Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation \begin{align*} \mathrm{d} x_t & = v_t \mathrm{d}t \\ m\mathrm{d} v_t& = -\nabla \Psi(x_t) \mathrm{d}t -\gamma v_t \mathrm{d}t + \sigma \mathrm{d} W_t \end{align*} where $x_t$ and $v_t$ are the position and velocity vectors, $\Psi(x)$ is a bell-shaped potential with a single minimum at $\Psi(0)$ and $W_t$ is a vector Wiener process. Now define the energy of the particle as \begin{align}\label{eq:energy} h(x_t, v_t) = \Psi(x_t) + \frac{1}{2}mv_t^2. \end{align} I am interested in calculating the time it takes for $\mathcal{P}$ to reach a certain energy $r$, i.e., the stopping time $$ \tau_h = \inf\{ t \geq 0: h(x_t, v_t) = r \}. $$ Calculating $\mathbb{E}(\tau_h)$ would be enough for now. So far I have tried calculating \begin{align}\label{eq:en_evolution} h(x_t, v_t) = h(x_0, v_0) + \int_0^t \mathrm{d} h(x_s, v_s), \end{align} and then, by taking expected values, I hoped I'd be able to solve by $\mathbb{E}(\tau_h)$. With the Wiener process I got the right and well-known result of the exit time from a ball of radius $r$ (this is, in fact, a naive way of using Dynkin's formula), but with the process I described before I didn't get too far. Assuming the initial conditions are distributed according to the invariant (Gibbs) measure, one gets, of course, that $$ \mathbb{E}[h(x_t, v_t)] = \mathbb{E}[\Psi(x_t)] + \frac{1}{2}k_B T, $$ which is true, and reassuring, but not very useful, since we lost $\mathbb{E}(\tau_h)$.

I'm not especially used to calculations like this, so any suggestions of "tricks" and/or common techniques for calculating hitting times would be appreciated. I would also be grateful for any references where similar exit time calculations are performed (for non trivial processes).


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