Citation: |
[1] |
H. Allouba and E. Nane, Interacting time-fractional and $\Delta^\nu$ PDEs systems via Brownian-time and Inverse-stable-Lévy-time Brownian sheets, 29 pages. Stoch. Dyn. doi: 10.1142/S0219493712500128. |
[2] |
H. Allouba, From Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky-variant interacting PDEs systems, Stoch. Anal. Appl., 29 (2011), 933-950. |
[3] |
H. Allouba, A Brownian-time excursion into fourth-order PDEs, linearized Kuramoto-Sivashinsky, and BTP-SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$, Stoch. Dyn., 6 (2006), 521-534. |
[4] |
H. Allouba, A linearized Kuramoto-Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process, C. R. Math. Acad. Sci. Paris, 336 (2003), 309-314. |
[5] |
H. Allouba, L-Kuramoto-Sivashinsky SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ via the imaginary-Brownian-time-Brownian-angle representation, In final preparation. |
[6] |
H. Allouba and J. Duan, Swift-Hohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors, In preparation. |
[7] |
H. Allouba and J. A. Langa, Nonlinear Kuramoto-Sivashinsky type SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors, In preparation. |
[8] |
H. Allouba and Y. Xiao, BTBM SIEs on $\mathbb R_{+}$ × $\mathbb R^d$: modulus of continuity, hitting probabilities, Hausdorff dimensions, and d-dependent variation, In preparation. |
[9] |
H. Allouba, SDDEs limits solutions to sublinear reaction-diffusion SPDEs, Electron. J. Differential Equations, (2003), 21 pp. (electronic). |
[10] |
H. Allouba, Brownian-time processes: the PDE connection II and the corresponding Feynman-Kac formula, Trans. Amer. Math. Soc., 354 (2002), 4627-4637. (electronic). |
[11] |
H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795. |
[12] |
H. Allouba, SPDEs law equivalence and the compact support property: applications to the Allen-Cahn SPDE, C. R. Acad. Sci. Paris S. I Math, 331 (2000), 245-250. |
[13] |
H. Allouba, Uniqueness in law for the Allen-Cahn SPDE via change of measure, C. R. Acad. Sci. Paris S. I Math, 330 (2000), 371-376. |
[14] |
H. Allouba, Different types of SPDEs in the eyes of Girsanov's theorem, Stochastic Anal. Appl., 16 (1998), 787-810. |
[15] |
H. Allouba, A non-nonstandard proof of Reimers' existence result for heat SPDEs, J. Appl. Math. Stochastic Anal., 11 (1998), 29-41. |
[16] |
R. Bass, "Probabilistic Techniques in Analysis," Springer, Berlin Heidelberg New York, 1995. |
[17] |
R. Bass, "Diffusions and Elliptic Operators," Springer, Berlin Heidelberg New York, 1997. |
[18] |
R. Bass and H. Tang, The martingale problem for a class of stable-like processes, Stochastic Process. Appl., 119 (2009), 1144-1167. |
[19] |
P. Carr and L. Cousot, A PDE approach to jump-diffusions, Quant. Finance, 11 (2011), 33-52. |
[20] |
R. Dalang, Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs, Electron. J. Probab, 4 (1999), 1-29.doi: 10.1214/EJP.v4-43. |
[21] |
R. Dalang and M. Sanz-Solé, Regularity of the sample paths of a class of second-order spde's, J. Funct. Anal., 227 (2005), 304-337. |
[22] |
R. Dalang and M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension 3, Mem. Amer. Math. Soc., 199 (2009), vi+70 pp. |
[23] |
G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal., 26 (1996), 241-263. |
[24] |
R. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab., 14 (2004), 1529-1558. |
[25] | |
[26] |
J. Duan and W. Wei, "Effective Dynamics of Stochasticpartial Differential Euations," To appear. |
[27] |
M. Foondun, and D. Khoshnevisan and E. Nualart, A local-time correspondence for stochastic partial differential equations, Trans. Amer. Math. Soc., 363 (2011), 2481-2515. |
[28] |
N. L. Garcia and T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281-303. (electronic). |
[29] |
N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusions," North-Holland Publishing Company, 1989. |
[30] |
I. Karatzas and S. Shreve, "Brownian Motion and Stochastic Calculus," Springer-Verlag, 1988.doi: 10.1007/978-1-4684-0302-2. |
[31] |
T. Kurtz, Particle representations for measure-valued population processes with spatially varying birth rates, Stochastic models (Ottawa, ON, 1998), CMS Conf. Proc., Amer. Math. Soc., Providence, RI., 26 (2000), 299-317. |
[32] |
J. Le Gall, A path-valued Markov process and its connections with partial differential equations, First European Congress of Mathematics, Vol. II (Paris, 1992). Progr. Math. Birkhäuser, 120 (1994), 185-212. |
[33] |
M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab., 37 (2009), 979-1007.doi: 10.1214/08-AOP426. |
[34] |
E. Nane, Stochastic solutions of a class of Higher order Cauchy problems in $\mathbb R^d$, Stoch. Dyn., 10 (2010), 341-366.doi: 10.1142/S021949371000298X. |
[35] |
M. Reimers, One dimensional stochastic partial differential equations and the branching measure diffusion, Probab. Theory Relat. Fields, 81 (1989), 319-340.doi: 10.1007/BF00340057. |
[36] |
R. Sowers, Short-time geometry of random heat kernels, (English summary) Mem. Amer. Math. Soc., 132 (1998), viii+130 pp. |
[37] |
R. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. |
[38] |
D. Stroock and S. Varadhan, "Multidimensional Diffusion Processes," Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006. xii+338 pp. ISBN: 978-3-540-28998-2; 3-540-28998-4. |
[39] |
D. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. H. Poincare Probab. Statist, 33 (1997), 619-649. |
[40] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. xxii+648 pp. ISBN: 0-387-94866-X |
[41] |
J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations," École d'été de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180. Springer, New York. 1986. |
[42] |
Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion, J. Theoret. Probab., 11 (1998), 383-408. |