Transformations of Markov processes

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.

We consider the space-time process. Let $V_{t}=V_{0}+t$. The process $V_{t}$ is simply the process that increases deterministically at unit speed. Thus $V_{t}$ can represent time. If $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process, show that $\left(\left(X_{t}, V_{t}\right), \mathbb {P}[(x, v)]\right)$ is also a Markov process. Is $\left(\left(X_{t}, V_{t}\right), \mathbb {P}[(x, v)]\right)$ necessarily a strong Markov process if $\left(X_{t}, \mathbb {P}[x]\right)$ is a strong Markov process?

For some applications, one lets $V_{t}=V_{0}-t$, and one thinks of time running backwards. Space-time processes are useful when considering parabolic partial differential equations.

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a strong Markov process and $f$ is a non-negative invariant function for $\left(X_{t}, \mathbb {P}[x]\right)$. Write $\mathbb {Q}^{x}$ for $\mathbb {P}_{f}^{x}$. Suppose $g$ is a non-negative invariant function for $\left(X_{t}, \mathbb {Q}^{x}\right)$. Show that $f g$ is a non-negative invariant function for $\left(X_{t}, \mathbb {P}[x]\right)$ and that $\mathbb {Q}_{g}^{x}=\mathbb {P}_{f g}^{x}$.

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

Here is an argument that the law of a Brownian motion conditioned to have a maximum at a certain level is a Bessel process of order 3.

Let $W$ be a one-dimensional Brownian motion killed on hitting 0. Let $S_{t}=\sup_{s \leq t} W_{s}$ be the maximum. By Exercise 19.1, $X=(W, S)$ is a Markov process. Determine the law of $X$ for $t \leq L$, where $L$ is the last time $X$ hits the diagonal. To define $L$ more precisely, let $D=\left\{ (w, s): w=s, w>0\right\}$ and $L=\sup \left\{ t \geq 0: X_{t} \in D\right\}$. $L$ is finite, a.s., because $W$ will hit 0 in finite time with probability one.