SOME LINEAR EQUIVALENT TRANSFORMATIONS OF GAUSIAN RANDOM FIELDS IN n-DIMENSIONAL EUCLIDEAN SPACE RnPART I: CONSTRUCTION OF ONE-TO-ONE LINEAR TRANSFORMATIONS FOR SOLVING DIRICHLET-NEYMANN BOUNDARY PROBLEM EXCITED BY GAUSSIAN RANDOM FIELDS IN THE EUCLIDEAN SPACE Rn
Let , , p - a fixed probabilistic space. In a measurable Euclidean space , n n R R , where Rn denotes the algebra of measurable subsets of the Euclidean space Rn , two boundary problems of Dirichle Neuman are considered. It is assumed that Q - some open connected area from the n -dimensional Euclidean space Rn , Q Rn with a homogeneous and isotropic medium. These Dirichlet Neuman problems are considered in the region Q for elliptic differential equations of the same order of 2k and excited by the same Gaussian field x, , x Rn with a nuclear correlation operator R , and the task of existence and uniqueness is raised with a probability of equal to the unit of their solution u x, and v x, , accordingly. For this, another Dirichlet Neuman problem is considered for the main differential order of 2k, for which the Green function 0G (x, y) is built, (see [3]), in space Rn , which in space 2L (Q) - in the space of real functions integrated with its square as Lebesba As the kernel, the integral operator of Hilbert Schmidt 0 G will generate (see [2,3]). Application, further, this operator 0 G to the differential equations considered in two boundary value problems and, given some properties of the Green function (see [3]), we reduce these differential equations of boundary value problems with two linear transformations L1(u(x, )) x, and 2L (v(x, )) x, with the same source of perturbation x, , where converted fields u(x, ) and v x, , if L1 , L2 mutually unequivocal transformations are the only solution of the regional-neimane boundary problems under consideration, and since these transformations are linear, it is obvious that their transformed quantities u(x, ) and v x, are also gaussian fields, so the original field x, , is Gaussian . Further, as is known from the theory of random functions and random processes, random fields u x, and v x, generate probabilistic measures u and v to algebra Rn and, accordingly, in this case, Gaussian measures. The second part of this work is fully devoted to the establishment of conditions that ensure absolute continuity u v = and equivalence measures u and v ~ u v and strict mathematical methods of evidence of some of the statements discussed in the first part of this work and will also be calculated by the corresponding density of radon-nicodem.