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1. Instantaneous everywhere-blowup of parabolic SPDEs

   https://doi.org/10.1007/s00440-024-01263-7  

   Foondun, Mohammud; Khoshnevisan, Davar; Nualart, Eulalia (October 2024, Probability Theory and Related Fields)

   Abstract We consider the following stochastic heat equation$$\begin{aligned} \partial _t u(t,x) = \tfrac{1}{2} \partial ^2_x u(t,x) + b(u(t,x)) + \sigma (u(t,x)) {\dot{W}}(t,x), \end{aligned}$$ $\begin{array}{c}{\partial }_{t}u(t,x)=\frac{1}{2}{\partial }_x^{2}u(t,x)+b\left(u(t,x)\right)+\sigma \left(u(t,x)\right)\dot{W}(t,x),\end{array}$defined for$$(t,x)\in (0,\infty )\times {\mathbb {R}}$$ $(t,x)\in (0,\infty )\times R$, where$${\dot{W}}$$ $\dot{W}$denotes space-time white noise. The function$$\sigma $$ $\sigma$is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the functionbis assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition$$\begin{aligned} \int _1^\infty \frac{\textrm{d}y}{b(y)}<\infty \end{aligned}$$ $\begin{array}{c}{\int }_1^{\infty }\frac{\text{d}y}{b\left(y\right)}<\infty \end{array}$implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that$$\textrm{P}\{ u(t,x)=\infty \quad \hbox { for all } t>0 \hbox { and } x\in {\mathbb {R}}\}=1.$$ $\text{P}\left\{u\right(t,x)=\infty \text{for all}t>0\text{and}x\in R\}=1.$The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022). 

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2. Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations

   https://doi.org/10.1007/s00220-025-05257-x  

   Chae, Dongho; Jeong, In-Jee; Na, Jungkyoung; Oh, Sung-Jin (April 2025, Communications in Mathematical Physics)

   Abstract We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,$$\begin{aligned} \begin{aligned} \partial _t \theta - \nabla ^\perp \log (10+(-\Delta )^{\frac{1}{2}})\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta -{\nabla }^{\perp }log(10+{(-\Delta )}^{\frac{1}{2}})\theta ·\nabla \theta =0,\end{array}\end{array}$and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities,arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the$$\delta $$ $\delta$-SQG equations, defined by$$\begin{aligned} \begin{aligned} \partial _t \theta + \nabla ^\perp (10+(-\Delta )^{\frac{1}{2}})^{-\delta }\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta +{\nabla }^{\perp }{(10+{(-\Delta )}^{\frac{1}{2}})}^{-\delta }\theta ·\nabla \theta =0,\end{array}\end{array}$for all sufficiently small$$\delta >0$$ $\delta >0$depending on the size of the initial data. For the same range of$$\delta $$ $\delta$, we establish global well-posedness of smooth solutions to the dissipative SQG equations. 

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3. Global well-posedness for 2D generalized parabolic Anderson model via paracontrolled calculus

   https://doi.org/10.1007/s40072-025-00358-z  

   Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan (May 2025, Stochastics and Partial Differential Equations: Analysis and Computations)

   Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ $R^{+}\times T^2$within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ $\begin{array}{c}({\partial }_t-\Delta )u=F\left(u\right)\eta \end{array}$where$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$with$$1/6>\kappa >0$$ $1/6>\kappa >0$, and$$F\in C_b^2(\mathbb {R})$$ $F\in C_b^2\left(R\right)$. Assume that$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 

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4. Measurement of CP violation in B0 → D+D− and $$ {\textrm{B}}_{\textrm{s}}^0 $$ → $$ {\textrm{D}}_{\textrm{s}}^{+}{\textrm{D}}_{\textrm{s}}^{-} $$ decays

   https://doi.org/10.1007/JHEP01(2025)061  

   Aaij, R; Abdelmotteleb, A_S W; Abellan_Beteta, C; Abudinén, F; Ackernley, T; Adefisoye, A A; Adeva, B; Adinolfi, M; Adlarson, P; Agapopoulou, C; et al (January 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> A time-dependent, flavour-tagged measurement ofCPviolation is performed withB⁰→ D⁺D⁻and$$ {B}_s^0 $$ $B_s^0$→$$ {D}_s^{+}{D}_s^{-} $$ $D_s^{+}{D}_s^{-}$decays, using data collected by the LHCb detector in proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 6 fb⁻¹. InB⁰→ D⁺D⁻decays theCP-violation parameters are measured to be$$ {\displaystyle \begin{array}{c}{S}_{D^{+}{D}^{-}}=-0.552\pm 0.100\left(\textrm{stat}\right)\pm 0.010\left(\textrm{syst}\right),\\ {}{C}_{D^{+}{D}^{-}}=0.128\pm 0.103\left(\textrm{stat}\right)\pm 0.010\left(\textrm{syst}\right).\end{array}} $$ $\begin{array}{c}{S}_{D^{+}{D}^{-}}=-0.552\pm 0.100\left(\text{stat}\right)\pm 0.010\left(\text{syst}\right),\\ C_{D^{+}{D}^{-}}=0.128\pm 0.103\left(\text{stat}\right)\pm 0.010\left(\text{syst}\right).\end{array}$ In$$ {B}_s^0 $$ $B_s^0$→$$ {D}_s^{+}{D}_s^{-} $$ $D_s^{+}{D}_s^{-}$decays theCP-violating parameter formulation in terms ofϕ_sand|λ|results in$$ {\displaystyle \begin{array}{c}{\phi}_s=-0.086\pm 0.106\left(\textrm{stat}\right)\pm 0.028\left(\textrm{syst}\right)\textrm{rad},\\ {}\mid {\lambda}_{D_s^{+}{D}_s^{-}}\mid =1.145\pm 0.126\left(\textrm{stat}\right)\pm 0.031\left(\textrm{syst}\right).\end{array}} $$ $\begin{array}{c}{\varphi }_s=-0.086\pm 0.106\left(\text{stat}\right)\pm 0.028\left(\text{syst}\right)\mathrm{rad},\\ \mid {\lambda }_{D_s^{+}{D}_s^{-}}\mid =1.145\pm 0.126\left(\text{stat}\right)\pm 0.031\left(\text{syst}\right).\end{array}$ These results represent the most precise single measurement of theCP-violation parameters in their respective channels. For the first time in a single measurement,CPsymmetry is observed to be violated inB⁰→ D⁺D⁻decays with a significance exceeding six standard deviations. 

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5. The Brown measure of the free multiplicative Brownian motion

   https://doi.org/10.1007/s00440-022-01142-z  

   Driver, Bruce K.; Hall, Brian; Kemp, Todd (October 2022, Probability Theory and Related Fields)

   Abstract The free multiplicative Brownian motion$$b_{t}$$ $b_t$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$ $\mathrm{GL}(N;C),$in the sense of$$*$$ $\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$ $b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$ ${\Sigma }_t$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$ $W_t$on$$\overline{\Sigma }_{t},$$ ${\overline{\Sigma }}_t,$which is strictly positive and real analytic on$$\Sigma _{t}$$ ${\Sigma }_t$. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $\begin{array}{c}{W}_t(r,\theta )=\frac{1}{r^2}{w}_t\left(\theta \right),\end{array}$where$$w_{t}$$ $w_t$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$ ${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$ $u_t$is a “shadow” of the Brown measure of$$b_{t}$$ $b_t$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results. 

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