Related papers: Remarks on parabolic Kolmogorov operator

We will show that the same type of estimates known for the fundamental solutions for scalar parabolic equations with smooth enough coefficients hold for the first order derivatives of fundamental solution with respect to space variables of…

We prove sharp two-sided estimates of the fundamental solution to the fractional Kolmogorov equation in $\mathbb{R}\times \mathbb{R}$ using Fourier methods. Additionally, we provide an explicit form of the fundamental solution in case of…

We consider a class of degenerate equations satisfying a parabolic H\"ormander condition, with coefficients that are measurable in time and H\"older continuous in the space variables. By utilizing a generalized notion of strong solution, we…

In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may…

Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution…

Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving…

In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is…

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…

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We…

The family of fifth order nonlinear evolution equations is studied. Some traveling wave elliptic solutions are found. The classification of these exact solutions is given.

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state.…

We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…

In this paper we solve Kolmogorov problem about existence of a function with given norms of derivatives for classes of multiple monotone functions and absolute monotone functions in the case of arbitrary number of norms. We also show the…

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem implies that the unique…

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities.…

The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on…

We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.

We consider stochastic equations for the class of formal mappings. Existence and uniqueness of solution, as well as evolution property are proved.

For a class of divergence type quasi-linear degenerate parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via nonlinear Wolff potentials.

We establish new intrinsic Strichartz estimates for solutions of the Cauchy problem for a class of possibly degenerate Schr\"odinger equations with a real drift.