Additive functionals of Markov processes

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a Brownian motion, $f$ a non-negative, bounded, Borel measurable function, and $A_{t}=\int_{0}^{t} f\left(X_{s}\right) d s$. Show that $\left(\left(X_{t}, A_{t}\right), \mathbb {P}[x]\right)$ is a Markov process.

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a one-dimensional Brownian motion, $L_{t}^{x}$ the local time of Brownian motion at $x$, and $m$ a positive finite measure on $\mathbb {R}$. Show that $A_{t}=\int L_{t}^{x} m(d x)$ is an additive functional.

Suppose $A$ and $B$ are additive functionals for a Markov process and $A$ and $B$ have continuous paths. Prove that if $\mathbb {E}_{x}\left[A_{t}\right]=\mathbb {E}_{x}\left[B_{t}\right]$ for all $x$ and $t$, then

for all $x$.

Hint: Show $A_{t}-B_{t}$ is a martingale.

Suppose $A$ and $B$ are additive functionals with continuous paths and suppose $\mathbb {E}_{x}\left[A_{\infty }\right]=\mathbb {E}_{x}\left[B_{\infty }\right]<\infty$ for each $x$. Show

for each $x$.

Hint: If $f(x)=\mathbb {E}_{x}\left[A_{\infty }\right]$, then

and similarly with $B$ in place of $A$. Then $A-B$ is a $\mathbb {P}[x]$ martingale for each $x$.

Let $A$ be an additive functional with continuous paths. Suppose there exists $K>0$ such that $\mathbb {E}_{x}\left[A_{\infty }\right] \leq K$ for each $x$. Prove that there exists a constant $c$ depending only on $K$ such that

Let $A_{t}$ be an additive functional (defined in (22.4)) and let $f(x)=\mathbb {E}_{x}\left[A_{\infty }\right]$. Show that $f$ is excessive.

For each $n$, let $A_{t}^{n}$ be an additive functional with continuous paths and suppose that $f_{n}(x)=\mathbb {E}_{x}\left[A_{\infty }^{n}\right]$ is finite for every $x$. Suppose $A_{t}$ is a continuous additive functional with $f(x)=\mathbb {E}_{x}\left[A_{\infty }\right]$ also finite for each $x$. Suppose $f_{n}$ converges to $f$ uniformly. Prove that for each $x$, with $\mathbb {P}[x]$-probability one, $A_{t}^{n}$ converges to $A_{t}$, uniformly over $t \geq 0$.

Hint: Use Proposition 9.11.

Show that in Lemma 19.32.a) the following stronger assertion holds: $\left(L_{t}\right)_{t \geqslant 0}$ is an additive functional, i.e. $L_{t+s} = L_{t}+L_{s} \circ \theta_{t}$ holds for all $s, t \geqslant 0$.