Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift
A talk in the SPDEvent series by
Lukas Gräfner from FU Berlin
| Abstract: | On $M \in \{\mathbb{T}^d, \mathbb{R}^d\}$ with $d \geq 2$ we study ${\mathrm d} X_t = b(t,X_t) {\mathrm d}t + \sqrt{2} {\mathrm d} B_t,$ where $b : \mathbb{R}_+ \rightarrow \mathcal{S}'$ is distributional and $B$ is a Brownian motion. We consider this equation in the scaling-supercritical regime using energy solutions and recent ideas for generators of singular SPDEs. For time-dependent $b = b_1 + b_2, \nabla \cdot b_1 =0$ we show weak well-posedness of energy solutions with initial ${\mathrm Leb}_M - L^p$ density under certain assumptions on the Besov- regularities of $b_i$ which allow $b_1$ to be distributional and supercritical, while $b_2$ is a function of critical regularity. For time-independent $b$ we show weak well-posedness of energy solutions under certain structural assumptions on $A$, where A is the matrix field describing the divergence-free part $b_1$ in the decomposition $b^i = b_1^i + b_2^i = \nabla \cdot A_i + b_2^i$. This includes the case that $A$ is locally diverging, which produces elements $A \in L^{\frac{2d}{d-2}-}, A \notin L_{\mathrm loc}^{\frac{2d}{d-2}}$ meaning that we obtain for any $p >2$ proper elements $b \in W^{-1,p-}$ in large enough dimensions. |