Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 989the general Chapman–Kolmogorov integro–differential equation∂ t P = − ∑ i+ 1 ∑2ij∂∂x i {A i[x(t), t] P }∂ 2∂x i ∂x j {B ij[x(t), t] P } +∫dx {W (x ′ |x ′′ , t) P − W (x ′′ |x ′ , t) P }including deterministic drift, diffusion fluctuations and discontinuousjumps (given respectively in the first, second and third terms on the r.h.s.).It is this general Chapman–Kolmogorov integro–differential equation,with its conditional probability density evolution, P = P (x ′ , t ′ |x ′′ , t ′′ ), thatwe are going to model by various forms of the Feynman path integral,providing us with the physical insight behind the abstract (conditional)probability densities.6.1.5 Quantum Probability ConceptAn alternative concept of probability, the so–called quantum probability, isbased on the following physical facts (elaborated in detail in this section):(1) The time–dependent Schrödinger equation represents a complex–valuedgeneralization of the real–valued Fokker–Planck equation for describingthe spatio–temporal probability density function for the system exhibitingcontinuous–time Markov ∫stochastic process.(2) The Feynman path integral Σ is a generalization of the time–dependentSchrödinger equation, including both continuous–time and discrete–time Markov stochastic processes.(3) Both Schrödinger equation and path integral give ‘physical description’of any system they are modelling in terms of its physical energy, insteadof an abstract probabilistic description of the Fokker–Planck equation.∫Therefore, the Feynman path integral Σ, as a generalization of the time–dependent Schrödinger equation, gives a unique physical description forthe general Markov stochastic process, in terms of the physically basedgeneralized probability density functions, valid both for continuous–timeand discrete–time Markov systems.Basic consequence: a different way for calculating probabilities. Thedifference is rooted in the fact that sum of squares is different from thesquare of sums, as is explained in the following text.