IS12: Superlinear stochastic partial differential equations

date: 7/14/2025, time: 14:00-15:30, room: ICS 139

Organizer: Mickey Salins (Boston University)

Chair: Mickey Salins (Boston University)

Malliavin differentiability of solutions to semilinear parabolic SPDEs

Carlo Marinelli (University College London)

Estimates on first-order pointwise Malliavin derivatives of mild solutions to a class of parabolic dissipative stochastic PDEs are discussed. Such equations, posed on bounded domains of \(\mathbb{R}^d\), are driven by multiplicative Wiener noise and their nonlinear drift term is the superposition operator associated to a locally Lipschitz-continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity techniques, and a comparison principle.

Stochastic reaction diffusion equations with superlinear coefficients

Tusheng Zhang (University of Manchester/USTC)

In this talk, we consider the stochastic reaction-diffusion equation with logarithmic nonlinearity driven by multiplicative noise. Well-posedness will be discussed both in the settling of finite interval and the whole space. Lower order moment estimates of stochastic convolution play an important role.

De Giorgi-Nash-Moser theory and quasilinear SPDEs with transport noise

Antonio Agresti (Sapienza University of Rome)

The De Giorgi–Nash–Moser (DGNM) estimates are one of the cornerstones of the regularity theory of PDEs, ensuring Hölder continuity for solutions to linear parabolic equations with bounded and measurable coefficients. In deterministic settings, they are particularly effective in establishing the existence of global smooth solutions of quasilinear parabolic PDEs in divergence form. However, their extension to SPDEs with transport noise remains a significant challenge. In this talk, I will present recent results establishing DGNM-type estimates for SPDEs driven by smooth transport noise. We show that even weak formulations suffice to obtain global smooth solutions for a broad class of quasilinear SPDEs, including Lotka–Volterra-type systems arising in population dynamics. I will also discuss the main challenges in extending these results to rough noise, which is essential for treating SPDEs with nonlinear transport noise.

Based on joint works with M. Sauerbrey (MPI MiS Leipzig) and M. Veraar (TU Delft).