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Recently, functional It\=o calculus has been introduced and developed infinite dimension for functionals of continuous semimartingales. With differenttechniques, we develop a functional It\=o calculus for functionals of Hilbertspacevalued diffusions. In this context, we first prove a path-dependentIt\=o's formula, then we show applications to classical solutions ofpath-dependent Kolmogorov equations in Hilbert spaces and derive a Clark-Oconetype formula. Finally, we explicitly verify that all the theory developed canbe applied to a class of diffusions driven by SDEs with a path-dependent drift(suitably regular) and constant diffusion coefficient.

Functional It\=o calculus in Hilbert spaces and application to path-dependent Kolmogorov equations