(Disclaimer: I've been working with SDEs for some years now but have not worked with general Markov processes before... so I'm trying to reconcile some ideas with this post.)
I recently read the definition of a Markov kernel $K$ on a measurable space $(\Omega,\mathcal{F})$ as being a map $K:\Omega,\times \mathcal{F}\rightarrow [0,1]$ satisfying:
- $\omega \mapsto K(\omega,A)$ is measurable, for any $A \in \mathcal{F}$,
- $K(\omega,\cdot)$ is a probability measure on $(\Omega,\mathcal{F})$ for any $\omega \in \Omega$.
Suppose that $(X_t)_{0\leq t\leq 1}$ is an $\mathbb{R}$-valued solution to the SDE: $$ X_t = \mu(t,X_t)dt + \sigma(t,X_t)dW_t, $$ for some Brownian motion $(W_t)_{0\leq t\leq 1}$ and some smooth Lipschitz functions $\mu$ and $\sigma$. I know that $(X_t)_{0\leq t\leq 1}$ has the Markov property, but how does this to relate this to a Markov kernel?
In other words... what is the Markov kernel determining $(X_t)_{0\leq t\leq 1}$.
(Excuse me if the question is silly, but I'm self-teaching Markov processes...)