Given the law of a diffusion $dX_t = \lambda_t(t,X_t) + \sigma dB_t$, can we find explicitly the drift?

Question

Let $(\Omega, \mathcal F, P)$ be a probability space. Suppose that we have a stochastic process of the form

$$d X_t = \lambda_t(t,X_t) + \sigma dB_t $$

where $\sigma >0$ is given and $B_t$ is a standard Brownian motion. We don't know the drift $\lambda$ but we know that $X_t \sim_P \mathcal N (\mu_t, \sigma_t)$ where $\mu_t,\sigma_t$ are given. Can we find $\lambda_t$ ?