I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align} \dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\ \dot{y}&(t)=v_0\cos{\beta(t)},\tag{2} \end{align}
where the over dot denotes time derivative, $\mathbf{x}=(x(t),y(t))\in\mathbb{R}^2$ is the position of the particle at time $t$, $v_0$ is a constant and $\beta$ is a compound Poisson process with rate $\lambda$
\begin{equation} \beta(t)=\sum_{i=1}^{N_\lambda(t)}{\beta_i}.\tag{3} \end{equation}
The counting process $N_\lambda$ has probability $P(N_\lambda(t)=n)=e^{-\lambda t}(\lambda t)^n/n!$ and $\beta_i$ are identical and independent random variables with equal probability of being $\pm\beta_0$, with $\beta_0$ constant. This is essentially a jump diffusion process.
I am able to obtain a Fokker-Planck-like equation for the evolution of the probability distribution $P(x,y,\beta,t)$ of finding the particle at position $(x,y)$ with orientation $\beta$ at time $t$
\begin{equation} \dot{P}(\mathbf{x},\beta,t)=-\dot{\mathbf{x}}\nabla P(\mathbf{x},\beta,t)+\frac{\lambda}{2}(P(\mathbf{x},\beta+\beta_0,t)+P(\mathbf{x},\beta-\beta_0,t)-2P(\mathbf{x},\beta,t)). \tag{4} \end{equation}
Expanding $P$ in Fourier modes
\begin{equation} P(x,y,\beta,t)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}{e^{in\beta-\lambda(1-\cos{(n\beta_0)})t}p_n(x,y,t)}.\tag{5} \end{equation}
I am then able to obtain the evolution equation for the marginal distribution $p_0(x,y,t)$ of finding the particle at position $(x,y)$ at time $t$, regardless of its orientation
\begin{equation} \ddot{p}_0+\hat{\lambda}\dot{p}_0-\frac{v_0^2}{2}\Delta p_0=0,\tag{6} \end{equation}
where $\hat{\lambda}=\lambda(1-\cos{\beta_0})$ and $\Delta=\partial^2/\partial x^2+\partial^2/\partial y^2$ is the Laplace operator. I am neglecting $p_{|i|\geq2}$, this step is a little bit dodgy, but this is a good approximation at late times. I can find the solution of this telegrapher equation with initial conditions $p_0(\mathbf{x},0)=\delta(\mathbf{x})$, $\dot{p}_0(\mathbf{x},0)=0$, as an expansion in plane waves
\begin{equation} p_0(\mathbf{x},t)=\int{G_q(t) e^{i\mathbf{q}\cdot\mathbf{x}}\text{d}\mathbf{q}}.\tag{7} \end{equation}
With this expression it is easy to calculate the moments of the displacement vector, in particular, I can show that the mean displacement $\langle\mathbf{x}\rangle$ vanishes and that the mean squared displacement $\langle|\mathbf{x}|^2\rangle$ has ballistic behavior for $\lambda t\ll 1$, $\langle|\mathbf{x}|^2\rangle\simeq v_0^2t^2$ and diffusive behavior for $1\ll\lambda t$, $\langle|\mathbf{x}|^2\rangle\simeq 2v_0^2t/\lambda$.
I would like now to bias the motion by assuming that the rate depends on the orientation of the particle at time $t$, that is $\lambda(t)=\lambda(1-\epsilon\dot{x}(t)/v_0)$, with $0<\epsilon\ll 1$. The problem is that I am not very familiar with stochastic processes, but here is what I have attempted.
1.- I believe Equation (4) is still valid, replacing $\lambda\mapsto\lambda(t)$
2.- Using a similar expansion as in Equation (5) but with $\lambda t\mapsto\int{\lambda(t')\text{d}t'}$, I obtain Equation (6) again, but with $\lambda\mapsto\lambda(t)$
3.- Since $\epsilon\ll 1$, I expand $p_0$ as follows $p_0=p_0^{(0)}+\epsilon p_0^{(1)}+\cdots$
4.- Inserting in Equation (6) and equating terms of the same order in $\epsilon$, I obtain $p_0^{(0)}$ as in Equation (7) and the first correction $p_0^{(1)}$ evolves according to
\begin{equation} \ddot{p}_0^{(1)}+\hat{\lambda}\dot{p}_0^{(1)}=\hat{\lambda}\frac{\dot{x}}{v_0}\dot{p}_0^{(0)}\tag{8} \end{equation}
Assuming $p_0^{(1)}$ is harmonic and has initial conditions $p_0^{(1)}=\dot{p}_0^{(1)}=0$.
5.- I can get a formal solution to this equation
\begin{equation} p_0^{(1)}=\frac{\hat{\lambda}}{v_0}\int_{0}^{t}{e^{-\hat{\lambda}(t-t')}\int_{0}^{t'}{\dot{x}(t'')\frac{\partial p_0^{(0)}}{\partial t''}\,\text{d}t''}\,\text{d}t'}\tag{9} \end{equation}
6.- I would like to use this equation to calculate the first correction to the moment of the displacement, but I don't seem to be able to deal with the fact that Equation (9) contains $\dot{x}$ evaluated at previous times. I have tried transforming Equation (8) into an equation for the evolution of the mean displacement $\langle x\rangle^{(1)}$, also without success.
Am I missing something? It seems a little bit odd that I can express $\dot{x}/v_0=\cos{\beta}$, in which case $p_0$ depends on $\beta$.
