1. BROWNIAN MOTION
A tutorial
Krzysztof Burdzy
University of Washington

2. A paradox
∞<′′→
∈
|)(|sup,]1,0[:
]1,0[
tfRf
t
(*)))((
2
1
exp)(
)10,)()((
1
0
2








′−≈
<<+<<−
∫ dttfc
ttfBtfP t
ε
εε

3. (*) is maximized by f(t) = 0, t>0
Microsoft stock
- the last 5 years
The most likely (?!?) shape of a
Brownian path:

4. Definition of Brownian motion
Brownian motion is the unique process
with the following properties:
(i) No memory
(ii) Invariance
(iii) Continuity
(iv) tBVarBEB tt === )(,0)(,00

5. Memoryless process
0t
2t1t 3t
,,, 231201 tttttt BBBBBB −−−
are independent

6. Invariance
The distribution of
depends only on t.
sst BB −+

7. Path regularity
(i) is continuous a.s.
(ii) is nowhere differentiable a.s.
tBt →
tBt →

8. Why Brownian motion?
Brownian motion belongs to several families
of well understood stochastic processes:
(i) Markov processes
(ii) Martingales
(iii) Gaussian processes
(iv) Levy processes

9. Markov processes
}0,|,{}|,{ suBstBBstB utst ≤≤≥=≥ LL
The theory of Markov processes uses
tools from several branches of analysis:
(i) Functional analysis (transition semigroups)
(ii) Potential theory (harmonic, Green functions)
(iii) Spectral theory (eigenfunction expansion)
(iv) PDE’s (heat equation)

10. Martingales are the only family of processes
for which the theory of stochastic integrals is
fully developed, successful and satisfactory.
Martingales
sst BBBEts =⇒< )|(
∫
t
ss dBX
0

11. Gaussian processes
nttt BBB ,,, 21
 is multidimensional
normal (Gaussian)
(i) Excellent bounds for tails
(ii) Second moment calculations
(iii) Extensions to unordered parameter(s)

12. The Ito formula
∑∫ =
+
∞→
−=
nt
k
nknknk
n
t
ss BBXdBX
0
//)1(/
0
)(lim
∫ ∫ ′′+′+=
t t
ssst dsBfdBBfBfBf
0 0
0 )(
2
1
)()()(

13. Random walk
Independent steps, P(up)=P(down)
{ } { }0,0,/ ≥ →≥ ∞→
tBtWa t
a
at
(in distribution)
tW
t

14. Scaling
Central Limit Theorem (CLT),
parabolic PDE’s
}10,{}10,{ / ≤≤=≤≤ tBatB at
D
t

15. The effect is the same as replacing
with
Multiply the probability of each Brownian path
by
Cameron-Martin-Girsanov formula
}10,{ ≤≤ tBt








′−′∫ ∫
1
0
1
0
2
))((
2
1
)(exp dssfdBsf s
}10,{ ≤≤ tBt }10),({ ≤≤+ ttfBt

16. Invariance (2)
}10,{}10,{ 11 ≤≤−=≤≤ − tBBtB t
D
t
Time reversal
0 1

17. Brownian motion and the heat equation
),( txu – temperature at location x at time t
),(
2
1
),( txutxu
t
x∆=
∂
∂
Heat equation:
dxxudx )0,()( =µ
)(),( dyBPdytyu t ∈= µForward
representation
)0,(),( yBEutyu t +=Backward representation
(Feynman-Kac formula)
0 t
µ
y

18. Multidimensional Brownian motion
,,, 321
ttt BBB - independent 1-dimensional
Brownian motions
),,,( 21 d
ttt BBB  - d-dimensional Brownian
motion

19. Feynman-Kac formula (2)
)(xf














−= ∫
τ
τ
0
)(exp)()( dsBVBfExu s
x
τB x
0)()()(
2
1
=−∆ xuxVxu

20. Invariance (3)
The d-dimensional Brownian motion is invariant
under isometries of the d-dimensional space.
It also inherits invariance properties of the
1-dimensional Brownian motion.
)2/)(exp(
2
1
)2/exp(
2
1
)2/exp(
2
1
2
2
2
1
2
2
2
1
xx
xx
+−=
−−
π
ππ

21. Conformal invariance
f
}0),()({ 0 ≥− tBfBf t
analytic
tB )( tBf
has the same distribution as
∫ ′=≥
t
stc dsBftctB
0
2
)( |)(|)(},0,{

22. If then
The Ito formula
Disappearing terms (1)
∫ ∫∆+∇+=
t t
ssst dsBfdBBfBfBf
0 0
0 )(
2
1
)()()(
∫∇+=
t
sst dBBfBfBf
0
0 )()()(
0≡∆f

