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Browniantime Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and latticelimits solutions and fourth order SPDEs links
1.  Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, United States 
References:
[1] 
H. Allouba and E. Nane, Interacting timefractional and $\Delta^\nu$ PDEs systems via Browniantime and InversestableLévytime Brownian sheets,, 29 pages. Stoch. Dyn., (). doi: 10.1142/S0219493712500128. 
[2] 
H. Allouba, From Browniantime Brownian sheet to a fourth order and a KuramotoSivashinskyvariant interacting PDEs systems, Stoch. Anal. Appl., 29 (2011), 933950. 
[3] 
H. Allouba, A Browniantime excursion into fourthorder PDEs, linearized KuramotoSivashinsky, and BTPSPDEs on $\mathbb R_{+}$ × $\mathbb R^d$, Stoch. Dyn., 6 (2006), 521534. 
[4] 
H. Allouba, A linearized KuramotoSivashinsky PDE via an imaginaryBrowniantimeBrownianangle process, C. R. Math. Acad. Sci. Paris, 336 (2003), 309314. 
[5] 
H. Allouba, LKuramotoSivashinsky SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ via the imaginaryBrowniantimeBrownianangle representation,, In final preparation., (). 
[6] 
H. Allouba and J. Duan, SwiftHohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., (). 
[7] 
H. Allouba and J. A. Langa, Nonlinear KuramotoSivashinsky 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 ddependent variation,, In preparation., (). 
[9] 
H. Allouba, SDDEs limits solutions to sublinear reactiondiffusion SPDEs, Electron. J. Differential Equations, (2003), 21 pp. (electronic). 
[10] 
H. Allouba, Browniantime processes: the PDE connection II and the corresponding FeynmanKac formula, Trans. Amer. Math. Soc., 354 (2002), 46274637. (electronic). 
[11] 
H. Allouba and W. Zheng, Browniantime processes: the PDE connection and the halfderivative generator, Ann. Probab., 29 (2001), 17801795. 
[12] 
H. Allouba, SPDEs law equivalence and the compact support property: applications to the AllenCahn SPDE, C. R. Acad. Sci. Paris S. I Math, 331 (2000), 245250. 
[13] 
H. Allouba, Uniqueness in law for the AllenCahn SPDE via change of measure, C. R. Acad. Sci. Paris S. I Math, 330 (2000), 371376. 
[14] 
H. Allouba, Different types of SPDEs in the eyes of Girsanov's theorem, Stochastic Anal. Appl., 16 (1998), 787810. 
[15] 
H. Allouba, A nonnonstandard proof of Reimers' existence result for heat SPDEs, J. Appl. Math. Stochastic Anal., 11 (1998), 2941. 
[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 stablelike processes, Stochastic Process. Appl., 119 (2009), 11441167. 
[19] 
P. Carr and L. Cousot, A PDE approach to jumpdiffusions, Quant. Finance, 11 (2011), 3352. 
[20] 
R. Dalang, Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs, Electron. J. Probab, 4 (1999), 129. doi: 10.1214/EJP.v443. 
[21] 
R. Dalang and M. SanzSolé, Regularity of the sample paths of a class of secondorder spde's, J. Funct. Anal., 227 (2005), 304337. 
[22] 
R. Dalang and M. SanzSolé, HölderSobolev 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 CahnHilliard equation, Nonlinear Anal., 26 (1996), 241263. 
[24] 
R. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab., 14 (2004), 15291558. 
[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 localtime correspondence for stochastic partial differential equations, Trans. Amer. Math. Soc., 363 (2011), 24812515. 
[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), 281303. (electronic). 
[29] 
N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusions," NorthHolland Publishing Company, 1989. 
[30] 
I. Karatzas and S. Shreve, "Brownian Motion and Stochastic Calculus," SpringerVerlag, 1988. doi: 10.1007/9781468403022. 
[31] 
T. Kurtz, Particle representations for measurevalued population processes with spatially varying birth rates, Stochastic models (Ottawa, ON, 1998), CMS Conf. Proc., Amer. Math. Soc., Providence, RI., 26 (2000), 299317. 
[32] 
J. Le Gall, A pathvalued Markov process and its connections with partial differential equations, First European Congress of Mathematics, Vol. II (Paris, 1992). Progr. Math. Birkhäuser, 120 (1994), 185212. 
[33] 
M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab., 37 (2009), 9791007. doi: 10.1214/08AOP426. 
[34] 
E. Nane, Stochastic solutions of a class of Higher order Cauchy problems in $\mathbb R^d$, Stoch. Dyn., 10 (2010), 341366. doi: 10.1142/S021949371000298X. 
[35] 
M. Reimers, One dimensional stochastic partial differential equations and the branching measure diffusion, Probab. Theory Relat. Fields, 81 (1989), 319340. doi: 10.1007/BF00340057. 
[36] 
R. Sowers, Shorttime geometry of random heat kernels, (English summary) Mem. Amer. Math. Soc., 132 (1998), viii+130 pp. 
[37] 
R. Sowers, Multidimensional reactiondiffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 20712121. 
[38] 
D. Stroock and S. Varadhan, "Multidimensional Diffusion Processes," Reprint of the 1997 edition. Classics in Mathematics. SpringerVerlag, Berlin, 2006. xii+338 pp. ISBN: 9783540289982; 3540289984. 
[39] 
D. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. H. Poincare Probab. Statist, 33 (1997), 619649. 
