Introduction to Brownian Motion

Brownian motion is the random motion of particles suspended in a fluid due to their collision with the fast-moving atoms or molecules in the gas or liquid.

Robert Brown, a Scottish botanist, first observed Brownian motion in 1827 while studying the movement of pollen grains in water.

The Wiener process is a continuous-time stochastic process that describes the evolution of a random variable over time. It is the mathematical model for Brownian motion.

Brownian motion is a continuous-time process, meaning that it can take on any value in a continuous interval. It has independent increments, meaning that the increments of the process are independent of each other. Its increments are normally distributed, meaning that the probability distribution of the increments is a normal distribution.

Brownian motion has applications in a wide variety of fields, including financial modeling, physics, and biology. In financial modeling, it is used to model the prices of stocks and other financial instruments. In physics, it is used to model the motion of particles in a fluid. In biology, it is used to model the motion of cells and other biological entities.

The diffusion equation is a partial differential equation that describes the evolution of the probability density function of Brownian motion. It is a second-order partial differential equation that can be used to solve for the probability density function of Brownian motion at any given time.

The Wiener process is a stochastic process that describes the evolution of a random variable over time. It is a continuous-time stochastic process that can be used to model the evolution of a random variable over time.