23. Brownian martingales
Theorem (Martingale representation theorem).
{Brownian martingales} = {stochastic integrals}
},{,)|(
0
tsBFMMMME
dBXM
s
B
ttsst
t
sst
≤=∈=
= ∫
σ

24. The Ito formula
Disappearing terms (2)
∫∫
∫
∂
∂
+
∂
∂
+
∂
∂
=−
t
s
t
s
t
sst
dsBsf
x
dsBsf
s
dBBsf
x
BtfBtf
0
2
2
0
0
0
),(
2
1
),(
),(),(),(
∫∫ ∂
∂
+
∂
∂
=
−
t
s
t
s
t
dsBsf
x
EdsBsf
s
E
BtEfBtEf
0
2
2
0
0
),(
2
1
),(
),(),(

25. Mild modifications of BM
Mild = the new process corresponds
to the Laplacian
(i) Killing – Dirichlet problem
(ii) Reflection – Neumann problem
(iii) Absorption – Robin problem

26. Related models – diffusions
dtXdBXdX tttt )()( µσ +=
(i) Markov property – yes
(ii) Martingale – only if
(iii) Gaussian – no, but Gaussian tails
0≡µ

27. Related models – stable processes
2/1
)(dtdB =
α/1
)(dtdX =
Brownian motion –
Stable processes –
(i) Markov property – yes
(ii) Martingale – yes and no
(iii) Gaussian – no
Price to pay: jumps, heavy tails, 20 ≤<α
220 ≤<

28. Related models – fractional BM
α/1
)(dtdX =
(i) Markov property – no
(ii) Martingale – no
(iii) Gaussian – yes
(iv) Continuous
∞<<α1
∞<< 21

29. Related models – super BM
Super Brownian motion is related to
2
uu =∆
and to a stochastic PDE.

30. Related models – SLE
Schramm-Loewner Evolution is a model
for non-crossing conformally invariant
2-dimensional paths.

31. (i) is continuous a.s.
(ii) is nowhere differentiable a.s.
(iii) is Holder
(iv) Local Law if Iterated Logarithm
Path properties
tBt →
tBt →
tBt → )2/1( ε−
1
|log|log2
suplim
0
=
↓ tt
Bt
t

32. Exceptional points
1
|log|log2
suplim
0
=
↓ tt
Bt
t
1
|)log(|log)(2
suplim =
−−
−
↓ stst
BB st
st
For any fixed s>0, a.s.,
),0(
)(2
suplim ∞∈
−
−
↓ st
BB st
st
There exist s>0, a.s., such that

33. Cut points
For any fixed t>0, a.s., the 2-dimensional
Brownian path contains a closed loop
around in every intervaltB ),( ε+tt
Almost surely, there exist
such that
)1,0(∈t
∅=∩ ])1,(()),0([ tBtB

34. Intersection properties
1))((.,.)4(
2))((.,.
2))((.,.)3(
))((.,.
..)2(
.,.)1(
14
13
13
12
≤∈∀=
≤∈∀
=∈∃=
∞=∈∃
≠≠∀∀=
=≠∃∀=
−
−
−
−
xBCardRxsad
xBCardRxsa
xBCardRxsad
xBCardRxsa
BBtssatd
BBtstsad
ts
ts

35. Intersections of random sets
∅≠∩
>+
BA
dBA

)dim()dim(
The dimension of Brownian trace is 2
in every dimension.

36. Invariance principle
(i) Random walk converges to Brownian
motion (Donsker (1951))
(ii) Reflected random walk converges
to reflected Brownian motion
(Stroock and Varadhan (1971) - domains,
B and Chen (2007) – uniform domains, not all
domains)
(iii) Self-avoiding random walk in 2 dimensions
converges to SLE (200?)
(open problem)
2
C

37. Local time
s
ts
t
Bt
BM
dsL s
≤
<<−
→
=
= ∫
sup
2
1
lim }{
0
t
0
1 εε
ε ε

38. Local time (2)
}10,|{|}10,{
}10,{}10,{
≤≤=≤≤−
≤≤=≤≤
tBtBM
tMtL
t
D
tt
t
D
t

39. Local time (3)
}{inf
0
tLs
s
t ≥=
>
σ
Inverse local time is a stable process
with index ½.

40. References
 R. Bass Probabilistic Techniques in
Analysis, Springer, 1995
 F. Knight Essentials of Brownian Motion
and Diffusion, AMS, 1981
 I. Karatzas and S. Shreve Brownian
Motion and Stochastic Calculus, Springer,
1988