AI 中文总结
针对有界域上的Stratonovich型抛物Anderson模型，计算小时间极限下总质量的期望与标准差的精确渐近，揭示三种机制的竞争及由Hurst指数影响产生的新相变现象。
AI 中文摘要
设D是ℝᵈ中的有界域，κ>0为固定常数，W是ℝ×ℝᵈ上的分数布朗片。考虑Stratonovich型抛物Anderson模型（PAM）∂ₜu_κ=(1/2Δ+κW')u_κ，其在D上满足Dirichlet边界条件，初始条件为平坦初值u_κ(0,·)=1_D。在W的所有Hurst指数均不小于1/2，且u_κ的矩在足够小的t>0时有限的假设下，我们计算了当t→0时，总质量∫_D u_κ(t,x)dx的期望与标准差的精确渐近行为。在此过程中，我们揭示这些渐近行为由三种机制的竞争决定：(1)几何：热通过边界∂D扩散的速率；(2)涨落：W的时间Hurst指数；(3)重整化：确定性Stratonovich修正的奇异性。结果，我们识别出由W的Hurst指数对这些贡献相对大小的影响所产生的新相变现象。
英文摘要
Let $D\subset\mathbb R^d$ be a bounded domain, let $κ>0$ be fixed, and let $W$ be a fractional Brownian sheet on $\mathbb R\times\mathbb R^d$. Consider the Stratonovich parabolic Anderson model (PAM) $\partial_tu_κ=(\frac12Δ+κW')u_κ$ with Dirichlet boundary condition on $D$ and the flat initial condition $u_κ(0,\cdot)=\mathbf 1_D$. We calculate exact asymptotics for the expectation and the standard deviation of the total mass $\int_Du_κ(t,x)~\mathrm d x$ as $t\to0$ under the assumption that $W$'s Hurst indices are all at least $1/2$ and that $u_κ$'s moments are finite for small enough $t>0$. In doing so, we uncover that these asymptotics are determined by a competition between three mechanisms: (1) $\mathbf{Geometry}$: The rate of heat diffusion through the boundary $\partial D$. (2) $\mathbf{Fluctuations}$: $W$'s time Hurst index. (3) $\mathbf{Renormalization}$: The singularity of deterministic Stratonovich corrections. As a result, we identify novel phase transition phenomena, which arise from the influence of $W$'s Hurst indices on the relative magnitudes of these contributions.
Comments46 pages, 1 figure
