My apologies for the naive question. As someone from a PDE background, I am just wondering what core things can be said using probabilistic methods about certain PDE problems that couldn't be said without probability. Perhaps my question can be summarised as: why should someone working in PDEs learn some stochastic analysis?
As a concrete example, given an elliptic operator in non-divergence form $$\mathcal L_t=a^{ij}(t,x)\partial_{ij}+b^i(t,x)\partial_i$$ my understanding is that given regularity on the coefficients, the solution to the Cauchy problem $\partial_tu+\mathcal L_tu=0$ with sufficiently regular initial data $u(0,\cdot)=u_0(\cdot)$ can be written as the integral of some functional on Wiener space $$u(t,x)=\mathbb E^x[u_0(X_t)],$$ where $X_t$ is the Markov process with generator $\mathcal L_t$. Disregarding the fact that this is only valid for second order operators, does this Markov process method allow to consider coefficients $a^{ij}, b^i$ beyond what's known with non-probabilistic methods?