On a family of complex-valued stochastic processes

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 $-\frac12\frac{d^2}{dx^2}$. For $\lambda=0$ the process $r_\lambda$ is real-valued and coincides with the Brownian local time process.