By Wolfgang Drechsler
Fiber package deal recommendations in gauge theories (Lecture notes in physics)
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2 The full linear stochastic equation This is the equation dx = −γ xdt + gxdW. 50) To solve it we can do one of two things. 2. 51) where we have been careful to expand the exponential to second order in dW . We now apply this relation repeatedly to x(0) to obtain the solution. 52) x(0). The random variable x(t) is therefore the exponential of a Gaussian random variable. We can use the same method to solve the linear stochastic equation when γ and g are functions of time, and we leave the details of this calculation as an exercise.
The difference equation for x is x(tn ) = x(tn ) t + f (tn ) t. 1) where f (tn ) t is the driving term. Given the value of x at time tn , the value at time tn+1 is then x(tn + t) = x(tn ) + x(tn ) = x(tn ) + x(tn ) t + f (tn ) t. 2) If we know the value of x at t = 0, then x( t) = x(0)(1 + 26 t) + f (0) t. 1 Introduction 27 Now, what we are really interested in is what happens if the driving term, f (tn ) t is random at each time tn ? This means replacing f (tn ) with a random variable, yn , at each time tn .
There are two exceptions to this. One is processes in which the random increment in an inﬁnitesimal time-step dt is not necessarily inﬁnitesimal. The sample paths of such processes make instant and discrete jumps from time to time, and are thus not continuous. These are called jump or point processes, and we consider them in Chapter 8. Jump processes are actually quite common and have many applications. The other exception, which is much rarer in nature, happens when the noise increments remain inﬁnitesimal, as in 32 Stochastic equations with Gaussian noise Gaussian noise, but are drawn from a probability density with an inﬁnite variance (one that avoids the central limit theorem).