Related papers: Martingale problems for some degenerate Kolmogorov…
We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case:…
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
In this paper we focus on the validity of some fundamental estimates for time-degenerate Schr\"{o}dinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison…
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] in unbounded domains of the form \[…
In this note we provide uniform a priori estimates for solutions to degenerate complex Hessian equations on compact hermitian manifolds. Our approach relies on the corresponding a priori estimates for Monge-Amp\`ere equations; it provides…
In this paper we analyze the oscillation of functions having derivatives in the H\"older or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the…
In this Note, we present a Calder\'on-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their…
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…
We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We…
We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form \begin{equation*} c|\xi A(\omega,x)|^p\leq…
This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The…
In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak H\"{o}rmander condition. The approach relies on the apriori…
Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted…
In this paper, we consider the problem of representing any polynomial in terms of the ordered Bell and degenerate ordered Bell polynomials, and more generally of the higher-order ordered Bell and higher-order degenerate ordered Bell…
We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD 2) + Az, D , where A, B are N x N matrices, z $\in$ R N , N $\ge$ 1, which satisfy the Kalman condition which is equivalent to the…
In this paper we investigate the well-posedness of the Cauchy problem for a Schr\"odinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core…
We obtain an alternative approach to recent results by M. Lacey \cite{La} and T. Hyt\"onen {\it et al.} \cite{HRT} about a pointwise domination of $\omega$-Calder\'on-Zygmund operators by sparse operators. This approach is rather elementary…
We prove sharp reverse H\"older inequalities for minima of multi-phase variational integrals and apply them to Calder\'on-Zygmund estimates for nonhomogeneous problems.
For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic…
Using Carleman estimates, we give a lower bound for solutions to the discrete Schr\"odinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of the solutions.