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Hitting probabilities for nonlinear systems of stochastic waves
About this Title
Robert C. Dalang, Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland and Marta Sanz-Solé, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 237, Number 1120
ISBNs: 978-1-4704-1423-8 (print); 978-1-4704-2507-4 (online)
DOI: https://doi.org/10.1090/memo/1120
Published electronically: January 22, 2015
Keywords: Hitting probabilities,
systems of stochastic wave equations,
spatially homogeneous Gaussian noise,
capacity,
Hausdorff measure
MSC: Primary 60H15, 60J45; Secondary 60G15, 60H07, 60G60
Table of Contents
Chapters
- 1. Introduction
- 2. Upper bounds on hitting probabilities
- 3. Conditions on Malliavin matrix eigenvalues for lower bounds
- 4. Study of Malliavin matrix eigenvalues and lower bounds
- A. Technical estimates
Abstract
We consider a $d$-dimensional random field $u = \{u(t,x)\}$ that solves a nonlinear system of stochastic wave equations in spatial dimensions $k \in \{1,2,3\}$, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent $\beta$. Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of $\mathbb {R}^d$, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when $d(2-\beta ) > 2(k+1)$, points are polar for $u$. Conversely, in low dimensions $d$, points are not polar. There is however an interval in which the question of polarity of points remains open.- Robert J. Adler and Peter Müller (eds.), Stochastic modelling in physical oceanography, Progress in Probability, vol. 39, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1383868
- Hermine Biermé, Céline Lacaux, and Yimin Xiao, Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields, Bull. Lond. Math. Soc. 41 (2009), no. 2, 253–273. MR 2496502, DOI 10.1112/blms/bdn122
- S. K. Biswas and N. U. Ahmed, Stabilization of systems governed by the wave equation in the presence of distributed white noise, IEEE Trans. Automat. Control 30 (1985), no. 10, 1043–1045. MR 804147, DOI 10.1109/TAC.1985.1103814
- Sandra Cerrai and Mark Freidlin, Smoluchowski-Kramers approximation for a general class of SPDEs, J. Evol. Equ. 6 (2006), no. 4, 657–689. MR 2267703, DOI 10.1007/s00028-006-0281-8
- Daniel Conus and Robert C. Dalang, The non-linear stochastic wave equation in high dimensions, Electron. J. Probab. 13 (2008), no. 22, 629–670. MR 2399293, DOI 10.1214/EJP.v13-500
- Mireille Chaleyat-Maurel and Marta Sanz-Solé, Positivity of the density for the stochastic wave equation in two spatial dimensions, ESAIM Probab. Stat. 7 (2003), 89–114. MR 1956074, DOI 10.1051/ps:2003002
- Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI 10.1214/EJP.v4-43
- Robert C. Dalang, The stochastic wave equation, A minicourse on stochastic partial differential equations, Lecture Notes in Math., vol. 1962, Springer, Berlin, 2009, pp. 39–71. MR 2508773, DOI 10.1007/978-3-540-85994-9_{2}
- Robert C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046, DOI 10.1214/aop/1022855416
- Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart, Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 231–271. MR 2365643
- Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart, Hitting probabilities for systems for non-linear stochastic heat equations with multiplicative noise, Probab. Theory Related Fields 144 (2009), no. 3-4, 371–427. MR 2496438, DOI 10.1007/s00440-008-0150-1
- Dalang, R. C., D. Khoshnevisan, E. Nualart, Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimensions $k\ge 1$. Journal of SPDE’s: Analysis and Computations 1-1 (2013), 94–151.
