77 research outputs found
Kernel estimates for nonautonomous Kolmogorov equations
Using 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
Analytic approach to solve a degenerate parabolic PDE for the Heston Model
We 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
On vector-valued Schrödinger operators with unbounded diffusion in Lp spaces
We prove generation results of analytic strongly continuous semigroups on Lp(Rd, Rm) (1 < p< ∞) for a class of vector-valued Schrödinger operators with unbounded coefficients. We also prove Gaussian type estimates for such semigroups
Global properties of Generalized Ornstein-Uhlenbeck Operators on L p (R N, R N) with more than linearly growing coefficients
Instantaneous blowup and singular potentials on Heisenberg groups
In this paper we generalize the instantaneous blowup result from [3] and [15]
to the heat equation perturbed by singular potentials on the Heisenberg group
Structured models of cell migration incorporating molecular binding processes
The dynamic interplay between collective cell movement and the various
molecules involved in the accompanying cell signalling mechanisms plays a
crucial role in many biological processes including normal tissue development
and pathological scenarios such as wound healing and cancer. Information about
the various structures embedded within these processes allows a detailed
exploration of the binding of molecular species to cell-surface receptors
within the evolving cell population. In this paper we establish a general
spatio-temporal-structural framework that enables the description of molecular
binding to cell membranes coupled with the cell population dynamics. We first
provide a general theoretical description for this approach and then illustrate
it with two examples arising from cancer invasion
Computational Approaches and Analysis for a Spatio-Structural-Temporal Invasive Carcinoma Model
Spatio-temporal models have long been used to describe biological systems of cancer, but it has not been until very recently that increased attention has been paid to structural dynamics of the interaction between cancer populations and the molecular mechanisms associated with local invasion. One system that is of particular interest is that of the urokinase plasminogen activator (uPA) wherein uPA binds uPA receptors on the cancer cell surface, allowing plasminogen to be cleaved into plasmin, which degrades the extracellular matrix and this way leads to enhanced cancer cell migration. In this paper, we develop a novel numerical approach and associated analysis for spatio-structuro-temporal modelling of the uPA system for up to two-spatial and two-structural dimensions. This is accompanied by analytical exploration of the numerical techniques used in simulating this system, with special consideration being given to the proof of stability within numerical regimes encapsulating a central differences approach to approximating numerical gradients. The stability analysis performed here reveals instabilities induced by the coupling of the structural binding and proliferative processes. The numerical results expound how the uPA system aids the tumour in invading the local stroma, whilst the inhibitor to this system may impede this behaviour and encourage a more sporadic pattern of invasion.PostprintPeer reviewe
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