Complex Analysis and Brownian Motion
                                                                   Yuxuan Zhang
                                                                    June 5, 2013
                                                                       Abstract
                                           This paper discusses some basic ideas of Brownian motion. Beginning
                                        from measure theory, this paper makes a brief introduction to stochastic
                                        process, stochastic calculus and Markov property, recurrence as well as
                                        martingale related to Brownian motion. Later, it shows an application
                                        of Brownian motion which applies Brownian motion to prove Liouville’s
                                        theorem in complex analysis.
                                  Contents
                                  1 Introduction                                                                     2
                                  2 Brownian Motion                                                                  3
                                     2.1   Sigma Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .     3
                                     2.2   Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . .    4
                                     2.3   Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .     4
                                     2.4   Some related properties for Brownian motion . . . . . . . . . . .         6
                                           2.4.1   Markov property . . . . . . . . . . . . . . . . . . . . . . .     6
                                           2.4.2   Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . .    6
                                           2.4.3   Martingale . . . . . . . . . . . . . . . . . . . . . . . . . .    8
                                  3 Stochastic Calculus                                                              9
                                     3.1   Introduction to Stochastic Calculus . . . . . . . . . . . . . . . . .     9
                                     3.2   Ito Process and Ito’s Formula . . . . . . . . . . . . . . . . . . . .    10
                                  4 The Application of Brownian Motion                                             12
                                     4.1   Proof of Liouville’s theorem from Brownian motion . . . . . . . .        12
                                                                           1
                   Complex Analysis and Brownian Motion          2
                   1  Introduction
                   The first observation to Brownian Motion was in 1827 by British botanist,
                   Robert Brown. When studying pollen grains under the microscope, he sur-
                   prisingly found that the pollen grains are not static but instead, move in some
                   irregular way. However, at that time, Brown cannot find out the mechanisms
                   which caused this strange motion. The puzzle existed for about 100 years un-
                   til Albert Einstein, who published a related paper in 1905, suggested that the
                   motion which Brown observed was caused by the impact from the moving wa-
                   ter molecules. Molecules constantly bombard the pollen grain and due to the
                   imbalance force from each direction, the pollen grain will move in a constantly
                   changing path. Brownian motion is a great example which we can directly
                   observe the consequence of the moving of unobservable particles and this obser-
                   vation provides a solid confirmation for the existence to the molecules and atoms.
                       Fig 1 A three dimensional Brownian Motion for times 0 ≤ t ≤ 2 [4].
                   Scientists have studied Brownian Motion for a long time, and mathematicians
                   also, from their perspective, provide their explanation and prediction result to
                   Brownian Motion. Brownian Motion now is not a mysterious observation to
                   confuse people any more, but instead, we apply what we get from the study to
                   Brownian Motion to help us determine more and more complex phenomenon
                   in this world. For example, the use of Brownian Motion to predict the Stock
                   market [5] and the application in the prediction of heat flow [1]. In this paper,
                   we will discuss the study of Brownian Motion structured in math related to
                   complex analysis and later, we will consider some examples related to Brownian
                   Motion.
                                Complex Analysis and Brownian Motion                                           3
                                2    Brownian Motion
                                In this section, we’ll cover up some definition and basic properties for Brownian
                                Motion.
                                2.1    Sigma Algebra
                                As the steal theoretical foundation of the modern probability, the measure the-
                                ory provides us a pure mathematical perspective of probability knowing from
                                the classical, frequency or subjective interpretations to probability from philos-
                                ophy. We’ll here only discuss some basic theorems building up the whole system.
                                   Definition 2.1.1 (σ − algebra) A collection F of subsets of a set X is
                                called a σ − algebra if
                                   • 1. ∅ ∈ F;
                                   • 2. if A ∈ B, then A′ ∈ F;
                                   • 3. if A , A , ... , A ,... ∈ B, then S∞ A ∈ F.
                                            1    2       n                 i=1  i
                                The pair (X,F) is a field of sets, called a measurable space.
                                   Next, we will give a formal definition of the probability space as well as the
                                probability measure.
                                   Definition 2.1.2 (Probability Measure) A probability measure on a given
                                probability space is function υ satisfying the following conditions:
                                   • 1. υ is a function map event space to unit interval [0, 1], or υ : Ω 7→ [0,1];
                                           S          P
                                   • 2. υ( i∈I Ei) =     i∈I υ(Ei)
                                   Definition 2.1.3 (Probability Space) A probability space is a triple (Ω, F,
                                P) consisting of:
                                   • the sample space Ω as an arbitrary non-empty set;
                                                          Ω
                                   • the σ-algebra F ⊆ 2    ;
                                   • the probability measure P : F 7→ [0,1] .
                                   And the following is the formal definition of filtration, which will be used
                                later when defined stopping time and martingale.
                                   Definition 2.1.4 (Filtration) A filtration on (Ω,F,P) is a collection mea-
                                surable sets F : t ≥ 0 which satisfies F ⊂ F ⊂ F if s < t.
                                              t                         s    t
                                    Complex Analysis and Brownian Motion                                                      4
                                    2.2     Stochastic Process
                                    Also called as random process, the stochastic process is often used to show the
                                    evolutionofsomerandomvaluebasedontime. Notlikehowdeterministicprocess
                                    works, which predicts the process can only develop in one way, stochastic pro-
                                    cess allows some indetermination.
                                        Definition 2.2.1 (Stochastic Process) Given a probability space (Ω,F,P)
                                    and a measurable space (S,Σ), an S-valued stochastic process is a collection of
                                    S-valued random variables on Ω, indexed by a totally ordered time set T. That
                                    is, a stochastic process B is a collection
                                                                           {Bt : t ∈ T}
                                    where each B is an S-valued random variable on Ω. The space S is then called
                                                    i
                                    the state space of the process.
                                    In the study of stochastic process, there is a important concept called stopping
                                    time. As a specific type of ”random time”, stopping time is a random variable
                                    related to time with respect to the event space Ω. The following is the formal
                                    definition of stopping time based on filtration.
                                        Definition 2.2.2 (Stopping time) A random variable T : Ω 7→ [0,∞] de-
                                    fined on a filtered probability space is called a stopping time with respect to the
                                    filtration F if the set x ∈ Ω : T(x) ≤ t ∈ Ft for all t.
                                    One of the example of stopping time is the first occasion of the expected event.
                                    From stopping time, we can decide whether T ≤ t simply by knowing the states
                                    of the stochastic process until time t [6].
                                    2.3     Brownian Motion
                                    Now, based on our theories above, we’ll be able to give the formal definition of
                                    Brownian motion based on measure theory.
                                        Definition 2.3.1 (d − dimensional Brownian Motion) A d-dimensional
                                    Brownian motion is a stochastic process Bt : Ω 7→ R from the probability space
                                    (Ω,F,P) to Rd such that the following properties hold:
                                        • 1. (Independent Increments) For any finite sequence of times t0 < t1 <
                                           ... < tn, the distributions Bt      −Bt for i = 1,...,n are independent,
                                                                           i+1       i
                                        • 2. For all ω ∈ Ω, the parametrization function t 7→ Bt(ω) is continuous,
                                        • 3. (Stationary) For any pair s,t ≥ 0, let B           −B ∈A,
                                                                                            s+t      s
                                                                                  Z       1      −|x|2/2t
                                                              P(Bs+t −Bs) = A (2πt)d/2e                  dx.