A priori bounds for 2-d generalised Parabolic Anderson Model
A talk in the Bielefeld Stochastic Afternoon series by
Guilherme De Lima Feltes
| Abstract: | We show a priori bounds for the solution to $(\partial_t - \Delta) u =\sigma (u) \xi$ in the framework of Hairer's regularity structures [Invent Math 198:269--504, 2014] in finite volume. We assume $\sigma \in C_b^2 (\mathbb{R})$ and that $\xi$ is of negative Hölder regularity of order $- 1 - \kappa$ and can be lifted to a model in the sense of regularity structures. Our main results guarantee non-explosion of the solution in finite time and a growth which is at most polynomial in $t > 0$, for the regime $\kappa \in (0, \bar{\kappa})$, for an explicit $\bar{\kappa}< 1/3$. Our estimates imply global well-posedness for the 2-d generalised parabolic Anderson model on the torus, as well as for the parabolic quantisation of the Sine-Gordon EQFT on the torus in the regime $\beta^2 \in (4 \pi, (1 + \bar{\kappa}) 4 \pi)$. Construction of the measure for the massive Sine-Gordon EQFT and of stochastic flows for random dynamical systems in the Da Prato-Zabczyk regime [$\textit{Stochastic Equations in Infinite Dimensions}$, Enc. Math. App., Cambridge Univ. Press, 1992] beyond the
trace-class case also follow from our results. This is joint work with A. Chandra (Imperial) and H. Weber (Münster). Within the CRC this talk is associated to the project(s): B8 |