Existence of nonnegative mild solutions of stochastic evolution inclusions via weak topology

Cretarola Alessandra , University of Perugia
Benedetti Irene, Department of Mathematics and Computer Science, University of Perugia

We provide sufficient conditions for the existence of mild solutions to stochastic differential inclusions in infinite dimensional Hilbert spaces driven by a cylindrical Wiener process. Inspired by [2] we develop an approximation procedure based on the weak topology. By applying this method, we can accomplish the dual objective of proving the existence of a solution over the entire real half-line while relaxing the commonly assumed hypotheses of Lipschitz continuity and compactness found in the literature on the subject. Moreover, the problem of the non negativity of the solution is addressed. Namely, assuming an additional sign condition that involves both the deterministic and stochastic nonlinear terms as in [1], we can ensure the nonnegativity of the solution starting from a nonnegative initial datum. The talk is based on a joint paper with Lucia Angelini and Irene Benedetti, University of Perugia. References [1] C. Marinelli, L. Scarpa, On the positivity of local mild solutions to stochastic evolution equations. In: International Conference on Random Transformations and Invariance in Stochastic Dynamics. Cham: Springer International Publishing, 2019. p. 231-245. [2] Y. Zhou, R-N Wang, L. Peng, Topological structure of the solution set for evolution inclusions. Springer Singapore, 2017.

Area: CS46 - Semilinear Stochastic Differential Equations in Infinite-Dimensional Spaces (Alessandra Cretarola)

Keywords: Stochastic Differential Inclusions; Weak Topology; Nonnegative Mild Solutions; Fixed Point Theory

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