The Experts below are selected from a list of 180936 Experts worldwide ranked by ideXlab platform
Zoran Vondraček - One of the best experts on this subject based on the ideXlab platform.
-
sharp Green Function estimates for delta delta alpha 2 in c 1 1 open sets and their applications
Illinois Journal of Mathematics, 2010Co-Authors: Zhenqing Chen, Renming Song, Zoran VondračekAbstract:We consider a family of pseudo differential operators {Δ+ aΔ; a ∈ [0,1]} on R that evolves continuously from Δ to Δ+Δ, where d≥ 1 and α ∈ (0,2). It gives rise to a family of Levy processes {X, a ∈ [0,1]}, where X is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green Function of the process X killed upon exiting a bounded C open set D ⊂ R. Our estimates are uniform in a ∈ (0,1] and taking a→ 0 recovers the Green Function estimates for Brownian motion in D. As a consequence of the Green Function estimates for X in D, we identify both the Martin boundary and the minimal Martin boundary of D with respect to X with its Euclidean boundary. Finally, sharp Green Function estimates are derived for certain Levy processes which can be obtained as perturbations of X. Received December 11, 2009; received in final form October 13, 2011. The first author was partially supported by NSF Grants DMS-0906743 and DMR1035196. The second author supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (0409-20110087). The fourth author was partially supported by the MZOS Grant 037-0372790-2801 of the Republic of Croatia. 2010 Mathematics Subject Classification. Primary 31A20, 31B25, 60J45. Secondary 47G20, 60J75, 31B05. 981 c ©2012 University of Illinois 982 Z.-Q. CHEN ET AL.
-
sharp Green Function estimates for delta delta alpha 2 in c 1 1 open sets and their applications
arXiv: Probability, 2009Co-Authors: Zhenqing Chen, Renming Song, Zoran VondračekAbstract:We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives rise to a family of L\'evy processes \{$X^a, a\in [0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green Function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset\R^d$. As a consequence, we identify the Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green Function estimates are derived for certain L\'evy processes which can be obtained as perturbations of $X^a$.
-
Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Lévy process in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaacq% WIDesOdaahaaWcbeqaaiaadsgaaaaaaa!3A16! $$\mathbb{R}^{d} $$ , % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaaca% WGKbGaeyyzImRaaG4maaaa!3AFC! $$d \geqslant 3$$ , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X .
-
Green Function estimates and harnack inequality for subordinate brownian motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Levy process in\(\mathbb{R}^{d} \), \(d \geqslant 3\), obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Levy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Levy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X.
L. Shuster - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotics on the Diagonal of the Green Function of a Sturm–Liouville Operator and its Applications
Journal of the London Mathematical Society, 2000Co-Authors: N. A. Chernyavskaya, L. ShusterAbstract:Asymptotics on the diagonal of the Green Function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm–Liouville operator are obtained.
-
Estimates for the Green Function of a general Sturm-Liouville operator and their applications
Proceedings of the American Mathematical Society, 1999Co-Authors: N. A. Chernyavskaya, L. ShusterAbstract:For a general Sturm-Liouville operator with nonnegative coefficients, we obtain two-sided estimates for the Green Function, sharp by order on the diagonal.
Renming Song - One of the best experts on this subject based on the ideXlab platform.
-
sharp Green Function estimates for delta delta alpha 2 in c 1 1 open sets and their applications
Illinois Journal of Mathematics, 2010Co-Authors: Zhenqing Chen, Renming Song, Zoran VondračekAbstract:We consider a family of pseudo differential operators {Δ+ aΔ; a ∈ [0,1]} on R that evolves continuously from Δ to Δ+Δ, where d≥ 1 and α ∈ (0,2). It gives rise to a family of Levy processes {X, a ∈ [0,1]}, where X is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green Function of the process X killed upon exiting a bounded C open set D ⊂ R. Our estimates are uniform in a ∈ (0,1] and taking a→ 0 recovers the Green Function estimates for Brownian motion in D. As a consequence of the Green Function estimates for X in D, we identify both the Martin boundary and the minimal Martin boundary of D with respect to X with its Euclidean boundary. Finally, sharp Green Function estimates are derived for certain Levy processes which can be obtained as perturbations of X. Received December 11, 2009; received in final form October 13, 2011. The first author was partially supported by NSF Grants DMS-0906743 and DMR1035196. The second author supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (0409-20110087). The fourth author was partially supported by the MZOS Grant 037-0372790-2801 of the Republic of Croatia. 2010 Mathematics Subject Classification. Primary 31A20, 31B25, 60J45. Secondary 47G20, 60J75, 31B05. 981 c ©2012 University of Illinois 982 Z.-Q. CHEN ET AL.
-
sharp Green Function estimates for delta delta alpha 2 in c 1 1 open sets and their applications
arXiv: Probability, 2009Co-Authors: Zhenqing Chen, Renming Song, Zoran VondračekAbstract:We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives rise to a family of L\'evy processes \{$X^a, a\in [0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green Function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset\R^d$. As a consequence, we identify the Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green Function estimates are derived for certain L\'evy processes which can be obtained as perturbations of $X^a$.
-
Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Lévy process in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaacq% WIDesOdaahaaWcbeqaaiaadsgaaaaaaa!3A16! $$\mathbb{R}^{d} $$ , % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaaca% WGKbGaeyyzImRaaG4maaaa!3AFC! $$d \geqslant 3$$ , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X .
-
Green Function estimates and harnack inequality for subordinate brownian motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Levy process in\(\mathbb{R}^{d} \), \(d \geqslant 3\), obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Levy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Levy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X.
-
sharp bounds on the density Green Function and jumping Function of subordinate killed bm
Probability Theory and Related Fields, 2004Co-Authors: Renming SongAbstract:Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2-sta- ble subordinator gives rise to a process Zt whose infinitesimal generator is −(−� |D) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green Function and jumping Function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.
N. A. Chernyavskaya - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotics on the Diagonal of the Green Function of a Sturm–Liouville Operator and its Applications
Journal of the London Mathematical Society, 2000Co-Authors: N. A. Chernyavskaya, L. ShusterAbstract:Asymptotics on the diagonal of the Green Function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm–Liouville operator are obtained.
-
Estimates for the Green Function of a general Sturm-Liouville operator and their applications
Proceedings of the American Mathematical Society, 1999Co-Authors: N. A. Chernyavskaya, L. ShusterAbstract:For a general Sturm-Liouville operator with nonnegative coefficients, we obtain two-sided estimates for the Green Function, sharp by order on the diagonal.
Murali Rao - One of the best experts on this subject based on the ideXlab platform.
-
Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Lévy process in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaacq% WIDesOdaahaaWcbeqaaiaadsgaaaaaaa!3A16! $$\mathbb{R}^{d} $$ , % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaaca% WGKbGaeyyzImRaaG4maaaa!3AFC! $$d \geqslant 3$$ , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X .
-
Green Function estimates and harnack inequality for subordinate brownian motions
Potential Analysis, 2006Co-Authors: Murali Rao, Renming Song, Zoran VondračekAbstract:Let X be a Levy process in\(\mathbb{R}^{d} \), \(d \geqslant 3\), obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Levy process with no continuous component. We study the asymptotic behavior of the Green Function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein Function we also describe the asymptotic behavior of the Green Function at infinity. With an additional assumption on the Levy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic Functions of X.
