Second order elliptic partial differential equations driven by Lévy white noise. (English) Zbl 1477.60094

In this paper, the authors deal with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise whose solutions are defined as a generalized random process. 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 are proved. Further, the authors discuss the generalized electric Schrödinger operator for different potential functions.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H40 White noise theory
35J15 Second-order elliptic equations
35J10 Schrödinger operator, Schrödinger equation
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[1] Barndorff-Nielsen, O.E., Basse-O’Connor, A.: Quasi Ornstein-Uhlenbeck processes. Bernoulli17, 916-941 (2011). MR2817611. · Zbl 1233.60020
[2] Berger, D.: Lévy driven Carma generalized processes and stochastic partial differential equations. Stoch. Process. Appl.130, 5865-5887 (2020). MR4140021. · Zbl 1455.60070
[3] 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. · Zbl 1447.60097
[4] Davey, B., Hill, J., Mayboroda, S.: Fundamental matrices and Green matrices for non-homogeneous elliptic systems. Publ. Mat.2, 537-614 (2018). MR3815288. · Zbl 1400.35098
[5] Fageot, J., Humeau, T.: The domain of definition of the Lévy white noise. Stoch. Process. Appl.135, 75-102 (2021). MR4222403. · Zbl 1469.60113
[6] Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press, New York and London (1964). MR0173945
[7] Grafakos, L.: Classical Fourier Analysis, Second Edition. Springer, New York (2008). MR2445437 · Zbl 1220.42001
[8] Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Springer, Berlin Heidelberg (2003). MR1996773. · Zbl 1028.35001
[9] 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 · Zbl 0116.30302
[10] Mayboroda, S., Poggi, B.: Exponential decay estimates for fundamental solutions of Schrödinger-type operators. Trans. Am. Math. Soc.372, 4313-4357 (2019). MR4009431. · Zbl 1432.35065
[11] Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Cambridge University Press (2007). MR2356959. · Zbl 1205.60122
[12] Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields82, 451-487 (1989). MR1001524. · Zbl 0659.60078
[13] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2013). MR3185174 · Zbl 1287.60003
[14] Shen, Z.: On fundamental solutions of generalized Schrödinger operators. J. Funct. Anal.167, 521-567 (1999). MR1716207. · Zbl 0936.35051
[15] Stein, E.: Harmonic Analysis. Princeton University Press (1993). MR1232192
[16] van Putten, M.: Maxwell’s equations in divergence form for general media with applications to MHD. Commun. Math. Phys.141, 63-77 (1991). MR1133260 · Zbl 0736.76071
[17] Walsh, J.B.: An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour1984, 265-439 (1986). MR0876085. · Zbl 0608.60060
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