Consider the following parabolic equation on $\mathbb{R}^d$:

\begin{equation} \partial_t\mu=\mathrm{div}(b\mu) + \mathrm{div}(D\nabla\mu), \end{equation} where the drift $b:\mathbb{R}^d\rightarrow\mathbb{R}$ is sufficiently smooth say $C^2$ and the diffusion matrix $D:\mathbb{R}^d\rightarrow \mathbb{R}^{d\times d}$ is symmetric, strictly positive definite with entries sufficiently smooth, say $C^2$ as well. Here notation $C^2$ means continously differentiable upto second order.

Such an equation arises as the forward-kolmogorov equation for an stochastic differential equation (SDE).

Question: Following the connection from SDEs assume that at time $t=0$, $\mu|_{t=0}=\nu$ where $\nu\in L^1(\mathbb{R}^d)$. How does one prove regularity for the solution of such a PDE?

I have heard that the solution of this equation should be smooth (upto C^2?), but I have no idea how to prove this. Any suggestions and references are very welcome.

Thanks in advance.

  • 1
    $\begingroup$ Well, you can read about this kind of parabolic pde in the book by Evans "Partial differential equations". Actually,if you assume $\nu$ to be smooth you can apply the standard energy methods straightforwardly to check that the solution is smooth. If the initial data is merely $L^1$, then you have to argue with parabolic smoothing. $\endgroup$ – guacho Mar 17 '15 at 6:30
  • $\begingroup$ What do you mean exactly by parabolic smoothing? Can you point me out to some reference? $\endgroup$ – ABC Mar 17 '15 at 8:22
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    $\begingroup$ I already did :-). Check chapter 7 in the Evans's book. By parabolic smoothing I mean that the solution will gain some space derivatives in some integral sense in time. Something like $$ \mu\in L^2_t H^1_x $$ $\endgroup$ – guacho Mar 17 '15 at 18:57

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