Motivated by a finite-dimensional path-dependent stochastic differential equation \[ dX=b_t(X_t)\,dt+\sigma\,dW,\qquad X_{t_0}=\gamma_{t_0} \] driven by an \(\mathbb R^d\)-valued Wiener process where \(\sigma\) is a diagonalizable \((d\times d)\)-matrix and the \(b_t\) map functions to vectors in \(\mathbb R^d\), the authors consider an infinite-dimensional stochastic differential equation \[ dY=[AY+B(t,Y(t))]\,dt+\Sigma\,d\beta,\qquad Y(s)=y \] in various Banach spaces, typically in a product of \(\mathbb R^d\) and a function space such as continuous, càdlàg or \(L^p\)-integrable functions. Here, \(A\) is an (unbounded) linear operator (typically, the derivative), \(B\) corresponds to the maps \(b_t\), \(\Sigma\) is a diffusion operator corresponding to \(\sigma\) and \(\beta\) is a finite-dimensional Brownian motion corresponding to \(W\).
The authors first prove the existence and uniqueness of mild solutions \(Y^{s,y}\) to the infinite-dimensional stochastic differential equation. The solutions \(Y^{s,y}\) are shown to be \(C^{2,\alpha}\)-continuous in \(y\) and continuous in \(s\), they satisfy the Markov property and \(u(t,y)=\mathbb E\,[\Phi(U^{t,y}(T))]\) satisfies the associated backward Kolmogorov equation in the respective Banach space, with the terminal condition \(\Phi\in C^{2,\alpha}_b\) at time \(T\).
Finally, the results are compared with the path-dependent calculus of Dupire, Cont and Fournié.