Functional SPDE with Multiplicative Noise and Dini Drift
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 519-537.

Dans cet article, nous établissons l’existence, l’unicité et la non-explosion de la solution douce pour une classe d’équations aux dérivées partielles stochastiques semi-linéaires dont le bruit est multiplicatif et le drift satisfait la condition de Dini. Dans le cas de dimension finie et du temps de retard borné, nous montrons l’inégalité de Harnack logarithmique et une estimée de gradient dans L 2 pour la solution douce. Comme le semi-groupe Markovien est associé à la solution fonctionnelle de l’équation, nous devons étudier l’analyse sur l’espace des chemins des solutions définies sur l’intervalle du temps de retard.

Existence, uniqueness and non-explosion of the mild solution are proved for a class of semi-linear functional SPDEs with multiplicative noise and Dini continuous drifts. In the finite-dimensional and bounded time delay setting, the log-Harnack inequality and L 2 -gradient estimate are derived. As the Markov semigroup is associated to the functional solution of the equation, one needs to make analysis on the path space of the solution in the time interval of delay.

Publié le :
DOI : https://doi.org/10.5802/afst.1544
Classification : 60H15,  60B10
Mots clés : Functional SPDE, Dini continuity, time delay, log-Harnack inequality, gradient estimate
@article{AFST_2017_6_26_2_519_0,
     author = {Huang, Xing and Wang, Feng-Yu},
     title = {Functional SPDE with Multiplicative Noise and Dini Drift},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {519--537},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     doi = {10.5802/afst.1544},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1544/}
}
Huang, Xing; Wang, Feng-Yu. Functional SPDE with Multiplicative Noise and Dini Drift. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 519-537. doi : 10.5802/afst.1544. http://www.numdam.org/articles/10.5802/afst.1544/

[1] Arnaudon, Marc; Thalmaier, Anton; Wang, Feng-Yu Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math., Volume 130 (2006) no. 3, pp. 223-233 | Article

[2] Arnaudon, Marc; Thalmaier, Anton; Wang, Feng-Yu Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stochastic Processes Appl., Volume 119 (2009) no. 10, pp. 3653-3670 | Article

[3] Arnaudon, Marc; Thalmaier, Anton; Wang, Feng-Yu Equivalent Harnack and gradient inequalities for pointwise curvature lower bound, Bull. Sci. Math., Volume 138 (2014) no. 5, pp. 643-655 | Article

[4] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages

[5] Da Prato, Giuseppe; Zabczyk, Jerzy Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, 44, Cambridge University Press, 1992, xviii+454 pages

[6] Es-Sarhir, Abdelhadi; von Renesse, Max-K.; Scheutzow, Michael Harnack inequality for functional SDEs with bounded memory, Electron. Commun. Probab., Volume 14 (2009), pp. 560-565 (electronic only) | Article

[7] Ondreját, Martin Uniqueness for stochastic evolution equations in Banach spaces, Diss. Math, Volume 426 (2004), pp. 1-63

[8] Prévôt, Claudia; Röckner, Michael A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, 2007, vi+144 pages

[9] Priola, Enrico; Wang, Feng-Yu Gradient estimates for diffusion semigroups with singular coefficients, J. Funct. Anal., Volume 236 (2006) no. 1, pp. 244-264 | Article

[10] Röckner, Michael; Wang, Feng-Yu Log-Harnack Inequality for Stochastic differential equations in Hilbert spaces and its consequences, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 13 (2010) no. 1, pp. 27-37 | Article

[11] Shao, Jinghai; Wang, Feng-Yu; Yuan, Chenggui Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients, Electron. J. Probab., Volume 17 (2012) (Paper no 100, 18 pages, electronic only) | Article

[12] Wang, Feng-Yu Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl., Volume 94 (2010) no. 3, pp. 304-321 | Article

[13] Wang, Feng-Yu Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab., Volume 39 (2011) no. 4, pp. 1449-1467 | Article

[14] Wang, Feng-Yu Harnack inequalities for stochastic partial differential equations, SpringerBriefs in Mathematics, Springer, 2013, x+125 pages

[15] Wang, Feng-Yu Analysis for Diffusion Processes on Riemannian Manifolds, Advanced Series on Statistical Science and Applied Probability, 18, World Scientific, 2014, xii+379 pages

[16] Wang, Feng-Yu Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differ. Equations, Volume 260 (2016) no. 3, pp. 2792-2829 | Article

[17] Wang, Feng-Yu; Zhang, Tu-Sheng Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients, J. Math. Anal. Appl., Volume 365 (2010) no. 1, pp. 1-11 | Article

[18] Yamada, Toshio; Watanabe, Shinzo On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., Volume 11 (1971), pp. 155-167 | Article