Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces

Middle Convolution and Harnad Duality

We interpret the additive middle convolution operation in terms of the Harnad duality, and as an application, generalize the operation to have a multi-parameter and act on irregular singular systems.Comment: 50 pages; v2: Submitted version once revised according to referees' comment

The Ising model and Special Geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries". Beyond the small order factor operators (occurring in the linear differential operators associated with $\chi^{(5)}$ and $\chi^{(6)}$), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group $Sp(12, \mathbb{C})$. The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group $SO(21, \mathbb{C})$.Comment: 33 page

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T], U(0) & = u_0. {aligned}. {equation} Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains $(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that $(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The functions $F$ and $B$ are nonlinear functions defined on certain interpolation spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical Brownian motion on a separable Hilbert space $H$. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk, to obtain space-time regularity results for the solution $U$ of \eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already been publishe

The Fuchsian differential equation of the square lattice Ising model χ(3)\chi(3) susceptibility

Using an expansion method in the variables $x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the 3-particle contribution $\chi^{(3)}$, using a $N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $\chi^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.Comment: 15 pages, no figures, submitted to J. Phys.

Sandhill Crane Activity in the Central Platte River Valley in Late May and Early June

In this report we discuss observations of Sandhill Cranes remaining in the Central Platte River Valley until early June 2018 and discuss potential explanations for this extended stay into the breeding season. We detected two pairs of adult Sandhill Cranes in two different locations on 15 May 2018 (Table 1). We then detected three injured adult Sandhill Cranes in a third location on 16 May 2018 (Table 1). Two of the three Sandhill Cranes each had a missing leg and the third crane’s leg was broken above the tibiotarsal joint; however, all were still capable of foraging and flight. Later, on 6 June 2018, a Sandhill Crane was reported to the Crane Trust Nature and Visitor Center north of our survey area near Grand Island, Nebraska (Sandpit Lake, Table 1). In all, Sandhill Cranes were located in nine distinct locations throughout the CPRV from 15 May 2018 to 07 June 2018 (Table 1). Considering their temporal and spatial occurrence, we estimate that there were seven to eight unique individual cranes present in the late May to early June period (Table 1). However, without making assumptions about their movements, there could have been a significantly higher number of Sandhill Cranes present (Table 1). They were detected in a diversity of habitats including lowland tallgrass prairie and wet meadow (Table 1; for habitat definitions see Currier 1982, Harner and Whited 2011). Sandhill Cranes were generally detected within 2 km of the Platte River (x̅+SD = 1.19+1.28 km)