| Abstract: |
| Joint work with Lukas Anzeletti, M\`at\`e Gerencs\`er, Alexander Shaposhnikov. We obtain strong uniqueness for SDEs in Hilbert spaces with irregular drift:
$$
dX_t= (A X_t + b(X_t))dt +(-A)^{-\gamma/2}dW_t, X_0=x\in H,
$$
where $H$ is a separable Hilbert space, $A$ is a self-adjoint negative definite operator, $W$ is a cylindrical Wiener process, $b$ is an $\alpha$-H\older continuous function $H\to H$, $\gamma\ge0$. We show that this equation has a unique strong solution provided that $\alpha > \alpha^*(\gamma)$, with an explicit function $\alpha^*$ that takes values in $(0,1)$ for all $\gamma\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010), as no structural assumption on $b$ is imposed. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining L\^e`s theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and the method of Lasry and Lions for approximation in Hilbert spaces. |
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