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DEPARTMENT of MathemATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY 1. Conceive a particle free to wander along the x-axis. (3) of the direction in which it wanders. Suppose the It may be shown that under these circumstances, the probability that after a time, t, it has wandered from the origin to a position lying between x=x。 and x=x1 is where t is the time and c is a certain constant which we can reduce to 1 by a proper choice of units. This choice we shall make in what follows. The exponential form of this integral needs no comment, while the mode in which t enters results from the fact that This identity will be presupposed in all that follows. If Let us now consider a particle wandering from the origin for a given period of time, say from t=0 to t=1. Its position will then be a function of the time, say x=f(t). There are certain quantities—functionals— which may depend on the whole set of values of ƒ from t=0 to t=1. we take a large number of particles (i.e. a large number of values of f) at random, it is natural to suppose that the average value of the functional will often approach a limit, which we may call the average value of the functional over its entire range. What will this average be, and how shall we find it? If F {f} is a functional depending on the values of ƒ for only a finite num ber of values of the argument of f-if F {f} is a function3 of ƒ(t1), ƒ(t1⁄2),..., f(t) of the form [ f(t), ..., f(t)]-then it is easy enough to give a natural definition of the average of F, which we shall write A {F}. We can reasonably say where P is a polynomial, and can be evaluated by means of the well known formulae: We can thus easily evaluate A{F} as a polynomial in tɩ, t2, ..., t„, which we shall call Pm, (t1, ..., t). It is easy to show that if Σm is odd, Pm,,..., m, (t1, ..., t2) = 0. To return to the more general functional: there is a large class of so-called analytic functionals, which may be expanded in the form of series such as 4 F{f} =ao+S* ƒ(x)41(x)dx + S. S'ƒ(x) f(v)4x(x,y)dxdy + ... and an even wider class of what may be called Stieltjes analyt ic functionals, in which the general term S'... S'of(x) is replaced by the Stieltjes integral5 In what follows, we shall confine our discussion to Stieltjes analytic functionals, which we shall call simply analytic. The problem with which we are now concerned is the definition of the average of an analytic functional. Now, the first property which any average ought to fulfil is that the average of the sum of two functionals should equal the sum of their averages. We should expect, therefore, that: (a) Over a wide range of cases, the average of a series should equal the series of the averages of the terms; (b) The average of a Stieltjes integral, single or multiple, of a given functional with respect to such parameters as it may contain, should be equal to the integral of the average; (c) A constant multiplied by the average of a functional should equal the average of a constant times the functional. In accordance with this, we get the following natural definition of the average of the analytic functional F. 1 A{F} = A{ao+S' f(x)dy; (x) +SS, ƒ(x)}ƒ(1)}d¥2(x,y) +........} 0 = ao + A{ S', f(x)d¥(x) } + A{S'S' f(x) f(y)d42(x,y)} + ... 0 0 0 = ao + +S* A{ƒ(x)}d4(x) +S*S*A{ƒ(x)ƒ(v)}d¥2(x,y) + .. 0 0 0 We have already seen how to determine A {f(x1)... f(x)} as a polynomial in the x's. Hence whenever the above series converges, we have now a way of obtaining a perfectly definite value for A{F}. It may be noted that every term in the above expression in which the sign of integration is repeated an odd number of times is identically zero. If A {F} is to behave as we should expect it to behave, there are certain properties which it must fulfil, at least over a large and important class of cases. Among these are the following: (4) If F, is a functional depending on the parameter x, and u(x) is a function of limited total variation, then S* A{F,}du = A{S*F,du} (5) Suppose F., (x1,...,x) be defined as Ffx1,...,xn (t) l1,..., tn for t<t<t-1. Then n 2 n 1 -1) ... Σ 2 (k-1) dx1... dxn, where the limit is taken as the t's increase in number in such a manner to divide the interval from 0 to 1 more and more finely. 2. The next task before us is to investigate hypotheses which are sufficient to guarantee the validity of propositions (1)–(5). Propositions (1) and (2) require indeed very little discussion, for they are always satisfied when the series for F1, F2, A{F1} and A{F2} converge. In (3), let F1,..., F,..., and the series ZF, all possess averages, and let where A{R} vanishes as m increases without limit. Then and (3) is proved. If ZF, converges and lim A{R}=0, we shall say ZF converges smoothly. Proposition (4) reduces to the ordinary inversion of a multiple Stieltjes integral when F. {f} is of the form |