Second order elliptic partial differential equations driven by Lévy white noise
Volume 8, Issue 2 (2021), pp. 179–207
Pub. online: 22 June 2021
Type: Research Article
Open Access
Open Access
Received
11 February 2021
11 February 2021
Revised
21 May 2021
21 May 2021
Accepted
21 May 2021
21 May 2021
Published
22 June 2021
22 June 2021
Abstract
This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. First, an existence theorem for integral transforms of Lévy white noise is derived and the existence of generalized and mild solutions of second order elliptic partial differential equations is proved. Further, the generalized electric Schrödinger operator for different potential functions V is discussed.
References
Barndorff-Nielsen, O.E., Basse-O’Connor, A.: Quasi Ornstein-Uhlenbeck processes. Bernoulli 17, 916–941 (2011). MR2817611. https://doi.org/10.3150/10-BEJ311
Berger, D.: Lévy driven Carma generalized processes and stochastic partial differential equations. Stoch. Process. Appl. 130, 5865–5887 (2020). MR4140021. https://doi.org/10.1016/j.spa.2020.04.009
Dalang, R.C., Humeau, T.: Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noise. Electron. J. Probab. 24, 1–28 (2019). MR3978210. https://doi.org/10.1214/19-EJP317
Davey, B., Hill, J., Mayboroda, S.: Fundamental matrices and Green matrices for non-homogeneous elliptic systems. Publ. Mat. 2, 537–614 (2018). MR3815288. https://doi.org/10.5565/PUBLMAT6221807
Fageot, J., Humeau, T.: The domain of definition of the Lévy white noise. Stoch. Process. Appl. 135, 75–102 (2021). MR4222403. https://doi.org/10.1016/j.spa.2021.01.007
Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press, New York and London (1964). MR0173945
Grafakos, L.: Classical Fourier Analysis, Second Edition. Springer, New York (2008). MR2445437
Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Springer, Berlin Heidelberg (2003). MR1996773. https://doi.org/10.1007/978-3-642-61497-2
Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 17, 43–77 (1963). MR0161019
Mayboroda, S., Poggi, B.: Exponential decay estimates for fundamental solutions of Schrödinger-type operators. Trans. Am. Math. Soc. 372, 4313–4357 (2019). MR4009431. https://doi.org/10.1090/tran/7817
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Cambridge University Press (2007). MR2356959. https://doi.org/10.1017/CBO9780511721373
Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 451–487 (1989). MR1001524. https://doi.org/10.1007/BF00339998
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2013). MR3185174
Shen, Z.: On fundamental solutions of generalized Schrödinger operators. J. Funct. Anal. 167, 521–567 (1999). MR1716207. https://doi.org/10.1006/jfan.1999.3455
Stein, E.: Harmonic Analysis. Princeton University Press (1993). MR1232192
van Putten, M.: Maxwell’s equations in divergence form for general media with applications to MHD. Commun. Math. Phys. 141, 63–77 (1991). MR1133260
Walsh, J.B.: An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour 1984, 265–439 (1986). MR0876085. https://doi.org/10.1007/BFb0074920
