We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the $d$-dimensional Schr\"odinger operator $L^{\gamma}$ with a one-point potential at the origin, where the driving family is the ground state of $L^{\gamma}$ and $d\in\{2,3\}$. We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time $T$, the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.
Let $P$ denote normalized nearest-neighbor averaging on $\mathbb{Z}^d$ and set $A=I-P$. We consider $\partial_tu+A^su=f(u),t>0, x\in\mathbb{Z}^d, 00$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixed level set lies between two constant multiples of $\exp(at/(d+2s))...
We prove that for every fixed inverse temperature $\beta<1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_{\beta}\left(n \log\left(n / \varepsilon...
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_...
Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The pro...
We consider the damped nonlinear Klein-Gordon equation \begin{align*} \partial_t^2u-\Delta u+2\alpha\partial_tu+u-|u|^{p-1}u=0 \end{align*} on $\mathbb{R}^d$, where $\alpha>0$, $2\leq d\leq5$, and $p>2$ is in the energy-subcritical range. We classify global solutions that converge to a superposition of two positive and...
I. Kenjiro· 0 citations
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