  We extend the Heston stochastic volatility model to a Hilbert space
framework. The tensor Heston stochastic variance process is defined as a tensor
product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. The
volatility process is then defined by a Cholesky decomposition of the variance
process. We define a Hilbert-valued Ornstein-Uhlenbeck process with Wiener
noise perturbed by this stochastic volatility, and compute the characteristic
functional and covariance operator of this process. This process is then
applied to the modelling of forward curves in energy markets. Finally, we
compute the dynamics of the tensor Heston volatility model when the generator
is bounded, and study its projection down to the real line for comparison with
the classical Heston dynamics.
