[확률론] Brownian Motion (Wiener process)

브라운 운동(Brownian motion)은 1827년 스코틀랜드 식물학자 로버트 브라운(Robert Brown)이 발견한, 액체나 기체 속에서 미소입자들이 불규칙하게 운동하는 현상이다. 수학 분야에서는 Random Walk로 모델링 될 수 있다.

(In mathematics, Brownian motion is described by the Wiener process)

Brownian motion is a fundamental concept in mathematics, particularly in probability theory and stochastic processes. It provides a mathematical model for the random motion of particles suspended in a fluid, but its applications extend far beyond physics.

Mathematical Definition

In mathematics, Brownian motion is defined as a continuous-time stochastic process with the following properties:

1. It starts at zero: $B(0) = 0$
2. It has independent increments
3. For $t > s$, the increment $B(t) - B(s)$is normally distributed with mean 0 and variance $t - s$
4. It has continuous sample paths

This definition formalizes the idea of a particle moving randomly in a fluid, with its position at any time being unpredictable but statistically characterized[1].

Key Properties

Continuity: Brownian motion paths are continuous but nowhere differentiable. This means that while the position of a Brownian particle changes smoothly over time, its velocity is undefined at any given instant[1].

Scaling Property: For any constant $a > 0$, the process ${aB(t/a^2)}_{t\geq0}$ is also a standard Brownian motion. This property, known as self-similarity, implies that Brownian motion looks statistically the same at all time scales[4].

Markov Property: The future behavior of a Brownian motion depends only on its current state, not on its past history. This property makes Brownian motion a Markov process[2].

Mathematical Tools

Several mathematical tools are crucial in the study of Brownian motion:

1. Stochastic Calculus: This branch of mathematics, including Itô calculus, provides tools for analyzing Brownian motion and related processes[2].
2. Partial Differential Equations: The behavior of Brownian motion is closely related to certain PDEs, particularly the heat equation[7].
3. Measure Theory: The rigorous construction of Brownian motion relies heavily on measure-theoretic probability[4].

Applications in Mathematics

Brownian motion serves as a building block for more complex stochastic processes and has numerous applications:

1. Stochastic Differential Equations: These equations, which incorporate Brownian motion, are used to model various phenomena in physics, biology, and finance[6].
2. Martingale Theory: Brownian motion is a fundamental example of a continuous-time martingale, a concept central to modern probability theory[2].
3. Functional Analysis: The study of Brownian motion has led to important developments in the theory of function spaces and operators[7].
4. Probability Theory: Brownian motion plays a crucial role in proving limit theorems and studying convergence of stochastic processes[4].

Conclusion

Brownian motion stands as a cornerstone of modern probability theory and stochastic processes. Its rich mathematical structure and wide-ranging applications make it an essential topic in mathematics, with implications spanning from theoretical research to practical modeling in various scientific disciplines[6][9].

Citations:
[1] https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Dahl.pdf
[2] https://repository.rit.edu/cgi/viewcontent.cgi?article=5989&context=theses
[3] https://en.wikipedia.org/wiki/Brownian_motion
[4] https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/18:_Brownian_Motion/18.01:_Standard_Brownian_Motion
[5] https://fiveable.me/key-terms/mathematical-probability-theory/brownian-motion
[6] https://www.vaia.com/en-us/explanations/physics/solid-state-physics/brownian-motion/
[7] http://galton.uchicago.edu/~lalley/Courses/313/WienerProcess.pdf
[8] https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms/14.4.373
[9] https://datascientest.com/en/brownian-motion-principle-and-practical-uses