February  2017, 14(1): 237-248. doi: 10.3934/mbe.2017015

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion

Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Received  October 29, 2015 Accepted  April 12, 2016 Published  October 2016

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Citation: 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
References:
[1]

K. Amano, Newton's method for stochastic differential equations and its probabilistic second-order error estimate, Electron. J. Differential Equations, 2012 (2012), 1-8. Google Scholar

[2]

Z. Brzeźniak and M. Ondreját, Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Part. Diff. Eq., 36 (2011), 1624-1653. Google Scholar

[3]

P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dynam., 5 (2005), 45-64. doi: 10.1142/S0219493705001286. Google Scholar

[4]

P.-L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168. Google Scholar

[5]

D. Conus and R. C. Dalang, The non-linear stochastic wave equation in high dimension, Electron. J. Probab., 13 (2008), 629-670. doi: 10.1214/EJP.v13-500. Google Scholar

[6]

R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous spdes, Electron. J. Probab., 4 (1999), 1-29. doi: 10.1214/EJP.v4-43. Google Scholar

[7]

R. C. Dalang, The stochastic wave equation, A Minicourse on Stochastic Partial Differential Equations, in Lecture Notes in Math., 1962 (2009), Springer Berlin, 39-71. doi: 10.1007/978-3-540-85994-9_2. Google Scholar

[8]

R. C. DalangC. Mueller and R. Tribe, A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.s, Trans. Amer. Math. Soc., 360 (2008), 4681-4703. doi: 10.1090/S0002-9947-08-04351-1. Google Scholar

[9]

R. C. Dalang and M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199 (2009), 1-70. doi: 10.1090/memo/0931. Google Scholar

[10]

E. Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58. doi: 10.1016/j.cam.2010.03.026. Google Scholar

[11]

J. HuangY. Hu and D. Nualart, On Hölder continuity of the solution of stochastic wave equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 353-407. doi: 10.1007/s40072-014-0035-5. Google Scholar

[12]

S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in Séminaire de Probabilités XXV, Lecture Notes in Math., 1485 (1991), Springer Berlin, 121-137. doi: 10.1007/BFb0100852. Google Scholar

[13]

J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2. Google Scholar

[14]

C. Marinelli and L. Quer-Sardanyons, Existence of weak solutions for a class of semilinear stochastic wave equations, Siam J. Math. Anal., 44 (2012), 906-925. doi: 10.1137/110826667. Google Scholar

[15]

A. Millet and M. Sanz-Solé, A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab., 27 (1999), 803-844. doi: 10.1214/aop/1022677387. Google Scholar

[16]

M. Nedeljkov and D. Rajter, A note on a one-dimensional nonlinear stochastic wave equation, Novi Sad Journal of Mathematics, 32 (2002), 73-83. Google Scholar

[17]

S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2 (2002), 383-394. doi: 10.1007/PL00013197. Google Scholar

[18]

L. Quer-Sardanyons and M. Sanz-Solé, Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal., 206 (2004), 1-32. doi: 10.1016/S0022-1236(03)00065-X. Google Scholar

[19]

L. Quer-Sardanyons and M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006), 303-332. doi: 10.1007/s11118-005-9002-0. Google Scholar

[20]

M. Sanz-Solé and A. Suess, The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity, Electron. J. Probab., 18 (2013), 1-28. doi: 10.1214/EJP.v18-2341. Google Scholar

[21]

J. B. Walsh, An introduction to stochastic partial differential equations, in: É cole d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Math, 1180 (1986), Springer Berlin, 265-439. doi: 10.1007/BFb0074920. Google Scholar

[22]

J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018. Google Scholar

[23]

M. Wrzosek, Newton's method for stochastic functional differential equations, Electron. J.Differential Equations, 2012 (2012), 1-10. Google Scholar

[24]

M. Wrzosek, Newton's method for parabolic stochastic functional partial differential equations, Functional Differential Equations, 20 (2013), 285-310. Google Scholar

[25]

