Vol. 5 No. 1 (2021): vol 5,Iss 1,2021
Articles

A study on stochastic maximal regularity for rough time-dependent problems

Govindaraju P
PG and Research Department of Mathematics, Islamiah College (Autonomous), Vaniyambadi, Tamil Nadu, India.
Sasikala V
PG and Research Department of Mathematics, Islamiah College (Autonomous),Vaniyambadi, Tamil Nadu, India.
Mohamed Ali A
PG and Research Department of Mathematics, Islamiah College (Autonomous), Vaniyambadi, Tamil Nadu, India.
Published June 18, 2021
Keywords
  • Stochastic PDEs, Maximal regularity, VMO coefficients, Measurable coefficients.
How to Cite
P, G., V, S., & A, M. A. (2021). A study on stochastic maximal regularity for rough time-dependent problems. Journal of Computational Mathematica, 5(1), 60 - 69. https://doi.org/10.26524/cm92

Abstract

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain Lp(Lq) estimates for all p > 2 and q 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain Lp(Lp) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces Tp,2 of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

 

Downloads

Download data is not yet available.