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We establish weak existence and uniqueness in law for stochastic Volterraequations (SVEs for short) with completely monotone kernels and non-degeneratenoise under mild regularity assumptions. In particular, our results reveal theregularization-by-noise effect for SVEs with singular kernels, allowing formultiplicative noise with H\"{o}lder diffusion coefficients. In order to proveour results, we reformulate the SVE into an equivalent stochastic evolutionequation (SEE for short) defined on a Gelfand triplet of Hilbert spaces. Weprove weak well-posedness of the SEE using stochastic control arguments, andthen translate it into the original SVE.

Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise