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A 1D diffusion process of the form

$$dX_t = -V'(X_t)dt + \sigma dW_t$$

can be used to describe the movement of a particle in potential $V(x)$. The stationary density is then $$\pi(x) = Me^{V(x)/\sigma^2}$$ where $M$ is a normalizing constant [1, p. 57]

To what extent and under which assumptions can this be generalized to multivariate diffusion processes?

I have some bivariate data that I assume is generated by a potential of no particular functional form. So, basically I have an approximate stationary density, and I would like to get a non-parametric estimate for the underlying potential landscape.

Source: [1] Iacus (2008) Simulation and inference of stocastic differential equations

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