Lee, Wha-SuckLe Roux, Christiaan2020-12-102020-12-102020Lee, W.-S. & Le Roux, C. 2020. Convolution algebra for extended Feller convolution. Semigroup forum, (In press). [https://doi.org/10.1007/s00233-020-10145-y]0037-19121432-2137 (Online)http://hdl.handle.net/10394/36516https://link.springer.com/article/10.1007/s00233-020-10145-yhttps://doi.org/10.1007/s00233-020-10145-yWe 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 empathyenConvolution empathyFeller convolutionExtended Chapman-Kolmogorov equationIntertwined homogeneous Markov processesImplicit Fokker-Planck equationsAdmissible homomorphismsConvolution algebraTwo-dimensional uni-directional stochastic kernelArticle