Let $U(\beta) = \exp(-f(\beta))$ where $f$ is $m$ strongly convex and $L$ Lipschitz smooth. I am trying to find an upper bound on $E[\|\beta_t \|^2]$ where $\beta_t \in \mathcal{R}^p$ is defined below

\begin{align*} d\beta_t = -\nabla U(\beta_t) dt + dW_t \end{align*}

In the above equation, $W_t$ is a standard $p$ dimensional Brownian motion. We also know $\beta_0 \sim N(0,\lambda I)$, so $E[\|\beta_0 \|^2] = p\lambda$ and $\beta_{\infty} \sim N(\mu,\Sigma)$. I think we will have to use Gronwall's inequality to find an upper bound, but I am not sure.


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