I am learning about Stochastic processes and an exercise that I am trying to complete asks me to find the Fokker-Planck equation for the position of a microswimmer in 2D that follows the following Langevin equations:
$$\mathbf{v}(t)=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=v\mathbf{n}(t)+\mathbf{u}(t),\quad \frac{\mathrm{d}\theta}{\mathrm{d}t}=\omega +\beta(t)$$
Where $\mathbf{n}(t) = (\cos(\theta),\sin(\theta))$ and $\mathbf{u}(t)$ and $\beta(t)$ are Gaussian white noise that have correlations:
$$\langle \beta(t)\rangle = 0,\quad \langle \beta(t)\beta(t^{\prime})\rangle = 2D_{r}\delta(t-t^{\prime}) \\ \langle u_{i}(t)\rangle = 0,\quad \langle u_{i}(t)u_{j}(t^{\prime})\rangle = 2D\delta_{ij}\delta(t-t^{\prime})$$
I have found the moments:
$$\langle \mathbf{v}(t)-\mathbf{v}(0)\rangle = 4D\delta(t) + v^{2}\cos(\omega t)\exp\left(-D_{r}t\right) \\ \langle [\mathbf{r}(t)-\mathbf{r}(0)]^{2}\rangle = 4Dt + \frac{2v^{2}D_{r}t}{D_{r}^{2}+\omega^{2}}+\frac{2v^{2}(\omega^{2} - D_{r}^{2})}{(D_{r}^{2} + \omega^{2})^{2}}+\frac{2v^{2}e^{-D_{r}t}}{(D_{r}^{2} + \omega^{2})^{2}}\left[(D_{r}^{2}-\omega^{2})\cos(\omega t) - 2\omega D_{r}\sin(\omega t)\right]$$
To derive the Fokker-Planck equation, I then use that:
$$\mathcal{P}(\mathbf{x},t) = \langle\delta^{(2)}(\mathbf{x}-\mathbf{r}(t))\rangle$$
Taylor expanding $\delta^{(2)}(\mathbf{x}-\mathbf{r}(t+\Delta t))$ gives:
$$\delta^{(2)}(\mathbf{x}-\mathbf{r}(t+\Delta t))=\delta^{(2)}(\mathbf{x}-\mathbf{r}(t))-\Delta r_{i}\partial_{i}\delta^{(2)}(\mathbf{x} - \mathbf{r}(t)) + \Delta r_{i}\Delta r_{j}\partial_{i}\partial_{j}\delta^{(2)}(\mathbf{x}-\mathbf{r}(t))$$
So we have:
$$\mathcal{P}(\mathbf{x},t+\Delta t) - \mathcal{P}(\mathbf{x},t) = -\Delta r_{i}\partial_{i}\delta^{(2)}(\mathbf{x} - \mathbf{r}(t)) + \Delta r_{i}\Delta r_{j}\partial_{i}\partial_{j}\delta^{(2)}(\mathbf{x}-\mathbf{r}(t))$$
However, I'm not entirely sure how to proceed from here?