By Fujitsuka M., Ito O.

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21). Hence in the limit of small f. 21) coincide (this is a slightly stronger statement than that their generators coincide). In order to obtain a relation between the Trotter formula and stochastic equations we can directly represent k on functions of the form 'l/J = exp(i/IiW) 4>. 8) (neglecting the o(lx - Y12) terms). 15) with the initial condition q(O) = x. 9) and its exact form in the continuum limit in Chapter 7. CHAPTER 5 The Feynman integral The Feynman integral [126] gives an intuitive picture of quantum mechanics as a classical mechanics with some quantum corrections describing interference of various classically inadmissible amplitudes.

Let n-l b(n) = L: r(k) k=l and Q(n) = q(n) - AO'fb(n) . 42) then reads (Q(n + 1) - Q(n)) exp (-Q(n)) = _,,(€2 exp (AcHb(n)) . s. 44) in the form (Q(n + 1) - Q(n)) exp (-Q(n)) exp (-Q(n)) - exp (-Q(n + 1)) - exp (-Q(n)) X (1- exp (Q(n) - Q(n + 1)) - Q(n + 1) + Q(n)) . s. 45) is of the order €4. We can treat this term in a systematic perturbation expansion. So we solve first the equation exp (-Q(n + 1)) - exp (-Q(n)) = €2"(exp (AO"€b(n)) . 44). 46) we can compute Q(n + 1)(1) - Q(n)(l) in order to see that this difference is of the order €2.

The explosion takes place at a random explosion time r(b). If there is an explosion then an analytic computation of correlation functions of exploding processes is rather complicated. In such a case we shall apply another method (we construct so called weak solution). Namely, we determine a measure corresponding to the process (related to the Feynman integral of subsequent sections) rather than studying solutions of the stochastic equations directly. In Chapter 20 we shall briefly discuss numerical methods for stochastic systems.