I am interested in the distribution of the stochastic process on the unit circle \begin{align*} Y_t=\exp\Big(i\mu\int_0^t e^{X_s}\text{d} s\Big) \quad , \end{align*} and more specifically, in its expectation in the case of the Ornstein-Uhlenbek process \begin{align*} \text{d} X_t=-rX_t \text{d} t+\sigma \text{d}W_t \quad , \end{align*} where $W_t$ is the Wiener process (or Brownian motion).
With the initial condition $X_0=0$, the Feynman-Kac formula suggests to solve \begin{align*} \mathbb{E}[Y_t|X_0=0]&=\int_{-\infty}^\infty w(x,t)\text{d}x\quad ,\\ \partial_t w &=\tfrac{1}{2}\partial_x^2 w+i\mu e^{x}w \\ w(x,0) &= \delta(x)\quad . \end{align*} Is it the right approach? Is there a closed-form solution?