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We investigate the probabilistic and analytic properties of Volterraprocesses constructed as pathwise integrals of deterministic kernels withrespect to the H\"older continuous trajectories of Hilbert-valued Gaussianprocesses. To this end, we extend the Volterra sewing lemma from\cite{HarangTindel} to the two dimensional case, in order to construct twodimensional operator-valued Volterra integrals of Young type. We prove that thecovariance operator associated to infinite dimensional Volterra processes canbe represented by such a two dimensional integral, which extends the currentnotion of representation for such covariance operators. We then discuss aseries of applications of these results, including the construction of a roughpath associated to a Volterra process driven by Gaussian noise with possiblyirregular covariance structures, as well as a description of the irregularcovariance structure arising from Gaussian processes time-shifted alongirregular trajectories. Furthermore, we consider an infinite dimensionalfractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can beseen as an extension of the volatility model proposed by Rosenbaum et al. in\cite{ElEuchRosenbaum}.

Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications