Markov Processes

Let $\left(X_{t}, t \in T\right)$ be a Markov process, $T \subset \mathbb {R}_+$. Suppose that for each $t \in T$ a measurable function $h_{t}: (\mathsf{S}, \mathscr {B}) \to (\mathsf{S}', \mathscr {B}')$ is given. Consider the random process $Y_{t}=\left(h_{t}\left(X_{t}\right), t \in T\right)$.

Let $\left(X_{t}, t \in T\right),\left(Y_{t}, t \in T\right)$ be independent Markov processes. Is it true that the process $\left(X_{t}+Y_{t}, t \in T\right)$ is also Markov?

Is it true that the total number of particles $Y = (Y_{n} = X_{0} + \ldots + X_{n}, n \in \mathbb {Z}_{+})$ of the Galton-Watson branching process $X = \left(X_{n}, n \in \mathbb {Z}_{+}\right)$ is a Markov process

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Is the process $X_{t}=W_{t}^{2}$ Markov?

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process, and let $\tau_{x}=\min \left\{ t : W_{t}=x\right\}$ for $x \geqslant 0$. Prove that the process $\tau =\left(\tau_{x}, x \geqslant 0\right)$ is Markov.

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Compute its transition density.

Let $\left(X_{n}, n \geqslant 0\right)$ be independent random variables with the uniform distribution on the set $\left\{ -1,0,1\right\}$. Consider the process

Prove that $X_{n}$ is a martingale with respect to the filtration

but is not a Markov process with respect to it.