# American Institute of Mathematical Sciences

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

 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: $$U(t,x)=\int_{{\mathbb R}^{d}}{{\mathbb K}}^{\text{BTBM}^d}_{t;x,y} u_0(y) dy+ \int_{{\mathbb R}^{d}}\int_0^t{{\mathbb K}}^{\text{BTBM}^d}_{t-s;x,y} a(U(s,y))\mathscr W(ds\times dy), (0.1)$$ 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.
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 & Continuous Dynamical Systems - A, 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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [5] H. Allouba, L-Kuramoto-Sivashinsky SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ via the imaginary-Brownian-time-Brownian-angle representation,, In final preparation., ().   Google Scholar [6] H. Allouba and J. Duan, Swift-Hohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., ().   Google Scholar [7] H. Allouba and J. A. Langa, Nonlinear Kuramoto-Sivashinsky type SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., ().   Google Scholar [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., ().   Google Scholar [9] H. Allouba, SDDEs limits solutions to sublinear reaction-diffusion SPDEs,, Electron. J. Differential Equations, (2003).   Google Scholar [10] H. Allouba, Brownian-time processes: the PDE connection II and the corresponding Feynman-Kac formula,, Trans. Amer. Math. Soc., 354 (2002), 4627.   Google Scholar [11] H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator,, Ann. Probab., 29 (2001), 1780.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] H. Allouba, Different types of SPDEs in the eyes of Girsanov's theorem,, Stochastic Anal. Appl., 16 (1998), 787.   Google Scholar [15] H. Allouba, A non-nonstandard proof of Reimers' existence result for heat SPDEs,, J. Appl. Math. Stochastic Anal., 11 (1998), 29.   Google Scholar [16] R. Bass, "Probabilistic Techniques in Analysis,", Springer, (1995).   Google Scholar [17] R. Bass, "Diffusions and Elliptic Operators,", Springer, (1997).   Google Scholar [18] R. Bass and H. Tang, The martingale problem for a class of stable-like processes,, Stochastic Process. Appl., 119 (2009), 1144.   Google Scholar [19] P. Carr and L. Cousot, A PDE approach to jump-diffusions,, Quant. Finance, 11 (2011), 33.   Google Scholar [20] R. Dalang, Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs,, Electron. J. Probab, 4 (1999), 1.  doi: 10.1214/EJP.v4-43.  Google Scholar [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.   Google Scholar [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).   Google Scholar [23] G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.   Google Scholar [24] R. DeBlassie, Iterated Brownian motion in an open set,, Ann. Appl. Probab., 14 (2004), 1529.   Google Scholar [25] J. Doob, "Stochastic Processes,", John Wiley and Sons, (1953).   Google Scholar [26] J. Duan and W. Wei, "Effective Dynamics of Stochasticpartial Differential Euations,", To appear., ().   Google Scholar [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.   Google Scholar [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.   Google Scholar [29] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusions,", North-Holland Publishing Company, (1989).   Google Scholar [30] I. Karatzas and S. Shreve, "Brownian Motion and Stochastic Calculus,", Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0302-2.  Google Scholar [31] T. Kurtz, Particle representations for measure-valued population processes with spatially varying birth rates,, Stochastic models (Ottawa, 26 (2000), 299.   Google Scholar [32] J. Le Gall, A path-valued Markov process and its connections with partial differential equations,, First European Congress of Mathematics, 120 (1994), 185.   Google Scholar [33] M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains,, Ann. Probab., 37 (2009), 979.  doi: 10.1214/08-AOP426.  Google Scholar [34] E. Nane, Stochastic solutions of a class of Higher order Cauchy problems in $\mathbb R^d$,, Stoch. Dyn., 10 (2010), 341.  doi: 10.1142/S021949371000298X.  Google Scholar [35] M. Reimers, One dimensional stochastic partial differential equations and the branching measure diffusion,, Probab. Theory Relat. Fields, 81 (1989), 319.  doi: 10.1007/BF00340057.  Google Scholar [36] R. Sowers, Short-time geometry of random heat kernels, (English summary), Mem. Amer. Math. Soc., 132 (1998).   Google Scholar [37] R. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations,, Ann. Probab., 22 (1994), 2071.   Google Scholar [38] D. Stroock and S. Varadhan, "Multidimensional Diffusion Processes,", Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, (1997), 978.   Google Scholar [39] D. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions,, Ann. Inst. H. Poincare Probab. Statist, 33 (1997), 619.   Google Scholar [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition. Applied Mathematical Sciences, (1997), 0.   Google Scholar [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, (1180).   Google Scholar [42] Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion,, J. Theoret. Probab., 11 (1998), 383.   Google Scholar

