Consider a time-homogeneous diffusion process on $\mathbb R^n$ $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ and suppose that it satisfies Hormander's Lie bracket condition. Suppose that it is stationary at a measure $\rho(x)dx$ with strictly positive density $\rho$.

Provided that $b, \sigma$ are sufficiently nice, the time-reversed process $Y_t := X_{T-t}$ satisfies

$$dY_t= -b(Y_t)dt + \sigma \sigma^T \nabla \log \rho(Y_t)dt + \sigma(Y_t)dW_t$$

Denote by $P^+, P^-$ the laws on path space $C([0,T], \mathbb R^n)$.

In the uniformly elliptic case ($\sigma \sigma^T \succ \lambda I, \lambda \in \mathbb R$), it is relatively easy to show, using Cameron-Martin-Girsanov's formula that $P^+$ and $P^-$ are equivalent. However, I have not been able to use the CMG formula when I drop the ellipticity assumption.

Is there another way to see whether these measures are equivalent in the hypoelliptic case? Any reference or hint would be of great help.


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