The Parabolic Anderson Model by Wolfgang König

Author:Wolfgang König
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Example 5.15 (Poisson shot-noise potential)

Another natural choice of the potential is a Poisson shot-noise potential , see Sect. 3.​5.​3 For such a potential, the almost-sure asymptotics of the total mass of the solution to the PAM have been derived in [GärKönMol00]. Recall from Sect. 3.​5.​3 the standard Poisson point process on with intensity and the non-negative cloud . Like in Example 5.13, we neglect issues about the decay of the cloud at and simply assume to be compactly supported; see Sect. 7.​5 for questions about the correlation length of this potential. We consider the potential , i.e., with the opposite sign as in the obstacle case of Sect. 5.11. A mild assumption on implies that V is Hölder-continuous. We assume that is strictly maximal in 0 with a strictly positive definite Hessian matrix . Clearly, .

Like in the Gaussian case of Example 5.13, we introduce the Legendre transform of H and define h t via . Then it is derived in [GärKönMol00] that the asymptotics in (5.15) literally hold true with the same value , where now , and is the Hessian of at zero. The interpretation of this result is the same as in the Gaussian case; we do not spell it out. Again, , but here the numerical value is . This asymptotics is too rough to replace h t in (5.15), as the error term of the first term would spoil the precision of the second. Sufficiently precise asymptotics for h t would have to depend on more details of the cloud in a neighbourhood of zero. ◊

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