M. Wrzosek, Newton's method for first-order stochastic functional partial differential equations, Commentationes Mathematicae, 54 (2014), 51-64. Google Scholar

show all references

References:
[1]

K. Amano, Newton's method for stochastic differential equations and its probabilistic second-order error estimate, Electron. J. Differential Equations, 2012 (2012), 1-8. Google Scholar

[2]

Z. Brzeźniak and M. Ondreját, Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Part. Diff. Eq., 36 (2011), 1624-1653. Google Scholar

[3]

P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dynam., 5 (2005), 45-64. doi: 10.1142/S0219493705001286. Google Scholar

[4]

P.-L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168. Google Scholar

[5]

D. Conus and R. C. Dalang, The non-linear stochastic wave equation in high dimension, Electron. J. Probab., 13 (2008), 629-670. doi: 10.1214/EJP.v13-500. Google Scholar

[6]

R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous spdes, Electron. J. Probab., 4 (1999), 1-29. doi: 10.1214/EJP.v4-43. Google Scholar

[7]

R. C. Dalang, The stochastic wave equation, A Minicourse on Stochastic Partial Differential Equations, in Lecture Notes in Math., 1962 (2009), Springer Berlin, 39-71. doi: 10.1007/978-3-540-85994-9_2. Google Scholar

[8]

R. C. DalangC. Mueller and R. Tribe, A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.s, Trans. Amer. Math. Soc., 360 (2008), 4681-4703. doi: 10.1090/S0002-9947-08-04351-1. Google Scholar

[9]

R. C. Dalang and M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199 (2009), 1-70. doi: 10.1090/memo/0931. Google Scholar

[10]

E. Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58. doi: 10.1016/j.cam.2010.03.026. Google Scholar

[11]

J. HuangY. Hu and D. Nualart, On Hölder continuity of the solution of stochastic wave equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 353-407. doi: 10.1007/s40072-014-0035-5. Google Scholar

[12]

S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in Séminaire de Probabilités XXV, Lecture Notes in Math., 1485 (1991), Springer Berlin, 121-137. doi: 10.1007/BFb0100852. Google Scholar

[13]

J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2. Google Scholar

[14]

C. Marinelli and L. Quer-Sardanyons, Existence of weak solutions for a class of semilinear stochastic wave equations, Siam J. Math. Anal., 44 (2012), 906-925. doi: 10.1137/110826667. Google Scholar

[15]

A. Millet and M. Sanz-Solé, A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab., 27 (1999), 803-844. doi: 10.1214/aop/1022677387. Google Scholar

[16]

M. Nedeljkov and D. Rajter, A note on a one-dimensional nonlinear stochastic wave equation, Novi Sad Journal of Mathematics, 32 (2002), 73-83. Google Scholar

[17]

S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2 (2002), 383-394. doi: 10.1007/PL00013197. Google Scholar

[18]

L. Quer-Sardanyons and M. Sanz-Solé, Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal., 206 (2004), 1-32. doi: 10.1016/S0022-1236(03)00065-X. Google Scholar

[19]

L. Quer-Sardanyons and M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006), 303-332. doi: 10.1007/s11118-005-9002-0. Google Scholar

[20]

M. Sanz-Solé and A. Suess, The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity, Electron. J. Probab., 18 (2013), 1-28. doi: 10.1214/EJP.v18-2341. Google Scholar

[21]

J. B. Walsh, An introduction to stochastic partial differential equations, in: É cole d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Math, 1180 (1986), Springer Berlin, 265-439. doi: 10.1007/BFb0074920. Google Scholar

[22]

J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018. Google Scholar

[23]

M. Wrzosek, Newton's method for stochastic functional differential equations, Electron. J.Differential Equations, 2012 (2012), 1-10. Google Scholar

[24]

M. Wrzosek, Newton's method for parabolic stochastic functional partial differential equations, Functional Differential Equations, 20 (2013), 285-310. Google Scholar

[25]

M. Wrzosek, Newton's method for first-order stochastic functional partial differential equations, Commentationes Mathematicae, 54 (2014), 51-64. Google Scholar

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