Stochastic Processes and their Applications · 2014 · 17 citations · 8 references
EngineeringPhysicsDynamical SystemsStochastic ProcessesThermodynamicsBoundary ConditionsContinuous-time Stochastic ProcessesDiffusion ProcessesStochastic CalculusStochastic ControlStochastic Differential EquationsBoundary Value Problems For Stochastic Differential EquationsDirichlet Boundary ConditionsDirichlet ConditionsNeumann Boundary ConditionsParabolic Partial Differential EquationsWhite Noise
We consider nonlinear parabolic SPDEs of the form ∂tu=Δu+λσ(u)ẇ on the interval (0,L), where ẇ denotes space–time white noise, σ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on σ, we show that the expected L2-energy is of order exp[const×λ4] as λ→∞. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result in Khoshnevisan and Kim.
8