  We introduce a framework that allows to employ (non-negative) measure-valued
processes for energy market modeling, in particular for electricity and gas
futures. Interpreting the process' spatial structure as time to maturity, we
show how the Heath-Jarrow-Morton approach can be translated to this framework,
thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an
analog to the HJM-drift condition and then treat in a Markovian setting
existence of non-negative measure-valued diffusions that satisfy this
condition. To analyze mathematically convenient classes we build on Cuchiero et
al. (2021) and consider measure-valued polynomial and affine diffusions, where
we can precisely specify the diffusion part in terms of continuous functions
satisfying certain admissibility conditions. For calibration purposes these
functions can then be parameterized by neural networks yielding measure-valued
analogs of neural SPDEs. By combining Fourier approaches or the moment formula
with stochastic gradient descent methods, this then allows for tractable
calibration procedures which we also test by way of example on market data. We
also sketch how measure-valued processes can be applied in the context of
renewable energy production modeling.