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##### 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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [5] H. Allouba, L-Kuramoto-Sivashinsky SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ via the imaginary-Brownian-time-Brownian-angle representation,, In final preparation., ().   Google Scholar [6] H. Allouba and J. Duan, Swift-Hohenberg SPDEs drivenon $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., ().   Google Scholar [7] H. Allouba and J. A. Langa, Nonlinear Kuramoto-Sivashinsky type SPDEs on $\mathbb R_{+}$ × $\mathbb R^d$ and their attractors,, In preparation., ().   Google Scholar [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., ().   Google Scholar [9] H. Allouba, SDDEs limits solutions to sublinear reaction-diffusion SPDEs,, Electron. J. Differential Equations, (2003).   Google Scholar [10] H. Allouba, Brownian-time processes: the PDE connection II and the corresponding Feynman-Kac formula,, Trans. Amer. Math. Soc., 354 (2002), 4627.   Google Scholar [11] H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator,, Ann. Probab., 29 (2001), 1780.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] H. Allouba, Different types of SPDEs in the eyes of Girsanov's theorem,, Stochastic Anal. Appl., 16 (1998), 787.   Google Scholar [15] H. Allouba, A non-nonstandard proof of Reimers' existence result for heat SPDEs,, J. Appl. Math. Stochastic Anal., 11 (1998), 29.   Google Scholar [16] R. Bass, "Probabilistic Techniques in Analysis,", Springer, (1995).   Google Scholar [17] R. Bass, "Diffusions and Elliptic Operators,", Springer, (1997).   Google Scholar [18] R. Bass and H. Tang, The martingale problem for a class of stable-like processes,, Stochastic Process. Appl., 119 (2009), 1144.   Google Scholar [19] P. Carr and L. Cousot, A PDE approach to jump-diffusions,, Quant. Finance, 11 (2011), 33.   Google Scholar [20] R. Dalang, Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs,, Electron. J. Probab, 4 (1999), 1.  doi: 10.1214/EJP.v4-43.  Google Scholar [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.   Google Scholar [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).   Google Scholar [23] G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.   Google Scholar [24] R. DeBlassie, Iterated Brownian motion in an open set,, Ann. Appl. Probab., 14 (2004), 1529.   Google Scholar [25] J. Doob, "Stochastic Processes,", John Wiley and Sons, (1953).   Google Scholar [26] J. Duan and W. Wei, "Effective Dynamics of Stochasticpartial Differential Euations,", To appear., ().   Google Scholar [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.   Google Scholar [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.   Google Scholar [29] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusions,", North-Holland Publishing Company, (1989).   Google Scholar [30] I. Karatzas and S. Shreve, "Brownian Motion and Stochastic Calculus,", Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0302-2.  Google Scholar [31] T. Kurtz, Particle representations for measure-valued population processes with spatially varying birth rates,, Stochastic models (Ottawa, 26 (2000), 299.   Google Scholar [32] J. Le Gall, A path-valued Markov process and its connections with partial differential equations,, First European Congress of Mathematics, 120 (1994), 185.   Google Scholar [33] M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains,, Ann. Probab., 37 (2009), 979.  doi: 10.1214/08-AOP426.  Google Scholar [34] E. Nane, Stochastic solutions of a class of Higher order Cauchy problems in $\mathbb R^d$,, Stoch. Dyn., 10 (2010), 341.  doi: 10.1142/S021949371000298X.  Google Scholar [35] M. Reimers, One dimensional stochastic partial differential equations and the branching measure diffusion,, Probab. Theory Relat. Fields, 81 (1989), 319.  doi: 10.1007/BF00340057.  Google Scholar [36] R. Sowers, Short-time geometry of random heat kernels, (English summary), Mem. Amer. Math. Soc., 132 (1998).   Google Scholar [37] R. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations,, Ann. Probab., 22 (1994), 2071.   Google Scholar [38] D. Stroock and S. Varadhan, "Multidimensional Diffusion Processes,", Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, (1997), 978.   Google Scholar [39] D. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions,, Ann. Inst. H. Poincare Probab. Statist, 33 (1997), 619.   Google Scholar [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition. Applied Mathematical Sciences, (1997), 0.   Google Scholar [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, (1180).   Google Scholar [42] Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion,, J. Theoret. Probab., 11 (1998), 383.   Google Scholar
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