Recent Results on a Generalization of the Laplacian

Authors

  • Alexandre B Simas Universidade Federal da Paraíba
  • Fábio J Valentim Universidade Federal do Espírito Santo

DOI:

https://doi.org/10.5540/tema.2015.016.02.0131

Abstract

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.

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Published

2015-09-07

How to Cite

Simas, A. B., & Valentim, F. J. (2015). Recent Results on a Generalization of the Laplacian. Trends in Computational and Applied Mathematics, 16(2), 131. https://doi.org/10.5540/tema.2015.016.02.0131

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