分数阶非线性偏微分方程的(α,d,β)超过程和奇异解的密度

IF 1.5 Q2 PHYSICS, MATHEMATICAL Annales de l Institut Henri Poincare D Pub Date : 2022-05-01 DOI:10.1214/21-aihp1180
Thomas Hughes
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引用次数: 3

摘要

摘要:考虑R d中α∈(0,2)且β < α d的临界(α, β)-超过程的密度Xt(x)。我们的出发点是PDE[2]的一个最新结果,该结果暗示了以下二分法:如果x∈R是固定的,并且β≤β∗(α):= α d+α,则Xt(x) > 0 a.s. on {Xt 6= 0};否则,当条件为{Xt 6= 0}时,Xt(x)为正的概率呈幂律衰减。我们强化了这一点,并从概率上证明了当β < β∗(α)且密度是连续的,当且仅当d = 1且α > 1+ β,则在{Xt 6= 0}上,对于所有x∈R, Xt(x) > 0。以上补充了一个经典的超过程结果,即如果Xt不为零,那么它几乎肯定会对每个开集收费。在{Xt 6= 0}上给出了μ(Xt):=∫Xt(x)μ(dx) > 0的近似尖锐条件,统一并推广了这些结果。我们的表征是基于供给的大小(μ),在豪斯多夫测度和维数的意义上。对于s∈[0,d],若β≤β∗(α, s) = α d−s+α,且supp(μ)具有正的x-Hausdorff测度,则μ(Xt) > 0 a.s. on {Xt 6= 0};当β > β∗(α, s)时,如果μ满足暗(supp(μ)) < s的均匀低密度条件,则P (μ(Xt) = 0 |Xt 6= 0) > 0。我们的方法也给出了对(α, β)超过程对偶的分数阶偏微分方程的新结果,即∂tu(t, x) =∆αu(t, x)−u(t, x) 1+β,定域(t, x)∈(0,∞)× R,其中∆α =−(−∆)α
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The density of the (α,d,β)-superprocess and singular solutions to a fractional non-linear PDE
Abstract: We consider the density Xt(x) of the critical (α, β)-superprocess in R d with α ∈ (0, 2) and β < α d . Our starting point is a recent result from PDE [2] which implies the following dichotomy: if x ∈ R is fixed and β ≤ β∗(α) := α d+α , then Xt(x) > 0 a.s. on {Xt 6= 0}; otherwise, the probability that Xt(x) is positive when conditioned on {Xt 6= 0} has power law decay. We strengthen this and prove probabilistically that if β < β∗(α) and the density is continuous, which holds if and only if d = 1 and α > 1+ β, then Xt(x) > 0 for all x ∈ R a.s. on {Xt 6= 0}. The above complements a classical superprocess result that if Xt is non-zero, then it charges every open set almost surely. We unify and extend these results by giving close to sharp conditions on a measure μ such that μ(Xt) := ∫ Xt(x)μ(dx) > 0 a.s. on {Xt 6= 0}. Our characterization is based on the size of supp(μ), in the sense of Hausdorff measure and dimension. For s ∈ [0, d], if β ≤ β∗(α, s) = α d−s+α and supp(μ) has positive x-Hausdorff measure, then μ(Xt) > 0 a.s. on {Xt 6= 0}; and when β > β ∗(α, s), if μ satisfies a uniform lower density condition which implies dim(supp(μ)) < s, then P (μ(Xt) = 0 |Xt 6= 0) > 0. Our methods also give new results for the fractional PDE which is dual to the (α, β)superprocess, i.e. ∂tu(t, x) = ∆αu(t, x)− u(t, x) 1+β with domain (t, x) ∈ (0,∞) × R, where ∆α = −(−∆) α
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