Comparison for solutions of a SPDE driven by martingale measure

Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 231-244 Printed June 1, 2005

Nhansook Cho Hansung University

Abstract : We derive a comparison theorem for solutions of the following stochastic partial differential equations in a Hilbert space $H.$ $$Lu^i=\a(u^i) \dot M(t,x)+\beta^i(u^i),\,\,\text{for }i=1,2,$$ where $Lu^i=\frac{\partial u^i}{\partial t} -Au^i,$ $A$ is a linear closed operator on $H$ and $ \dot M(t,x)$ is a spatially homogeneous Gaussian noise with covariance of a certain form. We are going to show that if $\beta^1\le \beta^2$ then $u^1\le u^2$ under some conditions.