On the behaviour of stochastic heat equations on bounded domains
On the behaviour of stochastic heat equations on bounded domains
Citació
- Foondun M, Nualart E. On the behaviour of stochastic heat equations on bounded domains. ALEA Lat Am J Probab Math Stat. 2015;12(2):551-71.
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Descripció
Resum
Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.