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February  2013, 33(2): 413-463. doi: 10.3934/dcds.2013.33.413

Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links

Hassan Allouba 1,

1. 

Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, United States

Received  January 2011 Revised  June 2012 Published  September 2012

We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density,$ \mathbb{K}^{{BTBM}^d}_{t;x,y}$ , on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE: \begin{equation} U(t,x)=\int_{{\mathbb R}^{d}}{{\mathbb K}}^{\text{BTBM}^d}_{t;x,y} u_0(y) dy+ \int_{{\mathbb R}^{d}}\int_0^t{{\mathbb K}}^{\text{BTBM}^d}_{t-s;x,y} a(U(s,y))\mathscr W(ds\times dy), (0.1) \end{equation} which we recently introduced in [3].In sharp contrast to traditional second order heat-operator-based SPDEs---whose real-valued mild solutions are confined to $d=1$---we prove the existence of solutions to (0.1) in $d=1,2,3$ with dimension-dependent and striking Hölder regularity, under both less than Lipschitz and Lipschitz conditions on $a$. In space, we show an unprecedented nearly local Lipschitz regularity for $d=1,2$---roughly, $U$ is spatially twice as regular as the Brownian sheet in these dimensions---and we prove nearly local Hölder $1/2$ regularity in $d=3$. In time, our solutions are locally $\gamma$-Hölder continuous with exponent $γ∈(0, \frac{4-d}{8})$,$1≤d≤3$. To investigate (0.1) under less than Lipschitz conditions on $a$, we (a) introduce the Brownian-time random walk---a special case of lattice processes we call Brownian-time chains---and we use it to formulate the spatial lattice version of (0.1); and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including (0.1) and the mild forms of many SPDEs of different orders on the lattice. Solutions to (0.1) are defined as limits of their lattice version. Along the way, we prove interesting aspects of Brownian-time random walk, including a fourth order differential-difference equation connection. We also prove existence, pathwise uniqueness, and the same Hölder regularity for (0.1), without discretization, in the Lipschitz case. The SIE (0.1) is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that (0.1) is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $ \mathbb {K}^{{BTBM}^d}_{t;x,y}$ , by the intimately connected kernel of our recently-introduced imaginary-Brownian-time-Brownian-angle process (IBTBAP), (0.1) becomes the mild form of a Kuramoto-Sivashinsky (KS) SPDE with linearized PDE part. Ideas and tools developed here are adapted in separate papers to give an entirely new approach, via our explicit IBTBAP representation, to many linear and nonlinear KS-type SPDEs in multi-spatial dimensions.
Keywords: Brownian-time chains, fourth order SPDEs, kernel stochastic integral equations, discretized SPDEs, $2$-Brownian-times random walk, lattice limits solutions, BTRW SIEs limits solutions, multiscales approach., BTRW SIEs, K-martingale approach, BTP SIEs, Brownian-time processes, $2$-Brownian-times Brownian motion, BTBM SIEs, Brownian-time random walks.
Mathematics Subject Classification: 60H20, 60H15, 60H30, 45H05, 45R05, 35R11, 35R60, 35G99, 60J45, 60J35, 60J60, 60J6.
Citation: Hassan Allouba. Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 413-463. doi: 10.3934/dcds.2013.33.413
References:
[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 $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$, 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 $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ via the imaginary-Brownian-time-Brownian-angle representation, In final preparation.

[6]

H. Allouba and J. Duan, Swift-Hohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ and their attractors, In preparation.

[7]

H. Allouba and J. A. Langa, Nonlinear Kuramoto-Sivashinsky type SPDEs on $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ and their attractors, In preparation.

[8]

H. Allouba and Y. Xiao, BTBM SIEs on $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$: 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]

J. Doob, "Stochastic Processes," John Wiley and Sons, 1953.

[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 $\mathbb mathbb{R}^{d}$, 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.

show all references

References:
[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 $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$, 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 $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ via the imaginary-Brownian-time-Brownian-angle representation, In final preparation.

[6]

H. Allouba and J. Duan, Swift-Hohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ and their attractors, In preparation.

[7]

H. Allouba and J. A. Langa, Nonlinear Kuramoto-Sivashinsky type SPDEs on $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$ and their attractors, In preparation.

[8]

H. Allouba and Y. Xiao, BTBM SIEs on $\mathbb R_{+}$ × $\mathbb mathbb{R}^{d}$: 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]

J. Doob, "Stochastic Processes," John Wiley and Sons, 1953.

[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 $\mathbb mathbb{R}^{d}$, 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.

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