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I'm looking at the Langevin dynamics described by the following SDE

$$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$

where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity conditions (let's say smooth and Lipschitz), and $B_t$ is a Brownian motion. $\Sigma$ can be thought to be identity.

I'm interested in the random variable $X(T)$, for a fixed $T.$ In particular, I would like to state that its probability density function is continuous. I believe I could prove such statement, given the regularity of my problem, but I cannot find proper reference for this.

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1 Answer 1

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In Theorem 1.2 of the paper Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift, the authors prove that the transition density of the SDE $$ \mathrm{d} X_t=b\left(t, X_t\right) \mathrm{d} t+\sigma\left(t, X_t\right) \mathrm{d} W_t, \quad t \geqslant 0, X_0=x $$ is continuous under some mild conditions.


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