Notation in expectation

Question

I know this question has been asked several times before, anyway I don't find the answer to my problem - on some lecture notes I read the following statement:

Let$ P_t $ be a Markov semigroup acting on the space of bounded measurable functions. Then $$ (P_t f)(x) = \mathbb{E}_x[f(X_t)] $$ For each bounded measurable function $f$. I don't understand how to interpret the right and side. Any guess?

Answer 1

Average over $x$. In other words, if $x$ has marginal CDF $P$, the right-hand side is $\int f(X_t) dP(x)$.

Answer 2

This should really have been defined in your lecture notes (if you can't find it, try harder), but when I took Markov processes, our notational convention was that

$$ \mathbb E_x[f(X_t)] = \mathbb E[f(X_t) \vert X_0 = x]. $$

The comment you left on J.G.'s answer seems consistent with this interpretation.

Answer 3

The $t$ index just means that the stochastic process is dependent on time; so think of the process at a fixed time.

Then

$$\mathbb{E}_x[f(X)] = \int f(x)p(x)dx \approx \frac{1}{n} \sum_{i}^{n} f(x_i) $$

where $x_i$ is sampled from $p(x)$ which is the distribution of the random variable $X$.

Also, the initial condition of the process $X$ can be a fixed state $X(0)=x$ or a distribution i.e.: $X(0)=N(\mu, \sigma)$; hence, the $F_0$ notation.