Long time behavior of heat kernels of operators with unbounded drift terms
type: Item
instance of: scholarly article
title: Long time behavior of heat kernels of operators with unbounded drift terms
zbMATH Open document ID: 1213.35110
author: E. M. Ouhabaz
author: Giorgio Metafune
author: Diego Pallara
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-02-22T00:00:00Z
review text: The authors consider the differential operator: \[ A=- \text{div}(a(x)\nabla)+ b\cdot\nabla \] on \(L^2(\mathbb{R}^d)\), under the following assumptions on the coefficients: \[ a_{kj}\in L^\infty(\mathbb{R}^d),\;\exists\upsilon> 0: \sum^d_{j,k=1} a_{kj}(x) \xi_j\xi_k\geq \upsilon|\xi|^2\;\forall x,\xi\in\mathbb{R}^d, \] \[ b= (b_j)\in C^1(\mathbb{R}^d,\mathbb{R}^d)\text{ with div\,}b\leq w, \] where \(w\) is a constant in \(\mathbb{R}\). By the use of the sesquilinear forms the authors obtain a realization of \(A\), which is (minus) the generator of a strongly continuous semigroup on \(L^2(\mathbb{R}^d)\). The corresponding semigroup is given by an integral kernel. The time behaviour of the integral kernel is studied. The estimates on its decay at infinity are obtained. For symmetric diffusion coefficients a local lower estimate is given. An example with the Ornstein-Uhlenbeck operator is considered.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5854867
zbMATH Keywords: parabolic operators
zbMATH Keywords: heat kernels
zbMATH Keywords: strongly continuous semigroups
zbMATH Keywords: Ornstein-Uhlenbeck operator
reviewed by: Miklavž Mastinšek
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.10.023
OpenAlex ID: W2036080632
cites work: Gaussian estimates for elliptic operators with unbounded drift
cites work: ON REGULARITY OF TRANSITION PROBABILITIES AND INVARIANT MEASURES OF SINGULAR DIFFUSIONS UNDER MINIMAL CONDITIONS
cites work: Conditional transformation of drift formula and potential theory for \(\Delta +b(\cdot)\cdot \nabla\)
cites work: On the Ornstein-Uhlenbeck operator in spaces of continuous functions
cites work: Q4041031
cites work: Analytic semigroups generated by elliptic operators in \(L^ 1\) and parabolic equations
cites work: Q4514190
cites work: Global properties of generalized Ornstein--Uhlenbeck operators on \(L^p(\mathbb R^N, \mathbb R^N)\)with more than linearly growing coefficients
cites work: Q3943431
cites work: Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients
cites work: Extra regularity for parabolic equations with drift terms
cites work: Q5480574
cites work: Functional calculus for the Ornstein-Uhlenbeck operator
cites work: Q5489500
cites work: Global Properties of Transition Probabilities of Singular Diffusions
cites work: Feller semigroups on \(\mathbb{R}^N\)
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q5480350
cites work: A Harnack inequality for the equation \(\nabla (a \nabla u) + b \nabla u = 0\), when \(| b | \in K_{n+1}\)
cites work: Gaussian bounds for the fundamental solutions of \(\nabla(A\nabla u)+B\nabla u-u_ t=0\)
DOI: 10.1016/J.JMAA.2010.10.023
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description: scientific article; zbMATH DE number 5854867