Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise

This paper provides necessary as well sufficient conditions on the Hurst parameters so that continuous time parabolic Anderson model $$\frac{\partial u}{\partial t}=\frac{1}{2}\Delta +u{\dot{W}}$$ $$[0, \infty )\times {{\mathbb {R}}}^d $$ with $$d\ge 1$$ has a unique random field solution, where W(t, x) is fractional Brownian sheet {R}}}^d$$ and formally $$\dot{W} =\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots x_d} x)$$ . When noise white in time, our condition both when initial data u(0, bounded between two positive constants. parameter $$H_0>1/2$$ , condition, which improves known results literature, different from one.