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\(p\)-adic Brownian motion over \(\mathbb Q_p\). (English. Russian original) Zbl 1185.60056

Proc. Steklov Inst. Math. 265, 115-130 (2009); translation from Tr. Mat. Inst. Steklova 265, 125-141 (2009).
The author generalizes a result of A. Kh. Bikulov and I. V. Volovich [Izv. Math. 61, No. 3, 537–552 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 3, 75–90 (1997; Zbl 0896.60044)] and constructs a \(p\)-adic Brownian motion over the \(p\)-adic numbers \(\mathbb Q_p\). For this, he first constructs a \(p\)-adic white noise over \(\mathbb Q_p\) by using a specific complete orthonormal system of \(\mathbb L^2 (\mathbb Q_p )\). A \(p\)-adic Brownian motion over \(\mathbb Q_p\) is then constructed by the Paley-Wiener method. Moreover, random walks over \(\mathbb Q_p\) are introduced, and it is shown that a \(p\)-adic Brownian motion can be approximated such such random walks.

MSC:

60G99 Stochastic processes

Citations:

Zbl 0896.60044
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Full Text: DOI

References:

[1] A. Kh. Bikulov, ”Stochastic p-adic Equations of Mathematical Physics,” Teor. Mat. Fiz. 119(2), 249–263 (1999) [Theor. Math. Phys. 119, 594–604 (1999)]. · Zbl 0951.60073 · doi:10.4213/tmf737
[2] A. Kh. Bikulov and I. V. Volovich, ”p-Adic Brownian Motion,” Izv. Ross. Akad. Nauk, Ser. Mat. 61(3), 75–90 (1997) [Izv. Math. 61, 537–552 (1997)]. · Zbl 0896.60044 · doi:10.4213/im126
[3] T. Hida, Brownian Motion (Springer, Berlin, 1980). · Zbl 0423.60063
[4] A. Yu. Khrennikov and S. V. Kozyrev, ”Ultrametric Random Field,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 199–213 (2006). · Zbl 1154.60323 · doi:10.1142/S0219025706002317
[5] K. Kamizono, ”Symmetric Stochastic Integrals with Respect to p-adic Brownian Motion,” Stochastics 79, 523–538 (2007). · Zbl 1129.60048 · doi:10.1080/17442500701433756
[6] V. S. Vladimirov, ”Generalized Functions over the Field of p-adic Numbers,” Usp. Mat. Nauk 43(5), 17–53 (1988) [Russ. Math. Surv. 43 (5), 19–64 (1988)].
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