Small deviations for fractional stable processes

Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter H > 0. Consider ||.|| a translation invariant, b-self-similar, and p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c = (H - b - 1/p), then x log P [ log ||R|| - k 0, with k finite in the Gaussian case a = 2. If a < 2, we prove that k is finite when R is continuous and H > (b + 1/p + 1/a). We also show that under the above assumptions, x log P [ log ||X|| - k 0, where k is finite and X is the linear a-stable fractional motion with Hurst parameter 0 < H < 1 (if a = 2, then X is the classical fractional Brownian motion). These general results recover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and Non-Gaussian frameworks.Comment: 30 page

On finite generation of the Johnson filtrations

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He.Comment: 32 pages. v2: very minor edits. Weaker versions of the results of this paper previously appeared in arXiv:1704.01529 and arXiv:1703.04190v

The Miura Map on the Line

The Miura map (introduced by Miura) is a nonlinear map between function spaces which transforms smooth solutions of the modified Korteweg - de Vries equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this paper we study relations between the Miura map and Schroedinger operators with real-valued distributional potentials (possibly not decaying at infinity) from various spaces. We also investigate mapping properties of the Miura map in these spaces. As an application we prove existence of solutions of the Korteweg - de Vries equation in the negative Sobolev space H^{-1} for the initial data in the range of the Miura map.Comment: 33 page

Simulating the Formation of Massive Protostars: I. Radiative Feedback and Accretion Disks

We present radiation hydrodynamic simulations of collapsing protostellar cores with initial masses of 30, 100, and 200 M$_{\odot}$. We follow their gravitational collapse and the formation of a massive protostar and protostellar accretion disk. We employ a new hybrid radiative feedback method blending raytracing techniques with flux-limited diffusion for a more accurate treatment of the temperature and radiative force. In each case, the disk that forms becomes Toomre-unstable and develops spiral arms. This occurs between 0.35 and 0.55 freefall times and is accompanied by an increase in the accretion rate by a factor of 2-10. Although the disk becomes unstable, no other stars are formed. In the case of our 100 and 200 M$_{\odot}$ simulation, the star becomes highly super-Eddington and begins to drive bipolar outflow cavities that expand outwards. These radiatively-driven bubbles appear stable, and appear to be channeling gas back onto the protostellar accretion disk. Accretion proceeds strongly through the disk. After 81.4 kyr of evolution, our 30 M$_{\odot}$ simulation shows a star with a mass of 5.48 M$_{\odot}$ and a disk of mass 3.3 M$_{\odot}$, while our 100 M$_{\odot}$ simulation forms a 28.8 M$_{\odot}$ mass star with a 15.8 M$_{\odot}$ disk over the course of 41.6 kyr, and our 200 M$_{\odot}$ simulation forms a 43.7 M$_{\odot}$ star with an 18 M$_{\odot}$ disk in 21.9 kyr. In the absence of magnetic fields or other forms of feedback, the masses of the stars in our simulation do not appear limited by their own luminosities.Comment: 24 pages, 14 figures. Accepted to The Astrophysical Journa