Stochastic \(p\)-adic equations of mathematical physics
The author introduces several classes of complex-valued stochastic processes with \(p\)-adic time parameters. The basic process is the \(p\)-adic white noise, a Gaussian generalized stochastic process \(\Phi (\varphi)\), \(\varphi \in \mathcal D(Q_p)\) (the Schwartz-Bruhat space of test functions), with zero mean and the covariation \[ \mathbf E\Phi (\varphi)\overline{\Phi (\psi)}=\int\limits_{Q_p}\varphi (t)\overline{\psi (t)} dt. \] An analogue of the Wiener process is defined as solution of the Cauchy problem for the equation \(D_t\Psi =\frac{p}{\sqrt{2(p+1)}}\Phi\), where \(D\) is a pseudo-differential operator with the symbol \(|\xi |_p\). Properties of this process are studied. Similarly, \(p\)-adic counterparts of the Brownian sheet and the Lévy Brownian motion are connected with appropriate two-dimensional pseudo-differential operators.
scientific article; zbMATH DE number 1465871