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ABSTRACT. We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
Abstract.A dual pair 9 and g* of smooth and generalized random variables, respectively, over the white noise probability space is studied. 9 is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator, g* is its dual. Sufficient' criteria are proved for when a function on S(IR) is the S-transform of an element in 9 or g*. Introd uetionDual pairs of spaces of random variables have been studied and applied to many problems of stochastic and infinite dimensional analysis in numerous papers. We refer the reader to [HKP 93] for the background of the subject and for an extensive list of references. The two most used pairs of spaces seem to be the Hida spaces (S(*)) and the Meyer-Watanabe spaces V(*).In [PS 91] a result was proved for the space (S)* of Hida distributions whichcharacterizes this space in terms of its S-transform, essentially a Gauß-Laplace transform. This result appears to be quite useful, as weIl from a struetural point of view as for applications. For example, this "charaeterization theorem" has been applied to: quantum field theory [PS 93]' Feynman integrals [FPS 91, KS 92, LLS 93], (anticipating) stochastic (Volterra, partial) differential equations [CLP 93, CP 92, KP 90, Po 92, Po 93], law of large numbers and central limit theorem of Donsker's delta functional [PS 93ab]. On the other hand, for applications in stochastic analysis the pair V and V* of Meyer-Watanabe seems to be more fitting: for example, the solutions of non-degenerate Ito equations belong to V [Wa 84] and not to (S). This is due to the fact that elements in (S) have a chaos decomposition with kernels in S(IR n ) while solutions of SDE's have kerneis which fail to be in S(IR n ). Unfortunately, there is no characterization-type theorem known for the pair (V*, V), and such a characterization appears to be a rather difficult problem. Therefore the basic idea of the present paper is to consider aspace of random variables bigger than (S) but smaller than V which would possibly contain solutions of Ho SDE's (in particular having kerneis of the chaos decomposition in L 2 (IR n )) but which at the same time allows for a characterization in terms of the S":'transform. 1In fact, there is a pair denoted by (9, g*) which is "between" the above mentioned pairs, namely we have the embeddings (1.1) Although we could not find an equivalent characterization of g* and 9 in terms of their S-transforrris, we prove in Section 4 a sujJicient condition for functions on S( IR) to be the S-transform of an element in 9 or in g*. On the other hand, 9 is large enough to contain at least some solutions of Ito-equations: indeed, we show in Section 2 that the solution of the SDE defining the stochastic exponential of Brbwnian motion belongs to g. Thus there is some hope that this dual pair is appropriate for the study of stochastic differential equations, and that for this the power ofcharacterization-type results becomes available.The space 9 is ,constructed by L 2 -norms with exponential weights of the OrnsteinUhlenbeck operator. Ithas b...
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