- Robert C. Dalang and Olivier Lévêque, Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere, Ann. Probab. 32 (2004), no. 1B, 1068–1099. MR 2044674, DOI 10.1214/aop/1079021472
- Robert C. Dalang and Olivier Lévêque, Second-order hyperbolic S.P.D.E.’s driven by homogeneous Gaussian noise on a hyperplane, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2123–2159. MR 2197451, DOI 10.1090/S0002-9947-05-03740-2
- Robert C. Dalang, C. Mueller, and L. Zambotti, Hitting properties of parabolic s.p.d.e.’s with reflection, Ann. Probab. 34 (2006), no. 4, 1423–1450. MR 2257651, DOI 10.1214/009117905000000792
- Robert C. Dalang and Eulalia Nualart, Potential theory for hyperbolic SPDEs, Ann. Probab. 32 (2004), no. 3A, 2099–2148. MR 2073187, DOI 10.1214/009117904000000685
- Robert C. Dalang and Marta Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc. 199 (2009), no. 931, vi+70. MR 2512755, DOI 10.1090/memo/0931
- Robert C. Dalang and Marta Sanz-Solé, Criteria for hitting probabilities with applications to systems of stochastic wave equations, Bernoulli 16 (2010), no. 4, 1343–1368. MR 2759182, DOI 10.3150/09-BEJ247
- Donald A. Dawson and Habib Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), no. 2, 141–180. MR 575923, DOI 10.1016/0047-259X(80)90012-3
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
- Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics; Mathematical Notes. MR 0599578
- Gonzalez, O. and J. H. Maddocks, Extracting parameters for base-pair level models of DNA from molecular dynamics simulations. Theoretical Chemistry Accounts 106 (2001), 76–82.
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- Richard A. Holley and Daniel W. Stroock, Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions, Publ. Res. Inst. Math. Sci. 14 (1978), no. 3, 741–788. MR 527199, DOI 10.2977/prims/1195188837
- Davar Khoshnevisan, Multiparameter processes, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002. An introduction to random fields. MR 1914748
- Davar Khoshnevisan and Yimin Xiao, Harmonic analysis of additive Lévy processes, Probab. Theory Related Fields 145 (2009), no. 3-4, 459–515. MR 2529437, DOI 10.1007/s00440-008-0175-5
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
- Miller, R. N., Tropical data assimilation with simulated data: The impact of the tropical ocean and global atmosphere thermal array for the ocean. J. Geophysical Research 95 (1990), 11,461–11,482.
- Annie Millet and Pierre-Luc Morien, On a stochastic wave equation in two space dimensions: regularity of the solution and its density, Stochastic Process. Appl. 86 (2000), no. 1, 141–162. MR 1741200, DOI 10.1016/S0304-4149(99)00090-3
- Annie Millet and Marta Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), no. 2, 803–844. MR 1698971, DOI 10.1214/aop/1022677387
- C. Mueller and R. Tribe, Hitting properties of a random string, Electron. J. Probab. 7 (2002), no. 10, 29. MR 1902843, DOI 10.1214/EJP.v7-109
- David Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1344217
- David Nualart, Analysis on Wiener space and anticipating stochastic calculus, Lectures on probability theory and statistics (Saint-Flour, 1995) Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 123–227. MR 1668111, DOI 10.1007/BFb0092538
- David Nualart and Lluís Quer-Sardanyons, Existence and smoothness of the density for spatially homogeneous SPDEs, Potential Anal. 27 (2007), no. 3, 281–299. MR 2336301, DOI 10.1007/s11118-007-9055-3
- Eulalia Nualart, On the density of systems of non-linear spatially homogeneous SPDEs, Stochastics 85 (2013), no. 1, 48–70. MR 3011911, DOI 10.1080/17442508.2011.653567
- Eulalia Nualart and Frederi Viens, The fractional stochastic heat equation on the circle: time regularity and potential theory, Stochastic Process. Appl. 119 (2009), no. 5, 1505–1540. MR 2513117, DOI 10.1016/j.spa.2008.07.009
- Szymon Peszat and Jerzy Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000), no. 3, 421–443. MR 1749283, DOI 10.1007/s004400050257
- Lluís Quer-Sardanyons and Marta Sanz-Solé, A stochastic wave equation in dimension 3: smoothness of the law, Bernoulli 10 (2004), no. 1, 165–186. MR 2044597, DOI 10.3150/bj/1077544607
- Marta Sanz-Solé, Malliavin calculus, Fundamental Sciences, EPFL Press, Lausanne; distributed by CRC Press, Boca Raton, FL, 2005. With applications to stochastic partial differential equations. MR 2167213
- Marta Sanz-Solé, Properties of the density for a three-dimensional stochastic wave equation, J. Funct. Anal. 255 (2008), no. 1, 255–281. MR 2417817, DOI 10.1016/j.jfa.2008.04.004
- S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 73, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by M. Gopalan Nair and B. Rajeev. MR 742628
- Jerzy Zabczyk, A mini course on stochastic partial differential equations, Stochastic climate models (Chorin, 1999) Progr. Probab., vol. 49, Birkhäuser, Basel, 2001, pp. 257–284. MR 1948300