Density transformation under gaussian pertubation

\(x_{t+1} \gets x_t - \eta \nabla f(x_t)dt + \sqrt{2\eta}dB_t\) For a small $\Delta t$, \(x_{t+1} \gets \mathcal{N}(x_t - \eta \nabla f(x_t), 2\eta )\) $$ \partial_t p = \int \mathcal{N}(x_t - \eta \nabla f(x_t), 2\eta )p(x_t)dx_t

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