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Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a random variable $Z$ and $$\left|\mu-\nu\right|:=\sup_{B\in\mathcal E}\left|\mu(B)-\nu(B)\right|$$ denote the total variation distance of probability measures $\mu,\nu$ defined on a common $\sigma$-algebra $\mathcal E$.

How can we show that $$\left|\mathcal L(X_s)-\mathcal (Y_s)\right|\le\left|\mathcal L\left(\left(X_{s+t}\right)_{t\ge0}\right)-\mathcal L\left(\left(Y_{s+t}\right)_{t\ge0}\right)\right|\tag1$$ for all $s\ge0$?

We know that $$\mathcal L\left(X_{t_0},\ldots,X_{t_n}\right)=\mathcal L\left(X_{t_0}\right)\otimes\bigotimes_{i=1}^n\kappa_{t_i-t_{i-1}}\tag2$$ for all $n\in\mathbb N_0$ and $0\le t_0\le\cdots\le t_n$, where the right-hand side of $(2)$ denote the product of transition kernels. Thus, $$\mathcal L(X_t)=\mathcal L(X_0)\kappa_t\;\;\;\text{for all }t\ge0\tag3,$$ where the right-hand side of $(3)$ denotes the composition of transition kernels. Moreover, $\mathcal L\left(\left(X_{s+t}\right)_{t\ge0}\right)$ is uniquely determined of the finite-dimensional distributions $(2)$. So, it intuitively seems to be reasonable that $(1)$ holds. But how can we prove it?

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  • $\begingroup$ I don't understand your notation. Is the LHS of (1) the total variation distance between the time $s$ distributions and the RHS the distance between the measures on the space of sample paths? $\endgroup$
    – R W
    Jan 19, 2019 at 1:10
  • $\begingroup$ Note that the $\sigma$-algebra $\sigma(X_s)$ of events depending only on the state at time $s$ is included in the $\sigma$-algebra $\sigma((X_{s+t})_{t\geq 0})$ of events depending on the states from time $s$ onward. What you want to show thus follows directly from the definition of total variation distance. $\endgroup$
    – Algernon
    Jan 19, 2019 at 2:00
  • $\begingroup$ @RW Yes, correctly. The left-hand side is the total variation distance on $\mathcal B(\mathbb R)$ and the right-hand side the total variation distance on $\mathcal B(\mathbb R)^{[s,\:\infty)}$. $\endgroup$
    – 0xbadf00d
    Jan 19, 2019 at 9:37

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