How to solve the following inverse-time non-linear vector SDE?

In our recent studies on the diffusion-based generative models, we need to solve an inverse-time process of the diffusion model. Specifically, the inverse-time process of interest can be formulated as the following SDE: $$\text{d}\vec{\mathbf{X}}_t= -e^{-t}\cdot\text{tanh}\left(e^{-t}\left<\vec{\mathbf{X}}_t,\vec{{\mathbf{\Theta}}}\right>\right)\cdot\vec{\mathbf{\Theta}}\text{d}t+\sqrt{2}\text{d}\vec{\mathbf{W}}_t,~~~~~~~~~~~(1)$$ with initialization at time $$T>0$$ given as $$\vec{\mathbf{X}}_{T}=\vec{\mathbf{x}}_{T}\in\mathbb{R}^{d}$$, where $$\vec{{\mathbf{\Theta}}}\in\mathbb{R}^{d}$$ is a given vector, and $$\vec{\mathbf{W}}_t$$ is a standard inver-time $$d$$-dimensional Wiener process.
Out goal is to get an analytic expression of $$\vec{\mathbf{X}}_{0}$$. However, we do not know much about how to solve the SDE in Eq. (1).

We instead tried to consider the corresponding Fokker-Planck equation of Eq. (1) as follows:

\begin{align} \frac{\partial p_t(\vec{\mathbf{X}})}{\partial t} =&e^{-2t}\|\vec{\mathbf{\Theta}}\|_2^2\cdot\text{sech}^2\left(e^{-t}\left<\vec{\mathbf{X}},\vec{\mathbf{\Theta}}\right>\right)p_t(\vec{\mathbf{X}})\\ &+e^{-t}\text{tanh}\left(e^{-t}\left<\vec{\mathbf{X}}_t,\vec{{\mathbf{\Theta}}}\right>\right)\left<\vec{{\mathbf{\Theta}}},\nabla_{\vec{\mathbf{X}}}p_t(\vec{\mathbf{X}})\right>+\sum_{i,j}\frac{\partial^2}{\partial x_i \partial x_j}p_t(\vec{\mathbf{X}}).~~~~~~~(2) \end{align}

However, this is a seemingly complicated second-order partial differential equation, and we have very little experience in solving it.

Could you please give us some hints or references that can help us solve Eqs. (1) or (2)?

• What did you try? Commented Jan 26 at 7:30
• The problem is raised from our recent study on diffusion process. I am new to SDE and totally do not know how to solve it. Commented Jan 26 at 8:08
• @AntonW at least provide some context from where the equation came from. Commented Jan 26 at 14:47
• In our study on the diffusion-based generative models, we need to solve an inverse-time process of the diffusion model formulated in the question. Commented Jan 27 at 6:21
• @AntonW Even in the case $d = 1$, it is less likely that closed form solution exists.
– NN2
Commented Jan 27 at 16:04