Far Eastern Mathematical Journal

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Some remarks on integral parameters of Wiener process

Vladimirov A.A.

2015, issue 2, . 156-165

It is shown that if generalized function $\rho\in W_2^{-1}[0,1]$ is a multiplier of trace-class from space $W_2^1[0,1]$ to space $W_2^{-1}[0,1]$ then the distribution of stochastic variable $\int_0^1\rho\xi^2\,dt$ (where $\xi$ is a Wiener process) is determined by the spectrum of the boundary problem $$-y''=\lambda\rho y,\qquad y(0)=y'(1)=0,$$ as in the case when $\rho$ is a measure. An example of generalized function $\rho\in W_2^{-1}[0,1]$ that is not a multiplier of trace-class from $W_2^1[0,1]$ to $W_2^{-1}[0,1]$ is also given.

generalized function, multiplier, Wiener process, operator of trace-class

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[1] I.I. Gihman, A.V. Skorohod, Vvedenie v teoriju sluchajnyh processov, Nauka, M., 1977.
[2] I.C. Gohberg, M.G. Krejn, Vvedenie v teoriju linejnyh nesamosoprjazhjonnyh operatorov v gil'bertovom prostranstve, Nauka, M., 1965.
[3] M.I. Nejman-zade, A.A. Shkalikov, "Operatory Shrjodingera s singuljarnymi potencialami iz prostranstv mul'tiplikatorov", Matem. zametki, 66:5 (1999), 723-733.
[4] A.A. Vladimirov, I.A. Shejpak, "Samopodobnye funkcii v prostranstve L2[0,1] i zadacha Shturma-Liuvillja s singuljarnym indefinitnym vesom", Matem. sb., 197:11 (2006), 13-30.
[5]. A.A. Vladimirov, I.A. Shejpak, "Indefinitnaja zadacha Shturma-Liuvillja dlja nekotoryh klassov samopodobnyh vesov", Trudy MIAN im. V.A. Stekloea, 255 (2006), 88-98.
[6] M.A. Lifshits, "On the lower tail probabilities of some random series", Ann. Prob., 25:1 (1997), 424-442.
[7] A.I. Nazarov, "Logarifmicheskaja asimptotika malyh uklonenij dlja nekotoryh gaussovskih processov v L2-norme otnositel'no samopodobnoj mery", Zapiski nauch. semin. POMI, 311 (2004), 190-213.

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