Suppose we have a Stochastic differential equation (SDE) as such $$ dX_t = b(t, X_t)dt + \sigma(t, X_t) dW_t $$ with $X_0 = x$ and $W_t$ being a Brownian motion. We futhere assume that $b$ and $\sigma$ are such that the solution to the SDE exists and it is unique a.e.
Conversely, if we mollify the differential equation; that is, $$ dY_t = b_\varepsilon(t, Y_t) dt + \sigma_\varepsilon(t, Y_t) dW_t $$ with $b_\varepsilon(t,X_t) := (b \ast \eta_\varepsilon)(t, X_t)$ and similarly for $\sigma_\varepsilon$. Is it true that $Y_t = X_t \ast \eta_\varepsilon$? Or that $Y_t \to X_t$ as $\varepsilon \to 0$ in $L^p_{\text{loc}}$ if $X \in L^p_{\text{loc}}$? In general, there's some connection between the two SDE?
There is no meaning to $X_{t}* \eta_{\epsilon}$ because for each realization $X_{t}$ is only a function of $t$. (But if you want, you could mollify the white noise i.e. $B_t=\int 1_{[0,t]}\xi(s)$ can be mollified as $B_t^{\epsilon}=\int 1_{[0,t]}\xi_{\epsilon}(s)$ for $\xi_{\epsilon}(s)=\int \phi_{\epsilon}(r-s)\xi(r)dr$).
The study of these approximations usually goes under the name of Wong-Zakai theorems "On the Convergence of Ordinary Integrals to Stochastic Integrals".
A good book reference is the Stroock-Varadhan book "Multidimensional Diffusion Processes" chapter on limit theorems 11.1.4 Theorem.
In terms of Lp-convergence see 11.4.2 Theorem.