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We compute the joint probability law of the principal moments of inertia of Brownian paths (open or closed), using constrained path integrals and random matrix theory. The case of two-dimensional paths is discussed in detail. in particular, we show that the ratio of the average values of the largest and smallest moments is equal to 4.99 (open paths) and 3.07 (closed paths). We also present results of numerical simulations, which include investigation of the relationships between the moments of inertia and the arithmetic area enclosed by a path.
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Fougere et al. (1993) studied this question.
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