By Meghnad Saha
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106) −∞ Due to the symmetry property Cαβ (t) = Cβα (−t), the Fourier transforms are ∗ (ω). The Sαβ (ω) are called the spectral functions self-adjoint; Sαβ (ω) = Sβα of the underlying processes. In addition to the classiﬁcation of correlation processes introduced above, the same properties might be investigated in the frequency domain. To this end, we consider again a stationary process in a one-dimensional state space. 105) the important relation τc = S(0). Therefore, we conclude that a convergent behavior of S(ω) in the low-frequency regions indicates a short-range correlation, while any kind of divergence is related to long-range correlations.
This means explicitly Yα (ti+1 ) = Yα (ti ) + aα (Y (ti ))dt + bα,k (Y (ti ))dWk (ti ). 149) k On the other hand, in the Stratonovich interpretation, we take the mean of Y (t) before and after the jump so that Y (τi ) = (Y (ti+1 ) + Y (ti ))/2; namely Yα (ti+1 ) = Yα (ti ) + aα (Y (ti ))dt Y (ti ) + Y (ti+1 ) + bα,k 2 dWk (ti ). 152) are satisﬁed. 139), the coeﬃcients aα (Y ) and bα,k (Y ) can be extended to explicitly time-dependent functions aα (Y, t) and bα,k (Y, t). Such an extension is motivated above all by the fact that possibly a part of the irrelevant variables possesses relatively slow timescales on the order of magnitude of the characteristic time of the relevant quantities.
Therefore, we could possibly use this equation to obtain operator L ˆ Markov . 73) |Y −Z|<ε where we have used the notation ∆Yα = Yα − Zα . 74) for |Y − Z| > ε. We will see later that these quantities were chosen in a very natural way. They can be obtained directly from observations or deﬁned by suitable model assumptions. ˆ Markov by the exclusive use of If we are able to build the Markovian, L these quantities, we have arrived at our goal. Note that possible higher-order coeﬃcients must vanish for ε → 0.