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Let $(X_n)$ be a Markov chain on a standard borelian space $\mathbb{X}$. Anystopping time $\tau$ such that $\mathbb{E}_x\tau$ is finite for all$x\in\mathbb{X}$ induces a Markov chain in $\mathbb{X}$. In this article, weshow that there is a bijection between the invariant measures for the originalchain and for the induced one. We then study drift functions and prove a fewrelations that link the Markov operator for the original chain and for theinduced one. The aim is to use this drift function and the induced operator tolink the solution to Poisson's equation $(I\_d-P)g=f$ for the original chainand for the induced one. We also see how drift functions can be used to controlexcursions of the walk and to obtain the law of large numbers, the centrallimit theorem and the law of the iterated logarithm for martingales. We usethis technique to study the random walk on $\mathbb{R}_+$ defined by $X_{n+1} =\max( X_n + Y_{n+1}, 0)$ where $(Y_n)$ is an iid sequence of law $\rho^{\otimes\mathbb{N}}$ for a probability measure $\rho$ having a finite first moment anda negative drift.

Induction of Markov chains, drift functions and application to the LLN, the CLT and the LIL with a random walk on $\mathbb{R}_+$ as an example