  This paper is concerned with the evolution dynamics of local times of a
spectrally positive stable process in the spatial direction. The main results
state that conditioned on the finiteness of the first time at which the local
time at zero exceeds a given value, the local times at positive half line are
equal in distribution to the unique solution of a stochastic Volterra equation
driven by a Poisson random measure whose intensity coincides with the L\'evy
measure. This helps us to provide not only a simple proof for the H\"older
regularity, but also a uniform upper bound for all moments of the H\"older
coefficient as well as a maximal inequality for the local times. Moreover,
based on this stochastic Volterra equation, we extend the method of duality to
establish an exponential-affine representation of the Laplace functional in
terms of the unique solution of a nonlinear Volterra integral equation
associated with the Laplace exponent of the stable process.
