Mathematical aspects of quantum field theory - download pdf or read online

By K. O. Friedrichs

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The fact that the variance of x(T ) is proportional to T is a result of the fact that we chose each of the Wiener increments to have variance t. Since there is one Wiener increment in each time step t, the variance of x grows by precisely t in each interval t, and is thus proportional to t. So what would happen if we chose V [ W ( t)] to be some other power of t? To find this out we set V [ W ( t)] = t α and calculate once again the variance of x(T ) (before taking the limit as N → ∞). This gives N −1 V [ Wi ] = N( t)α = N V (x(T )) = i=0 T N α = N (1−α) T α .

83) we obtain N dyi = j =1 + 1 2 ∂yi ∂xj Gj k (x, t)dWk + fj (x, t)dt + k N N k=1 j =1 ∂yi dt ∂t M ∂ 2 yi ∂xk ∂xj Gj m Gkm dt, i = 1, . . , L. 87) n=0 for some function f (t), where t = t/N. In the above summation f (n t) is the value of the function f (t) at the start of the time interval to which the Wiener increment Wn corresponds. As we will see below, this fact becomes important when evaluating multiple stochastic integrals. When solving Ito equations that have multiple variables, or multiple noise processes, the solutions are in general multiple integrals that involve one or more Wiener processes.

74) dt, where C = η 1 − η2 . 75) Since the means of dV1 and dV2 are zero, and because we have pulled out the factor of dt in Eq. 74), C is in fact the correlation coefficient of dV1 and dV2 . The set of Ito calculus relations for dV1 and dV2 are given by essentially the same calculation: (dV1 )2 dV1 dV2 dV1 dV2 (dV2 )2 = dW1 dW2 = 1 C MM T C 1 (dW1 , dW2 ) dt. 78) where we have defined dW = (dW1 , dW2 )T . More generally, we can always write N correlated Gaussian noise processes, dV, in terms of N independent Wiener processes, dW.

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