0.

diagram x against P(x): linear graph from P(0)=1 to P(L)
so the probability only depends linearly on how far away you are from each of the boudaries

diagram T(x) = upside down u shape from T(0) =0 to T(L)=0
max in middle?

Mean time is 0 if i already start at the end points