Resolvent stochastic processes

We introduce a family $r_\lambda, \lambda\in\mathbb{C}$ of complex-valued stochastic processes giving a possibility to construct a probabilistic representation for the resolvent of the operator $-\frac{1}{2}\frac{d^2}{dx^2}$. For $\lambda=0$ the process $r_\lambda$ is real-valued and coincides with the Brownian local time process.