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For Komatu-Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of \cite{S} to find the SDEs satisfied by the induced motion $\xi(t)$ on $\partial\HH$ and the slit motion $\s(t)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogenous functions. Next with solutions of such SDEs, we study the corresponding stochastic Komatu-Loewner evolution, denoted as ${\rm SKLE}_{\alpha,b}$. We introduce a function $b_{\rm BMD}$ measuring the discrepancy of a standard slit domain from $\HH$ relative to BMD. We show that ${\rm SKLE}_{\sqrt{6},-b_{\rm BMD}}$ enjoys a locality property.

Stochastic Komatu-Loewner evolution, Brownian motion with darning, Komatu-Loewner equation for slits, SDE with homogeneous coefficients, generalized Komatu-Loewner equation for image hulls, BMD domain constant, locality property