| Research Abstract |
My aim is to extend the theory of canonical representation of a Gaussian process X(t), t 0, to the case where the parameter of X is multi-dimensional x<not a member of>R^n. In the course of extension difficulties will come from the fact that we cannot use any reasonable concept of order on R^n. Instead of the order structure, we therefore depend on the geometry (R^n,d) proposed by Hilbert as his famous fourth problem. We consider a Gaussian random field B(x) such that the restriction of B to each straight line is a process with independent increments. Such a B is called a Le^^'vy's Brownian motion in the wide sense, which constitutes a nice class of basic random fields including Le^^'vy's original Brownian motion B_0 (x). The variance d(x,y) of B(x)-B(y) then becomes a projective metric treated as Hilbert's fourth problem. We can apply the generalized Crofton formula and see that such a metric is indeed L^1-embeddable via the equation d(x,y)=m(U_x U_y), where H is the set of all half-spaces h in R^n, m is a measure on H and U_x={h<not a member of>H; x<not a member of>h}. We thus arrive at the desired expression B(x)=W(U_x) in terms of a Gaussian random measure W based on (H,m). Now, a general framework of representation of X is proposed. Namely, introducing a kernel F(x,h), we form X(x)=<integ>U_xF(x,h)dW(h), and expect that important Gaussian random fields encountered in various fields can be expressed in this form. We investigate X as well as the generalized Radon transform R associated with the above expression, from several points of view --invariance under transformations like rotations, the non-deterministic property, null spaces of R reflection positivity and so forth. In addition, we can go further from x<not a member of>^1n to l , to get an infinite-dimensional version of our representation and show the deterministic property discovered by Le^^'vy in case X=B_0, the original Brownian mwhen t
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