Let X be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of X, and prove a parabolic Harnack inequality for nonnegative parabolic functions of X.

We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and “saturation” properties of several generalizations of the Littlewood–Richardson numbers.

We formulate a quasilinear parabolic equation describing the behavior of global-in-time solution to semilinear equation. study this in accordance with blow-up and quenching patterns original This is new theory partial differential equations presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis viscosity solution.

We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.

We describe some recent results on the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplacian. More precisely we focus on Carleson-type estimates and boundary Harnack principles.

We investigate a class of over-determined parabolic problems involving a non-constant boundary condition. The Weinstein’s technique known for the elliptic problems is extended to the parabolic one by means of auxiliary functions and Green classical formula.

Talbot’s method for the numerical inversion of the Laplace transform consists of numerically integrating the Bromwich integral on a special contour by means of the trapezoidal or midpoint rules. In this paper we address the issue of parameter selection in the method, for the particular situation when parabolic PDEs are solved. In the process the well-known subgeometric convergence rate O(exp(−c...

Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.

For continuous time Markov chains on a countable state space, we derive a parabolic Harnack inequality using probabilistic methods. We derive some consequences of this inequality for the compactness of parabolic (i.e. spacetime harmonic) functions of the process.

We prove that the only complex parabolic geometries on Calabi–Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.