The typical example of path dependant derivatives are knock-ins and knock-outs. At the same time vanilla American options can also be considered to be highly path dependant.
Does a more or less formal defintion/classification of "path dependency" exist ?
For practitioners, a derivative is not path-dependent if its value can be expressed as an expectation of discounted future values at some specific tenor $T$
$$ V(0) = E\left[ \left. V(T) \exp{\left(-\int_0^T r(s)ds\right)} \right| {\cal{I}_0} \right] $$
Obviously this is convenient when it happens because one only needs to worry about probability densities inside the expectation for the single tenor $T$.
American-exercise options fail to meet this criterion since their value depends on an exercise strategy, written here as a stopping time $\tau$
$$ A(0) = \sup_{\tau \leq T} E\left[ \left. V(\tau) \exp{\left(-\int_0^\tau r(s)ds\right)} \right| {\cal{I}_0} \right] $$
Other options fail to meet the criterion because their value depends on fixings $t<T$, or barrier conditions, etc. etc. In many such cases, the path dependence is weak, in the sense that one can introduce a single extra dimension to the SDE/PDE, for example a current tabulation of running average price, and solve accordingly.
A stochastic process that is non-ergodic is inherently "path dependent." You can think of this a number of ways, but for me, most intuitively in the MC context.
A Markov chain is non-ergodic if it is not positive recurrent or if it is periodic. Alternatively, as the transitions in the chain increase, a non-ergodic chain's probability measure will not be greater than zero AND independent of its initial probability distribution (more on this in the famed Feller II). That is, an ergodic MC can eventually reach any other state with some positive probability.
Another useful source might be Omri Sarig's notes on ergodicity, depending on level of formality you are looking for.
