Almost sure asymptotic stability of fractional stochastic nonlinear heat equation

Recently, the fractional stochastic nonlinear heat equation in the Hilbert space L2(0; 1), driven by the fractional power of the Laplacian and perturbed by a trace-class noise has been studied by the first and the last authors. They have proved the wellposedness, the pth-moment exponential stability and the almost surely exponential stability of such problem in the semigroup framework. The current work is considered as a continuation of the previousely mentioned paper. More particularly, we establish the almost sure asymptotic stability under the same conditions imposed in our recent work, besides a regularity of the initial condition. Finally, some examples are provided to illustrate the obtained theory.