BROWNIAN MOTION A tutorial

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BROWNIAN MOTIONA tutorial

• Krzysztof Burdzy
• University of Washington

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A paradox
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() is maximized by f(t) 0, tgt0
The most likely (?!?) shape of a Brownian path
Microsoft stock - the last 5 years
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Definition of Brownian motion
Brownian motion is the unique process with the
following properties (i) No memory (ii)
Invariance (iii) Continuity (iv)
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Memoryless process
are independent
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Invariance
The distribution of depends only on t.
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Path regularity

• (i) is continuous a.s.
• (ii) is nowhere differentiable
  a.s.

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Why Brownian motion?

• Brownian motion belongs to several families
• of well understood stochastic processes
• Markov processes
• Martingales
• Gaussian processes
• Levy processes

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Markov processes

• The theory of Markov processes uses
• tools from several branches of analysis
• Functional analysis (transition semigroups)
• Potential theory (harmonic, Green functions)
• Spectral theory (eigenfunction expansion)
• PDEs (heat equation)

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Martingales
Martingales are the only family of processes for
which the theory of stochastic integrals is
fully developed, successful and satisfactory.
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Gaussian processes
is multidimensional normal (Gaussian)

1. Excellent bounds for tails
2. Second moment calculations
3. Extensions to unordered parameter(s)

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The Ito formula
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Random walk
Independent steps, P(up)P(down)
(in distribution)
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Scaling
Central Limit Theorem (CLT), parabolic PDEs
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Cameron-Martin-Girsanov formula
Multiply the probability of each Brownian path
by
The effect is the same as replacing
with
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Invariance (2)
Time reversal
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Brownian motion and the heat equation
temperature at location x at time t
Heat equation
Forward representation
Backward representation (Feynman-Kac formula)
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Multidimensional Brownian motion

• independent 1-dimensional
• Brownian motions

- d-dimensional Brownian motion
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Feynman-Kac formula (2)
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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.
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Conformal invariance
analytic
has the same distribution as
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The Ito formula Disappearing terms (1)
If then
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Brownian martingales
Theorem (Martingale representation
theorem). Brownian martingales stochastic
integrals
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The Ito formula Disappearing terms (2)
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Mild modifications of BM

• Mild the new process corresponds
• to the Laplacian
• Killing Dirichlet problem
• Reflection Neumann problem
• Absorption Robin problem

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Related models diffusions

1. Markov property yes
2. Martingale only if
3. Gaussian no, but Gaussian tails

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Related models stable processes
Brownian motion Stable processes

• Markov property yes
• Martingale yes and no
• Gaussian no
• Price to pay jumps, heavy tails,

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Related models fractional BM

1. Markov property no
2. Martingale no
3. Gaussian yes
4. Continuous

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Related models super BM
Super Brownian motion is related to
and to a stochastic PDE.
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Related models SLE
Schramm-Loewner Evolution is a model for
non-crossing conformally invariant 2-dimensional
paths.
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Path properties

• (i) is continuous a.s.
• is nowhere differentiable a.s.
• is Holder
• Local Law if Iterated Logarithm

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Exceptional points
For any fixed sgt0, a.s.,
There exist sgt0, a.s., such that
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Cut points
For any fixed tgt0, a.s., the 2-dimensional Brownia
n path contains a closed loop around in
every interval
Almost surely, there exist such that
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Intersection properties
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Intersections of random sets
The dimension of Brownian trace is 2 in every
dimension.
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Invariance principle

• 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)

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Local time
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Local time (2)
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Local time (3)
Inverse local time is a stable process with index
½.
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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