| Abstract: |
| In this talk, we will discuss pathwise uniqueness for mild solutions to stochastic PDEs with drift given in differential form. The key example that we want to study is the following SPDE evolving in $H=L^2([0,1]^d)$ with $d\in \{1,2,3\}$
\[
dX(t) + A^\gamma X(t) \, dt=B(X(t))\, d t
+A^{-\rho}\, d W(t)\,, \qquad X(0)=x\,,
\]
where $\{W(t)\}_{t\geq 0}$ is an $H$-cylindrical Wiener process, $-A$ is a suitable realization of the Laplacian in $H$, $B:D(A^\mu)\rightarrow D(A^{-\nu})$ is locally $\theta$-Holder continuous with $\theta\in (0,1)$, $\gamma>0$ and $\mu, \nu, \rho\geq 0$ are given constants. Under suitable assumptions on $\gamma>0$ and $\mu, \nu, \rho\geq 0$, we will show that the pathwise uniqueness holds in $D(A^\mu)$. The singularity of the drift perturbation $B$ allows to achieve novel pathwise uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models. |
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