Kolmogorov Forward Equations (other)
Kolmogorov forward equations may refer to: * Kolmogorov equations (Markov jump process), relating to discrete processes * Fokker–Planck equation In statistical mechanics, 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 drag forces and random forces, as ...

Kolmogorov Equations (Markov Jump Process)
In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evolution of the process's distribution. This article, as opposed to the article titled Kolmogorov equations, focuses on the scenario where we have a continuous-time Markov chain (so the state space \Omega is countable). In this case, we can treat the Kolmogorov equations as a way to describe the probability P(x,s;y,t), where x, y \in \Omega (the state space) and t > s, t,s\in\mathbb R_ are the final and initial times, respectively. The equations For the case of a countable state space we put i,j in place of x,y. The Kolmogorov forward equations read : \frac(s;t) = \sum_k P_(s;t) A_(t) , where A(t) is the transition rate matrix (also known as the generator matrix), while the Kolmogorov backward equations are : \frac(s;t) = -\ ...
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