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1. The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting

   https://doi.org/10.1353/ajm.2024.a917539  

   Morgan, Katrina (February 2024, American Journal of Mathematics)

   abstract: The current work considers solutions to the wave equation on asymptotically flat, stationary, Lorentzian spacetimes in $(1+3)$ dimensions. We investigate the relationship between the rate at which the geometry tends to flat and the pointwise decay rate of solutions. The case where the spacetime tends toward flat at a rate of $$|x|^{-1}$$ was studied by Tataru (2013), where a $$t^{-3}$$ pointwise decay rate was established. Here we extend the result to geometries tending toward flat at a rate of $$|x|^{-\kappa}$$ and establish a pointwise decay rate of $$t^{-\kappa-2}$$ for $$\kappa\in\Bbb{N}$$ with $$\kappa\ge 2$$. We assume a weak local energy decay estimate holds, which restricts the geodesic trapping allowed on the underlying geometry. We use the resolvent to connect the time Fourier Transform of a solution to the Cauchy data. Ultimately the rate of pointwise wave decay depends on the low frequency behavior of the resolvent, which is sensitive to the rate at which the background geometry tends to flat. 

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2. KPZ equation with a small noise, deep upper tail and limit shape

   https://doi.org/10.1007/s00440-022-01185-2  

   Gaudreau_Lamarre, Pierre Yves; Lin, Yier; Tsai, Li-Cheng (April 2023, Probability Theory and Related Fields)

   In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter \sqrt{\varepsilon} in front of the noise and let \varepsilon \to 0. We prove that the one-point large deviation rate function has a \frac{3}{2} power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the KPZ equation as \varepsilon \to 0. This confirms the physics prediction in Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016), and Le Doussal, Majumdar, Rosso, and Schehr (2016). 

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3. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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4. Rough hypoellipticity for the heat equation in Dirichlet spaces

   https://doi.org/10.1002/mana.202100014  

   Hou, Qi; Saloff‐Coste, Laurent (January 2023, Mathematische Nachrichten)

   Abstract This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut‐off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off‐diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; structure results for ancient (local weak) solutions of the heat equation. 

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5. Random quantum graphs

   https://doi.org/10.1090/tran/8584  

   Chirvasitu, Alexandru; Wasilewski, Mateusz (April 2022, Transactions of the American Mathematical Society)

   We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples ( X 1 , ⋯ , X d ) (X_1,\cdots ,X_d) of traceless self-adjoint operators in the n × n n\times n matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: 2 ≤ d ≤ n 2 − 3 2\le d\le n^2-3 . Moreover, the automorphism group is generically abelian in the larger parameter range 1 ≤ d ≤ n 2 − 2 1\le d\le n^2-2 . This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of X i X_i ’s (mimicking the Erdős-Rényi G ( n , p ) G(n,p) model) has trivial/abelian automorphism group almost surely. 

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