# U.A. Alekseeva. On the definition of a Brownian sheet ... P.3-11

We study the properties of a Brownian sheet, which is a random field generalizing the Brownian motion. It is demonstrated that different sets of properties can be used to define this random function, just as in the case of the Brownian motion. We formulate four definitions of the Brownian motion and, based on them, four definitions of a Brownian sheet. An interesting property of the Brownian motion, which is important for our discussion, is the fact that a process with continuous trajectories and independent increments starting from zero is Gaussian (J. Doob’s theorem). In the present paper, we generalize this statement to the case of random fields, which allows us to prove the equivalence of the formulated definitions of a Brownian sheet.

Keywords: Brownian sheet, Brownian motion, Wiener process, Gaussian process, Wiener–Chentsov random field.

The paper was received by the Editorial Office on March 19, 2018.

Funding Agency: Ministry of Education and Science of the Russian Federation (Grant Number 02.A03.21.0006).

Uliana Alekseevna Alekseeva, Cand. Sci. (Phys.-Math), Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: uliana.alekseeva@urfu.ru.

REFERENCES

1.   Allen E.J. Modeling with Ito Stochastic Differential Equations. Dordrecht: Springer, 2007, 228 p. doi: 10.1007/978-1-4020-5953-7 .

2.   Bovkun V.A. On models that lead to an infinite-dimensional stochastic Cauchy problem. Teor. Veroyatnost. i Primenen., 2017, vol. 62, no. 4, pp. 803–804 (in Russian). doi: 10.4213/tvp5151 .

3.   Gikhman I.I., Skorokhod A.V. Stochastic differential equations. Berlin: Springer, 1972, 356 p. ISBN: 0-387-05946-6 . Original Russian text published in Gikhman I.I., Skorokhod A.V. Stokhasticheskie differentsial’nye uravneniya. Kiev: Naukova Dumka, 1968, 355 p.

4.   It$\hat{\mathrm{o}}$ K., Mckean H.P. Diffusion processes and their sample paths. Berlin: Springer, 1965, 338 p. ISBN: 3540606297 . Translated to Russian under the title Diffuzionnye protsessy i ikh traektorii. Moscow, Mir Publ., 1968, 395 p.

5.   Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov [Theory of Stochastic Processes]. Moscow, Fizmatlit Publ., 2005, 400 p. ISBN: 5-9221-0335-0 .

6.    Da Prato G., Zabczyk J. Stochastic equations in infinite dimensions. Cambridge: Cambridge Univ. Press, 2014, 380 p. ISBN: 9781107295513 .

7.   Melnikova I.V. Stochastic cauchy problems in infinite dimensions. Regularized and generalized solutions. Boca Raton; London: CRC Press, Taylor & Francis Group, 2016, 300 p. ISBN: 1482210509 .