The Kolmogorov forward equation, also known as the Fokker-Planck equation, describes the time evolution of the probability density function of a stochastic process. It is fundamental in the study of continuous-time Markov processes and is widely used in fields such as statistical physics, finance, and biology to model systems with random dynamics.

Kolmogorov Forward Equation

Kolmogorov forward equation

The Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of forces and diffusion. It is widely used in statistical mechanics to model various systems, including Brownian motion and other processes driven by random forces.

Fokker-Planck equation

Time evolution refers to the change of a system's state over time, governed by specific dynamical laws or equations, such as the Schrödinger equation in quantum mechanics or Newton's laws in classical mechanics. Understanding time evolution is crucial for predicting future states of a system and analyzing its behavior under varying conditions.

time evolution

A Probability Density Function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value, where the area under the curve represents the probability of the variable falling within a given range. The total area under the PDF curve equals one, ensuring that it accounts for all possible outcomes of the variable.

probability density function

A stochastic process is a collection of random variables representing the evolution of a system over time, where the future state depends on both the present state and inherent randomness. It is widely used in fields like finance, physics, and biology to model phenomena that evolve unpredictably over time.

stochastic process

continuous-time Markov processes

Statistical physics is a branch of physics that uses statistical methods to explain and predict the behavior of systems composed of a large number of particles. It bridges the microscopic laws of physics and macroscopic observations, providing insights into phenomena like phase transitions and thermodynamic properties.

statistical physics

Finance is the science of managing money, including the processes of acquiring, investing, and utilizing funds to achieve personal, corporate, or governmental goals. It encompasses various activities such as budgeting, saving, lending, investing, and forecasting to optimize financial resources and ensure stability and growth.

finance

Biology is the scientific study of life and living organisms, encompassing their structure, function, growth, evolution, distribution, and taxonomy. It is a vast and interdisciplinary field that connects with other sciences to understand the complex interactions within ecosystems and the molecular mechanisms of life.

biology

Random dynamics refers to the study of systems where the evolution is influenced by random processes or stochastic elements, leading to outcomes that are not entirely predictable. This field explores how randomness affects the behavior of dynamical systems, often using probabilistic methods to understand stability, long-term behavior, and chaos in various contexts.

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