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Space-time fractional evolution equations are a powerful tool to modeldiffusion displaying space-time heterogeneity. We prove existence, uniquenessand stochastic representation of classical solutions for an extension of Caputoevolution equations featuring nonlocal initial conditions. We discuss theinterpretation of the new stochastic representation. As part of the proof a newresult about inhomogeneous Caputo evolution equations is proven.

Stochastic solutions for space-time fractional evolution equations on bounded domain