Related papers: Singular SPDEs on Homogeneous Lie Groups

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting…

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is…

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(\omega,t)u_{x^ix^j}+f)dt + (\sigma^{ik}(\omega,t)u_{x^i}+g^k)dw^k_t,…

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical…

We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$,…

Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\star…

We show that analytic pseudodifferential and Fourier integral operators behave well for ultradifferentiable classes satisfying minimal regularity properties. As an application we investigate the ultradifferentiable regularity properties of…

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but,…

In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes…

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and…

We introduce a calculus for parameter-dependent singular Green operators on the half-space $\mathbb{R}^n_+$ that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's…

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow…

Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $$ \Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in…

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian…

We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic $p$ with respect to the adjoint action of a Chevalley generator. In…

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of…

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Fr\"olicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2},…

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $su(n,n)$. Earlier were given the main multiplets of indecomposable elementary…

We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} \beta, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$,…

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -\Delta u = {u^{-\gamma}} \ \text{in}\…