138 research outputs found
Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential
No full textIn this article, we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi-linear equations perturbed by a singular Hardy potential on the whole Euclidean space depending on the controllability of the given singular potential. To control the singular potential we use a weighted Hardy inequality with an optimal constant, which was recently discovered in Hauer and Rhandi (2013). Our results in this paper extend the ones in Goldstein et al. (2012) concerning a linear Kolmogorov operator significantly in several ways: firstly, by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of p-Laplace type with a nonlinear convection term for 1 < p < ∞, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions
On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates
No full textWe consider the Schroedinger type operator , for and . We prove that, for any , the minimal realization of operator in generates a strongly continuous analytic semigroup .
For and , we then prove some upper estimates for the heat kernel associated to the semigroup
.
As a consequence we obtain an estimate for large of the eigenfunctions of . Finally, we extend such estimates to a class of divergence type elliptic operators
Semigroup applications everywhere
No full textMost dynamical systems arise from differential equations that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions the theory of one-parameter operator semigroups is one of the most powerful tools. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and partials differential equations, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, and stochastic processes. This theme issue includes papers on such semigroups and their many applications
Analytic approach to solve a degenerate parabolic PDE for the Heston Model
No full textWe present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane
The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions
No full text
Kernel estimates for nonautonomous Kolmogorov equations
No full textUsing time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients
- …
