By Piotr Garbaczewski, Marek Wolf, Aleksander Weron

The research of chaotic behaviour of dynamical platforms has caused new efforts to reconcile deterministic and stochastic methods, in addition to classical and quantum physics. Efforts are being made to appreciate complicated and unpredictable behaviour. this article is an outline of those actions.

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**Example text**

P ( x n 1 x n ) p ( x n ). qtn 1 ,tn ( x n 1 , x n )q ( x n ), where qti 1 ,ti ( xi 1 , xi ) is the transition probability density, 1 i n and t i 1 t i . e. qti 1 ,ti ( xi 1 , xi ) For deriving density the stochastic 1 e iu ( xi 1 , xi ) Ee iu ( xi 1 xi ) du. 2 equation, we consider the conditional (11) probability p ( x1 x 2 ), where p ( x1 , x 2 ) p ( x1 x 2 ) p ( x 2 ). After integrating over the variable x 2 , the above equation leads to p ( x1 ) q t1 ,t2 ( x1 , x 2 ) p ( x 2 )dx 2 .

The Fokker-Planck equation is also known as the Kolmogorov forward equation. The Fokker-Planck equation is a special case of the stochastic equation (kinetic equation) as well. The stochastic equation is about the evolution of the conditional probability for given initial states for non-Markov processes. The stochastic equation is an infinite series. Here, we explain how the Fokker-Planck equation becomes a special case of the stochastic equation. , x n ).... p ( x n 1 x n ) p ( x n ). , x n ) p ( x1 x 2 ) p ( x 2 x3 )....

Pesquera and M. ), World Scientific, Heyden, Philadelphia, pp. 69–95. , Talkner, P. & Schimansky-Geier, L. (1989). Colored noise driven systems with inertia, Z. Physik B, 77, 471–483. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory, Academic Press, New York and London. Kloeden, P. E. and Platen, E. (1991). The Numerical Solutions of Stochastic Differential Equations (Applications of Mathematics), Springer, New York. Kowalski, K. & Steeb W-H. (1991). Non-Linear Dynamical Systems and Carleman Linearization, World Scientific, Singapore, New Jersy.