On lp-theory for stochastic parabolic integro-differential equations

Stochastics and Partial Differential Equations: Analysis and Computations

Volumn 1, Issue 2, 2013, Pages 282-324

  Mikulevicius, R ^a   Pragarauskas, H ^b  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Levy processes; Lp theory; Stochastic integrodifferential equations; Zakai equation

Indexed keywords

EID: 84890874700     PISSN: 21940401     EISSN: 2194041X     Source Type: Journal    
DOI: 10.1007/s40072-013-0008-0     Document Type: Article

References (20)

1. Bergh, J., Löfstrom, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)
2. Banuelos, R., Bogdan, K.: Levy processes and Fourier multipliers. J. Funct. Anal. 250, 197–213 (2007)
3. Chang, T., Lee, K.: On a stochastic partial differential equation with a fractional Laplacian operator. Stoch. Process. Appl. 122, 3288–3311 (2012)
4. Dellacherie, C., Meyer, P.: Probabilities and Potential. North-Holland Mathematics Studies 29. North-Holland, Amsterdam (1978)
5. p-estimates for a class of non-local elliptic equations. J. Funct. Anal. 262(3), 1166–1199 (2012)
6. Grigelionis, B.: Reduced stochastic equations of the nonlinear filtering of random processes. Lithuan. Math. J. 16, 348–358 (1976)
7. Jacod, J.: Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)
8. p-theory of stochastic parabolic equations with the random fractional Laplacian driven by Levy processes. Stoch. Trans. Am. Math. Soc. 363(9), 5021–5055 (2012)
9. p-theory of stochastic partial differential equations. SIAM J. Math. Anal. 27, 313–340 (1996)
10. Mikulevicius, R.: Properties of solutions of stochastic differential equations. Lithuan. Math. J. 23(4), 367–376 (1983)
11. Mikulevičius, R., Pragarauskas, H.: On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. Lithuan. Math. J. 32(2), 238–264 (1992)
12. Mikulevičius, R., Pragarauskas, H.: On Hölder solutions of the integro-differential Zakai equation. Stoch. Proc. Appl. 119, 3319–3355 (2009)
13. Mikulevičius, R., Pragarauskas, H.: On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem. arXiv:1112.4467v4 [math.AP] (2011)
14. Mikulevičius, R., Pragarauskas, H.: Model problem for integro-differential Zakai equation with discontinuous observation process. Math. Appl. Optim. 64, 37–69 (2011)
15. p-estimates of some singular integrals related to jump processes. SIAM J. Math. Anal. 44(4), 2305–2328 (2012)
16. p-theory for systems of SPDEs. Electron. J. Probab. 6(12), 1-35 (2001)
17. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, London (1994)
18. Strichartz, R.S.: Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060 (1967)
19. Triebel, H.: Theory of Function Spaces I. Birkhaueser, Basel (1983)
20. Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrsch. 11, 230–243 (1969)