Recent Results on a Generalization of the Laplacian
DOI:
https://doi.org/10.5540/tema.2015.016.02.0131Abstract
In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a function$W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$, where each $W_k: \bb R \to \bb R$ is a right continuous with left limits and strictly increasing function.Using $W$, we construct the generalized laplacian $\mc L_W = \sum_{i=1}^d \partial_{x_i}\partial_{W_i}$, where $\partial_{W_i}$ is a generalized differentialoperator induced by the function $W_i$.We present results on spectral properties of $\mc L_W$, Sobolev spaces induced by $\mc L_W$ ($W$-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations andstochastic homogenization.References
bibitem{dyn2} E. B. Dynkin, {em Markov processes}. Volume II.
Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences], 122. Springer-Verlag, Berlin,
bibitem{E} L. Evans, {em Partial differential equation}. AMS, 1998.
bibitem{f} A. Faggionato, {em Random walks and exclusion processs among random
conductances on random infinite clusters: Homogenization and hydrodynamic limit}.arXiv:0704.3020v3 .
bibitem{fjl} A. Faggionato, M. Jara, C. Landim, {em Hydrodynamic
behavior of one dimensional subdiffusive exclusion processes with
random conductances}. Probability Theory and Related Fields. v. 144, p. 633-667, 2009.
bibitem{fsv} J. Farfan, A. B. Simas, F. J. Valentim, {em Equilibrium fluctuations for exclusion processes with conductances in random environments}
, Stochastic Processes and their Applications, v. 120, p. 1535-1562, 2010.
bibitem{f1} W. Feller, {em On Second Order Differential Operators}. Annals of Mathematics, 61,n.1, 90-105, (1955).
bibitem{f2} W. Feller, {em Generalized second order differential operators and their lateral conditions}. Illinois J. Math. Vol. 1, Issue 4, 459-504, (1957).
%bibitem{feller} W. Feller. {em On second order differential operators.} Ann. Math., 55, 468-519. 1952.
bibitem{TC} T. Franco, C. Landim, { em Hydrodynamic limit of gradient exclusion processes with conductances}. Archive for Rational Mechanics and Analysis (Print), v. 195, p. 409-439, 2009.
bibitem{gj} P. Gonçalves, M. Jara. {em Scaling Limits for Gradient Systems in Random Environment.} J. Stat. Phys., 131, 691-716. 2008.
bibitem{kp} G. Kallianpur, V. Perez-Abreu, {em Stochastic Evolution equations Driven by Nuclear-space-Valued Martingale}. Applied Mathematics and Optimization. 17, 237-272. 1988.
bibitem{kl} C. Kipnis, C. Landim, {em Scaling limits of interacting
particle systems}. Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences], 320.
Springer-Verlag, Berlin, 1999.
bibitem{jl} M. Jara, C. Landim, {em
Quenched nonequilibrium central limit theorem for a tagged particle
in the exclusion process with bond disorder}. arXiv: math/0603653. Ann. Inst. H. Poincar'e,
Probab. Stat. 44, 341-361, (2008).
bibitem{jlt} Jara, M., Landim, C., Teixeira, A., {em Quenched scaling limits of trap models}. Annals of Probability, v. 39, p. 176-223, 2011.
bibitem{liggett} T.M. Liggett. emph{Interacting Particle Systems}. Springer-Verlag, New York. 1985.
bibitem{lo1} J.-U. L"obus, {em Generalized second order differential
operators}. Math. Nachr. {bf 152}, 229-245 (1991).
bibitem{m} P. Mandl, {em Analytical treatment of one-dimensional
{M}arkov processes}, Grundlehren der mathematischen
Wissenschaften, 151. Springer-Verlag, Berlin, 1968.
bibitem{papa} G. Papanicolaou, S.R.S. Varadhan, emph{Boundary value problems with rapidly oscillating random coefficients}, Seria Coll. Math. Soc. Janos Bolyai vol. 27, North-Holland (1979).
bibitem{pr} A. Piatnitski, E. Remy, {em Homogenization of Elliptic Difference Operators}, SIAM J. Math. Anal. Vol.33, pp. 53-83, (2001).
bibitem{SV} A.B. Simas, F.J. Valentim, {em $W$-Sobolev spaces}. Journal of Mathematical Analysis and Applications V. 382, 1, 214-230, 2011.
bibitem{SVII} A.B. Simas, F.J. Valentim,{em Homogenization of second-order generalized elliptic operators}, submitted for publication.
bibitem{spitzer} F. Spitzer. {em Interacting of Markov processes}. Adv. Math, 5, 246-290. 1970.
bibitem{v} F.J. Valentim, {em Hydrodynamic limit of a $d$-dimensional exclusion process with conductances.}.Ann. Inst. H. Poincar'e Probab. Statist. V. 48, 1, 188-211, 2012.
bibitem{z} E. Zeidler, {em Applied Functional Analysis. Applications
to Mathematical Physics.}. Applied Mathematical Sciences, 108. Springer-Verlag, New York, 1995.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.