For regular Volterra kernels the square-root process obeys a time-dependent Feller condition and stays positive; for rough regularly-varying kernels it hits zero with positive probability and carries an atom at the boundary.
Comparison principles for solutions to the fractional differential inequalities with the general fractional derivatives and their applications.J
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Boundary behaviour of the Volterra square-root process
For regular Volterra kernels the square-root process obeys a time-dependent Feller condition and stays positive; for rough regularly-varying kernels it hits zero with positive probability and carries an atom at the boundary.