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In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr X)\to(-\infty,\infty] $$ so that

  1. for each $x\in \mathscr X$, $P(x,\cdot)$ is a probability measure on $(\mathscr X,\mathscr B(\mathscr X))$ and

  2. for each $B\in \mathscr B(\mathscr X)$, $P(\cdot,B)$ is a measurable function.

In further analysis one deals with other kernels of the kind $$ K:\mathscr X\times \mathscr B(\mathscr X)\to(-\infty,\infty] $$ e.g. the potential kernel $G = \sum\limits_{n\geq 0}P^n$ where a certain definition of the product of kernels is used.

There is a closely related object, called the harmonic measure: given the Markov process $X$ on the space $(\mathscr X,\mathscr B(\mathscr X))$, the initial point $x$ and the set $A$ the harmonic measure of $B\subset A^c$ is defined by: $$ \alpha(x,A,B) = \mathsf P_x\{X_\tau\in B,\tau<\infty\} $$ where $\tau$ is the first time $X$ leaves the set $A$, i.e. $\tau = \inf\{n\geq 0:X_n\in A^c\}$. E.g. when the process $X$ is a Brownian motion, $\alpha$ is a classical harmonic measure supported on the boundary of the bounded domain $A$.

Since I use these objects in my work, I prefer to use the common name for them. So my question is rather notational: would it be appropriate call $\alpha$ the harmonic kernel? In my case the dependence on $x$ and $A$ is crucial.

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