• Matrix-valued exponential martingale

Let $X$ be a matrix-valued Ito process $$dX_{t}=u_{t}dt+v_{t}dW_{t}$$ where $u$ and $v$ are $d\times d$-matrices, with $d\ge 2$, and $W$ is a real Brownian motion. Consider the exponential $$Q_{t}=\text{Exp}(X_{t}).$$ Under what conditions on $u,v$ is $Q$ a local martingale? ...

Open

Posted online: 2019-01-03 08:28:23Z by Andrea Pascucci13

• Probability
• Frames from rescaled iterative system

Let $A$ be a bounded linear operator on Hilbert space $\mathcal{H}$. For some fixed $x\in \mathcal{H}$, consider a system of vectors $$\left\{\frac{\|A^n x\|}{\|x\|} \right\}_{n=0,1,2,\dots}.$$ There are bounded (not-normal) operators for which a system of above form is an orthonormal basis (e.g. shift operator on $\mathcal{l}(\mathbb{N})$). ...

Open

Posted online: 2018-12-31 01:50:36Z by Armenak Petrosyan11

• Functional Analysis
• Uniqueness of energy maximizers

Let $$\begin{cases} -\Delta u_f=f & \text{in }D,\\ u_f=0 &\text{on }\partial D. \end{cases}$$ It is well-known that the maximization of the functional $$\Phi(f)=\int_D|\nabla u_f|^2dx,$$ over the set $$\bar{\mathcal{R}}_\beta=\{f\,\,|\,\, 0\leq f\leq 1,\,\,\text{and}\,\, \int_Dfdx=\beta\}$$ has a solution $$\hat{f}=\chi_{\{\hat{u}>\alpha\}}\in \mathcal{R}_\beta=\{\chi_E\,\,|\,\, |E|=\beta\},$$ for some $\alpha>0$, and $\hat{u}=u_{\hat{f}}$. ...

Open

Posted online: 2018-12-18 10:25:24Z by Hayk Mikayelyan15

• Analysis of PDEs
• Classical Analysis and ODEs
• Obstacle Problem: $C^{1,\alpha}$-Regularity of Lipschitz FB with continuous right hand side.

Let $v$ be a solution to the obstacle problem $$\Delta v = h\chi_{\{v > 0\}}, \qquad v \geq 0 \qquad \hbox{in } B_1.$$ We assume $h \geq c_0 >0$ is Lipschitz, and a Dirichlet data on $\partial B_1$ has been prescribed. ...

Open

Posted online: 2018-12-11 12:45:15Z by Henrik Shahgholian25

• Analysis of PDEs
• Entropy of Schur-Weyl measures at critical scale

Consider the probability measure $\mathbb{P}_N^n(\lambda)=\frac{\dim E_\lambda}{N^n}$ on the set of Young diagrams with $n$ cells and at most $N$ rows, where $\dim E_\lambda$ is the dimension of the isotypic component corresponding to the Young diagram $\lambda$ in the tensor representation $(\mathbb{C}^N)^{\otimes n}$ of the group $GL(N,\mathbb{C})$. ...

Open

Posted online: 2018-11-20 03:42:50Z by Sevak Mkrtchyan147

• Probability
• Threshold for pinned similar configurations

If $E$ is a compact subset of ${\Bbb R}^d$, of Hausdorff dimension greater than $\frac{d+1}{2}$, then there exists a non-empty open interval $I$ such that, for any $t \in I$, there exist $x^1, x^2, \dots, x^{k+1} \in E$ such that $|x^{j+1}-x^j|=t$, $1 \leq j \leq k$. ...

Open

Posted online: 2018-11-20 03:19:16Z by Sevak Mkrtchyan23

• Classical Analysis and ODEs
• Separation in low temperature beta-ensembles

The following result is known for a repelling system of $N$ particles in a plane, at inverse temperature $\beta$: if we zoom at a suitable point at a natural microscopic scale, and if $\beta$ is at least proportional to $\log N$, then the particles become (almost-certainly) separated from each other by a certain fixed distance, as $N->\infty$. ...

Open

Posted online: 2018-11-19 08:33:35Z by Yacin Ameur20

• Mathematical Physics
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