Lee, Wha-SuckLe Roux, Christiaan2021-04-192021-02Lee, WS., Le Roux, C. Convolution algebra for extended Feller convolution. Semigroup Forum 102, 184–216 (2021). https://doi.org/10.1007/s00233-020-10145-y.0037-1912 (print)1432-2137 (online)10.1007/s00233-020-10145-yhttp://hdl.handle.net/2263/79492We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of C2-valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.en© 2020, Springer Science Business Media, LLC, part of Springer Nature. The original publication is available at http://link.springer.com/journal/233.Convolution empathyFeller convolutionExtended Chapman–Kolmogorov equationIntertwined homogeneous Markov processesImplicit Fokker–Planck equationsAdmissible homomorphismsConvolution algebraTwo-dimensional uni-directional stochastic kernelPostprint Article