Ivancevic_Applied-Diff-Geom

988 Applied Differential Geometry: A Modern Introductionor corresponding Ito stochastic integral equationx i (t) = x i (0) +∫ t0ds A i [x i (s), s] +∫ t0dW j (s) B ij [x i (s), s],in which x i (t) is the variable of interest, the vector A i [x(t), t] denotes deterministicdrift, the matrix B ij [x(t), t] represents continuous stochastic diffusionfluctuations, and W j (t) is an N−variable Wiener process (i.e., generalizedBrownian motion) [Wiener (1961)], and dW j (t) = W j (t + dt) − W j (t).Now, there are three well–known special cases of the Chapman–Kolmogorov equation (see [Gardiner (1985)]):(1) When both B ij [x(t), t] and W (t) are zero, i.e., in the case of puredeterministic motion, it reduces to the Liouville equation∂ t P (x ′ , t ′ |x ′′ , t ′′ ) = − ∑ i∂∂x i {A i[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )} .(2) When only W (t) is zero, it reduces to the Fokker–Planck equation∂ t P (x ′ , t ′ |x ′′ , t ′′ ) = − ∑ i+ 1 ∑2ij∂∂x i {A i[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )}∂ 2∂x i ∂x j {B ij[x(t), t] P (x ′ , t ′ |x ′′ , t ′′ )} .(3) When both A i [x(t), t] and B ij [x(t), t] are zero, i.e., the state–space consistsof integers only, it reduces to the Master equation of discontinuousjumps∫∂ t P (x ′ , t ′ |x ′′ , t ′′ ) =dx {W (x ′ |x ′′ , t) P (x ′ , t ′ |x ′′ , t ′′ ) − W (x ′′ |x ′ , t) P (x ′ , t ′ |x ′′ , t ′′ )} .The Markov assumption can now be formulated in terms of the conditionalprobabilities P (x i , t i ): if the times t i increase from right to left,the conditional probability is determined entirely by the knowledge of themost recent condition. Markov process is generated by a set of conditionalprobabilities whose probability–density P = P (x ′ , t ′ |x ′′ , t ′′ ) evolution obeys