= Solution
The <Markov property> and the <Chapman-Kolmogorov equation> give
$$
\mathbb P(X_{2n+2}=j\mid X_{2n}=i,X_{2n-2},\ldots,X_0)
=\sum_{k\in S}p_{ik}p_{kj}=(P^2)_{ij}.
$$
For each intermediate state, one first uses the next-step <transition probability> and then the following one; the sampled history adds no information once $X_{2n}$ is known. Thus $W$ is a homogeneous <Markov chain> with <transition matrix> $P^2$ and initial law $\lambda$. In particular,
$$
\boxed{\mathbb P(W_1=0)=\sum_{i\in S}\lambda_i\sum_{j\in S}p_{ij}p_{j0}.}
$$
If zero is not a state of $S$, this probability is zero.
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