[40] 
R. Temam, "InfiniteDimensional Dynamical Systems in Mechanics and Physics," Second edition. Applied Mathematical Sciences, 68. SpringerVerlag, New York, 1997. xxii+648 pp. ISBN: 038794866X 
[41] 
J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations," École d'été de Probabilités de SaintFlour 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), 383408. 
show all references
References:
[1] 
H. Allouba and E. Nane, Interacting timefractional and $\Delta^\nu$ PDEs systems via Browniantime and InversestableLévytime Brownian sheets,, 29 pages. Stoch. Dyn., (). doi: 10.1142/S0219493712500128. 
[2] 
H. Allouba, From Browniantime Brownian sheet to a fourth order and a KuramotoSivashinskyvariant interacting PDEs systems, Stoch. Anal. Appl., 29 (2011), 933950. 
[3] 
H. Allouba, A Browniantime excursion into fourthorder PDEs, linearized KuramotoSivashinsky, and BTPSPDEs on $\mathbb R_{+}$ × $\mathbb R^d$, Stoch. Dyn., 6 (2006), 521534. 
[4] 
H. Allouba, A linearized KuramotoSivashinsky PDE via an imaginaryBrowniantimeBrownianangle process, C. R. Math. Acad. Sci. Paris, 336 (2003), 309314. 
[5] 
H. Allouba, LKuramotoSivashinsky SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ via the imaginaryBrowniantimeBrownianangle representation,, In final preparation., (). 
[6] 
H. Allouba and J. Duan, SwiftHohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., (). 
[7] 
H. Allouba and J. A. Langa, Nonlinear KuramotoSivashinsky 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 ddependent variation,, In preparation., (). 
[9] 
H. Allouba, SDDEs limits solutions to sublinear reactiondiffusion SPDEs, Electron. J. Differential Equations, (2003), 21 pp. (electronic). 
[10] 
H. Allouba, Browniantime processes: the PDE connection II and the corresponding FeynmanKac formula, Trans. Amer. Math. Soc., 354 (2002), 46274637. (electronic). 
[11] 
H. Allouba and W. Zheng, Browniantime processes: the PDE connection and the halfderivative generator, Ann. Probab., 29 (2001), 17801795. 
[12] 
H. Allouba, SPDEs law equivalence and the compact support property: applications to the AllenCahn SPDE, C. R. Acad. Sci. Paris S. I Math, 331 (2000), 245250. 
[13] 
H. Allouba, Uniqueness in law for the AllenCahn SPDE via change of measure, C. R. Acad. Sci. Paris S. I Math, 330 (2000), 371376. 
[14] 
H. Allouba, Different types of SPDEs in the eyes of Girsanov's theorem, Stochastic Anal. Appl., 16 (1998), 787810. 
[15] 
H. Allouba, A nonnonstandard proof of Reimers' existence result for heat SPDEs, J. Appl. Math. Stochastic Anal., 11 (1998), 2941. 
[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 stablelike processes, Stochastic Process. Appl., 119 (2009), 11441167. 
[19] 
P. Carr and L. Cousot, A PDE approach to jumpdiffusions, Quant. Finance, 11 (2011), 3352. 
[20] 
R. Dalang, Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs, Electron. J. Probab, 4 (1999), 129. doi: 10.1214/EJP.v443. 
[21] 
R. Dalang and M. SanzSolé, Regularity of the sample paths of a class of secondorder spde's, J. Funct. Anal., 227 (2005), 304337. 
[22] 
R. Dalang and M. SanzSolé, HölderSobolev 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 CahnHilliard equation, Nonlinear Anal., 26 (1996), 241263. 
[24] 
R. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab., 14 (2004), 15291558. 
[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 localtime correspondence for stochastic partial differential equations, Trans. Amer. Math. Soc., 363 (2011), 24812515. 
[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), 281303. (electronic). 
[29] 
N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusions," NorthHolland Publishing Company, 1989. 
[30] 
I. Karatzas and S. Shreve, "Brownian Motion and Stochastic Calculus," SpringerVerlag, 1988. doi: 10.1007/9781468403022. 
[31] 
T. Kurtz, Particle representations for measurevalued population processes with spatially varying birth rates, Stochastic models (Ottawa, ON, 1998), CMS Conf. Proc., Amer. Math. Soc., Providence, RI., 26 (2000), 299317. 
[32] 
J. Le Gall, A pathvalued Markov process and its connections with partial differential equations, First European Congress of Mathematics, Vol. II (Paris, 1992). Progr. Math. Birkhäuser, 120 (1994), 185212. 
[33] 
M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab., 37 (2009), 9791007. doi: 10.1214/08AOP426. 
[34] 
E. Nane, Stochastic solutions of a class of Higher order Cauchy problems in $\mathbb R^d$, Stoch. Dyn., 10 (2010), 341366. doi: 10.1142/S021949371000298X. 
[35] 
M. Reimers, One dimensional stochastic partial differential equations and the branching measure diffusion, Probab. Theory Relat. Fields, 81 (1989), 319340. doi: 10.1007/BF00340057. 
[36] 
R. Sowers, Shorttime geometry of random heat kernels, (English summary) Mem. Amer. Math. Soc., 132 (1998), viii+130 pp. 
[37] 
R. Sowers, Multidimensional reactiondiffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 20712121. 
[38] 
D. Stroock and S. Varadhan, "Multidimensional Diffusion Processes," Reprint of the 1997 edition. Classics in Mathematics. SpringerVerlag, Berlin, 2006. xii+338 pp. ISBN: 9783540289982; 3540289984. 
[39] 
D. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. H. Poincare Probab. Statist, 33 (1997), 619649. 
[40] 
R. Temam, "InfiniteDimensional Dynamical Systems in Mechanics and Physics," Second edition. Applied Mathematical Sciences, 68. SpringerVerlag, New York, 1997. xxii+648 pp. ISBN: 038794866X 
[41] 
J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations," École d'été de Probabilités de SaintFlour 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), 383408. 
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