On a Family of Complex-Valued Stochastic Processes

Academician of the RAS I. A. Ibragimov^a^,^b^,*, N. V. Smorodina^a^,^b^,**, and M. M. Faddeev^b^,***

^aSt. Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, 191023 Russia

^bSt. Petersburg State University, St. Petersburg, 199034 Russia

Correspondence to: *e-mail: ibr32@pdmi.ras.ru
Correspondence to: **e-mail: smorodina@pdmi.ras.ru
Correspondence to: ***e-mail: m.faddeev@spbu.ru

Received 14 August, 2021

Abstract—We introduce a family ${{r}_{\lambda }},\lambda \in \mathbb{C}$ of complex-valued stochastic processes making it possible to construct a probabilistic representation for the resolvent of the operator $ - \frac{1}{2}\frac{{{{d}^{2}}}}{{d{{x}^{2}}}}$. For $\lambda = 0$ the process ${{r}_{\lambda }}$ is real-valued and coincides with the Brownian local time process.

Keywords: random processes, local time

DOI: 10.1134/S1064562421060077