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

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.$T.$ 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.

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.

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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Simone256
Simone256
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Solution of SDE at finite time, continuity of pdf

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.