Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(E,\mathcal E)$ be a measurable space
- $(Y_n)_{n\in\mathbb N_0}$ and $(\tilde Y_n)_{n\in\mathbb N_0}$ be time-homogeneous Markov chains on $(\Omega,\mathcal A,\operatorname P)$
- $\kappa$ denote the transition kernel of $Y$
- $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$ to which $\tilde Y$ is adapted
- $\tau$ be an $\mathcal F$-stopping time with $$1_{\left\{\:n\:<\:\tau\:\right\}}Y_n=1_{\left\{\:n\:<\:\tau\:\right\}}\tilde Y_n\;\;\;\text{for all }n\in\mathbb N_0$$ and$^1$$^2$ $$\operatorname P\left[\tau<\infty,\left(\tilde Y_{\tau+n_1},\ldots,\tilde Y_{\tau+n_k}\right)\mid\mathcal F_\tau\right]=1_{\left\{\:\tau\:<\:\infty\:\right\}}\bigotimes_{i=1}^k\kappa^{n_i-n_{i-1}}(Y_\tau,B)\tag1$$ almost surely for all $k\in\mathbb N_0$, $n_0,\ldots,n_k\in\mathbb N_0$ with $0=n_0<\cdots<n_k$ and $B\in\mathcal E^{\otimes k}$
Are we able to conclude that $Y$ and $\tilde Y$ have the same distribution?
It's clear to me that the distribution is uniquely determined by the finite-dimensional distributions. For simplicity and as a first step, I've tried to prove $$\operatorname P\left[Y_n\in B\right]=\operatorname P\left[\tilde Y_n\in B\right]\tag2$$ for some fixed $n\in\mathbb N_0$ and $B\in\mathcal E$. My idea is to write $$\operatorname P\left[\tilde Y\in B\right]=\operatorname P\left[n<\tau,\tilde Y\in B\right]+\operatorname P\left[n\ge\tau,\tilde Y\in B\right].\tag3$$ By $(1)$, it's sufficient to show the second term on the right-hand side of $(3)$ is equal to $\operatorname P\left[n\ge\tau,Y\in B\right]$ in order to conclude $(2)$. Noting that $\left\{n\ge\tau\right\}=\biguplus_{m=0}^n\left\{\tau=m\right\}$, we obtain \begin{equation}\begin{split}\operatorname P\left[n\ge\tau,\tilde Y\in B\right]&=\sum_{m=0}^n\operatorname P\left[\tau=m,\tilde Y_{\tau+(n-m)}\in B\right]\\&=\sum_{m=0}^n\operatorname E\left[1_{\left\{\:\tau\:=\:m\:\right\}}\kappa^{n-m}(Y_\tau,B)\right]\\&=\operatorname E\left[1_{\left\{\:n\:\ge\:\tau\:\right\}}\kappa^{n-\tau}(Y_\tau,B)\right]\end{split}\tag4\end{equation} by $(1)$.
However, I don't know how to conclude $(2)$ from $(4)$. And I don't see how this approach can be generalized in order to show the actual claim.
(By the way, I would be interested in a solution which would still work if the index set $\mathbb N_0$ is replaced by something uncountable like $[0,\infty)$ - but that's not mandatory; if you only know a solution for the countable case that would be fine for me).
$^1$ $\mathcal F_\tau$ denotes the $\sigma$-algebra of $\tau$-past.
$^2$ $\bigotimes$ denotes the product of transition kernels.