We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$\mathbb{C}^2$$ -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.

中文翻译：

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扩展 Feller 卷积的卷积代数

我们以 $$\mathbb{C}^2$$ 值的可允许同态卷积代数的形式应用最近引入的可允许同态框架来处理二维单向齐次随机核。代数乘积是 Feller 卷积的非交换扩展，用于对此类内核进行充分的算子表示：一对由扩展的 Chapman-Kolmogorov 方程单向交织的齐次转换函数是卷积移情；相关的 Fokker-Planck 方程被重写为隐式柯西方程，用可允许的同态表示。这种隐式进化方程的可解性条件源于对卷积移情生成器的考虑。