Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in $\mathbb {R}^{d}$ ℝ d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

On the stochastic nonlinear Schrödinger equations at critical regularities

We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by Bényi et al. (Trans Am Math Soc Ser B 2:1–50, 2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.

Optimizing the Fractional Power in a Model with Stochastic PDE Constraints

We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.