Expected value in case of Langevin dynamics

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.