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: \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.
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

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.  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

[1]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[2]

Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245

[3]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[4]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[5]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

[6]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[7]

Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077

[8]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[9]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[10]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[11]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[12]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001

[13]

Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248

[14]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084

[15]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015

[16]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[17]

Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095

[18]

Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034

[19]

Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial & Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461

[20]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]