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- W2108357131 abstract "to investigate the existence of local time so the question is: When does Le converge as e → 0+? The condition for L2-convergence in terms of covariation is relatively easy to obtain (see (2),(9)). We consider the convergence in some Sobolev space D2,α defined in the book of Watanabe (10). The interest in this topic arises from the possibility to define the local time as an element of the Sobolev space with negative α, while it does not exist in the usual sense. It is possible to determine the exact smoothness of a local time in the sense of spaces D2,α. Sometimes the local time does not exist as an element of any Sobolev space, but if we modify the approximation by subtracting its mathematical ex- pectation, we obtain the convergence to the so-called renormalized local time. A classical example of renormalization is the renormalization of the self-intersection local time for a two-dimensional Brownian motion. This result can be found, for instance, in (7). We introduce some kind of renormalization, and our definition includes this classical setup. In (5), Imkeller, Perez-Abreu, and Vives found an explicit form of the Ito-Wiener expansion for an approximation of the k-fold self-intersection local time of a Brown- ian motion. They estimated Hermite polynomials in this expansion and obtained the convergence of the approximation in some Sobolev space. Albeverio, Hu and Zhou (8) employed a similar approach, but their result is opposite. They proved that the renor- malized self-intersection local time of a planar Brownian motion is not differentiable in the Watanabe-Meyer sense, i.e. it does not belong to the space D2,1. Another question was considered in (11) where Nualart and Hu deal with the renormalization of the 2-fold self-intersection local time for a fractional Brownian motion. They proved the conver- gence of this renormalization in L2 (which is same as in D2,0) under some conditions." @default.
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- W2108357131 date "2007-01-01" @default.
- W2108357131 modified "2023-10-16" @default.
- W2108357131 title "LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE" @default.
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