Question on Markov property for Itô diffusion : does it implies time homogeneity as well? Let $(X_t)$ a diffusion process. I denote $\mathbb P^x$ the measure : $$\mathbb P(C(t_1,...,t_n,A_1,...,A_n)\mid X_0=x),$$
for all cylinders $$C(t_1,...,t_n,A_1,...,A_n)=\{X_{t_1}\in A_1,...,X_{t_n}\in A_n\}.$$
So, I know that $(X_t)$ has Markov property if for all $h&gt;0$, $$\mathbb P\{X_{s+h}\in A\mid \mathcal F_s\}=\mathbb P\{X_{s+h}\in A\mid X_s\},$$
for all Borel set $A$. In my lecture, it's denoted by $$\mathbb P^0(X_t\in A\mid \mathcal F_s)=\mathbb P^{X_s}(X_h\in A),\tag{1}$$
which I interpret as $$\mathbb P^0\{X_{s+h}\in A\mid \mathcal F_s\}(\omega )=\mathbb P\{X_h\in A\mid X_0=X_s(\omega )\}\tag{2}$$
The thing it's that $(1)$ and $(2)$ looks rather to be time homogeneity+Markov property than Markov process only. Indeed,
$$\mathbb P(X_{s+h}\in A\mid \mathcal F_s)(\omega )=\mathbb P(X_{s+h}\in A\mid X_s=X_s(\omega ))=\mathbb P(X_h\in A\mid X_0=X_s(\omega )),$$
where the first equality is Markov property and the second one is time homogeneity.
Q1) Am I right ?
Q2) For Itô diffusion, does Markov property means Markov property+ Time homogeneity ?
This looks like the strong Markov property. Under certain natural conditions, having the Markov property and time homogeneity implies the strong Markov property: see Exercise 13.1 of this; it seems you also gave a proof.
The answer to your question (2) is no if you define Markov property without the time homogeneity assumptions. There's just no reason that all Itô diffusion processes with the Markov property must be time homogeneous.
This looks like the strong Markov property. Under certain natural conditions, having the Markov property and time homogeneity implies the strong Markov property: see Exercise 13.1 of this; it seems you also gave a proof.
The answer to your question (2) is no if you define Markov property without the time homogeneity assumptions. There's just no reason that all Itô diffusion processes with the Markov property must be time homogeneous.