Stochastic version of the Kirchoff circuit law

I assume this question could be written in a non-technical jargon, but I will try to be as simple as possible.

The Kirchoff circuits law assert that the sum of inward and outward currents at a node is exactly 0. This is a view of the classical charge conservation principle.

From a statistical perspective, however, I'd expect this law microscopically refers to the expected values of the sum of inward and outward currents.

Here I am wondering how this law can be stated probabilistically. In mathematical terms, what if we allow the sum of inward and outwards current to be a standard normal random variate with zero mean and unit variance (a noisy term, like a white noise process).

For example if electrical currents are seen as discrete electric charges moving toward and outward nodes, I'd expect the Kirchoff law to become correct only in the very large N limit.

Did someone ever tried to model the temporal evolution of a discrete circuit with dynamics in the regime of stochastic nodes?