Example: Markov Property of the Modulus of a Brownian Motion with Drift

Modulus of Standard $d$-dimensional Brownian Motion

Let $(B_t)_{t \geq 0}$ be a standard $d$-dimensional Brownian motion. It is a well known fact that the one-dimensional process $t \mapsto |B_t|$ is a Bessel process of dimension $d$. In particular this means that it is a Markov process. Most importantly, $t \mapsto |B_t|$ is Markov for any choice of the starting distribution of $B_0$.

This property is destroyed by adding a drift to the original process. More precisely, if $\mu \in \mathbb{R}^d$ then the process $t \mapsto |B_t + \mu t|$ is Markov only for certain choices of the distribution of $B_0$. The Pitman-Rogers theorem gives a nice way of finding such distributions.

Pitman-Rogers for the Modulus of $d$-dimensional Brownian Motion with Drift

In this example the state space of the original process is $S = \mathbb{R}^d$ and the state space of the projected process is $S' = [0, \infty)$. The projection mapping we are studying is $$ \varphi(x) = |x|, $$ where $|x|$ denotes the Euclidean norm (modulus). We need to come up with a candidate for the Markov kernel $\mathbf{\Lambda}$. For any $r \geq 0$ the measure $\mathbf{\Lambda}(r, \cdot)$ should be supported on the $d$-dimensional sphere of radius $r$: $$ \mathcal{S}_r = { \mathbf{x} \in \mathbb{R}^d : |x| = r }. $$ Intuitively one expects that $\mathbf{\Lambda}(r, \cdot)$ reflects the conditional distribution of the $d$-dimensional Brownian motion with drift given that is norm is currently $r$. This distribution should reflect the fact that the $d$-dimensional process tends to move in the direction $\mu$, and so the measure $\mathbf{\Lambda}(r, \cdot)$ should be biased in the direction $\mu$.

Here is the answer: choose $\mathbf{\Lambda}(r, \cdot)$ to be the uniform measure on the sphere of radius $r$, biased by the factor $e^{\mu \cdot \mathbf{x}}$. Symbolically we can write this as \begin{equation}\label{eq:VM_dist} \mathbf{\Lambda}(r, d \mathbf{x}) = \frac{1}{Z_{r, \mu}} e^{\mu \cdot \mathbf{x}} \sigma_r(d \mathbf{x}), \end{equation} where $\sigma_r$ is the uniform probability measure on the sphere of radius $r$ and $Z_{r, \mu}$ is the normalizing factor for the biased measure, i.e. $$ Z_{r, \mu} = \int_{\mathcal{S}_r} e^{\mu \cdot \mathbf{x}} \sigma_r(d \mathbf{x}). $$ The distribution \eqref{eq:VM_dist} is a special type of von Mises-Fisher distribution on the sphere (more precisely it is a scaling transformation of the usual form of this distribution). Exact formulas for the normalizing constant $Z_{r, \mu}$ are available, in terms of Bessel functions of the first kind.

Proof of the Intertwining Relation

We want to check that $\mathbf{\Lambda} P_t = Q_t \mathbf{\Lambda}$ for the kernel $\mathbf{\Lambda}$ given in \eqref{eq:VM_dist}. Here $P_t$ is the transition semi-group of the process $t \mapsto B_t + \mu t$ taking values in $\mathbb{R}^d$. For $B_0 = \mathbf{x} \in \mathbb{R}^d$ the distribution of $B_t + \mu t$ is that of the $d$-dimensional Gaussian with mean $\mathbf{x} + \mu t$ and covariance matrix $t I$, hence it follows that $$ (P_t f)(\mathbf{x}) = \int_{\mathbb{R}^d} f(\mathbf{y}) \varrho_d(t, \mathbf{y} - \mathbf{x} - \mu t) , d \mathbf{y}, \quad f : \mathbb{R}^d \to \mathbb{R}, $$ where $\varrho_d(t, \mathbf{z})$ is the heat kernel on $\mathbb{R}^d$ given by $$ \varrho_d(t, \mathbf{z}) = (2 \pi t)^{-d/2} \exp { -|z|^2/2t }. $$ Now we can use our guess for $\mathbf{\Lambda}$ to construct a formula for $Q_t$, and then check if this formula satisfies the intertwining relation. We expect that for $g : [0, \infty) \to \mathbb{R}$ there is a kernel $q$ such that $$ (Q_t g)(r) = \int_0^{\infty} g(\ell) q(t, \ell, r) d \ell. $$ Of course $q$ is the density of $|B_t + \mu t|$ when $|B_0| = r$. To guess a formula for it $|B_t + \mu t|$ when $B_0 \sim \mathbf{\Lambda}(r, \cdot)$. We can compute the law of $B_t + \mu t$ when $B_0 \sim \mathbf{\Lambda}(r, \cdot)$, we have $$ \mathbb{P}_{\mathbf{\Lambda}(r, \cdot)}(B_t + \mu t \in d \mathbf{y}) = \int_{\mathcal{S}_r} \varrho_d(t, \mathbf{y} - \mathbf{x} - \mu t) , \mathbf{\Lambda}(r, d \mathbf{x}) $$ From this we can compute the density of $|