These notes form a brief introductory tutorial to elements of Gaussian noise analysis and basic stochastic partial differential equations (SPDEs) in general, and the stochastic heat equation, in particular. The chief aim here is to get to the heart of the matter quickly. We achieve this by studying a few concrete equations only. This chapter provides sufficient preparation for learning more advanced theory from the remainder of this volume.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Louis Bachelier (1900). Théorie de la Spéculation, Ann. Sci. École Norm. Sup. 17, 21–86. [See also the 1995 reprint. Sceaux: Gauthier—Villars.]
V. V. Baklan (1965). The existence of solutions for a class of equations involving variational derivatives, Dopovidi Akad. Nauk UkraÄn. RSR 1965, 554–556 (Ukranian. Russian, English summary)
D. L. Burkholder (1971). Martingale inequalities, In: Lecture Notes in Math. 190, 1–8 Springer-Verlag, Berlin
E. M. Cabaña (1970). The vibrating string forced by white noise, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 15, 111–130
Robert C. Dalang (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.', Electron. J. Probab. 4, no. 6, 29 pages (electronic)
Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart (2007). Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, Latin American J. Probab. and Math. Statist. (or Alea; http://alea.impa.br/english), Vol. III, 231–371
Ju. L. DaleckiÄ(1967). Infinite-dimensional elliptic operators and the corresponding parabolic equations, Uspehi Mat. Nauk 22 (4) (136), 3–54 (In Russian) [English translation in: Russian Math. Surveys 22 (4), 1–53, 1967]
D. A. Dawson (1975). Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5, 1–52
D. A. Dawson (1972). Stochastic evolution equations, Math. Biosci. 15, 287–316
J. L. Doob (1942). The Brownian movement and stochastic equations, Ann. of Math. 43 (2), 351–369
R. M. Dudley (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, 1, 290–330
X. Fernique (1975). Regularité des trajectoires des fonctions aléatoires gaussiennes, In: Lecture Notes in Math. 480, 1–96 Springer-Verlag, Berlin (in French )
Tadahisa Funaki (1984). Random motion of strings and stochastic differential equations on the space C([0, 1], Rd), In: Stochastic Analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32, 121–133, North-Holland, Amsterdam
Tadahisa Funaki (1983). Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89, 129–193
Kiyosi Itô (1944). Stochastic integral, Proc. Imp. Acad. Tokyo 20, 519–524
Kiyosi Itô (1950). Stochastic differential equations in a differentiable manifold, Nagoya Math. J. 1, 35–47
Kiyosi Itô (1951). On a formula concerning stochastic differentials, Nagoya Math. J. 3, 55–65
A. Ya. Khintchine (1933). Asymptotische Gesetz der Wahrscheinlichkeitsrechnung, Springer, Berlin
A. N. Kolmogorov (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung,Springer, Berlin
N. V. Krylov and B. L. RozovskiÄ (1979a). Itô equations in Banach spaces and strongly parabolic stochastic partial differential equations, Dokl. Akad. Nauk SSSR 249 (2), 285–289 (in Russian)
N. V. Krylov and B. L. RozovskiÄ (1979b). Stochastic evolution equations, In: Current Problems in Mathematics, Vol. 14 Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 71–147, 256 (in Russian)
N. V. Krylov, N. V. and B. L. RozovskiÄ (1977). The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat. 41 (6),1329–1347, 1448 (in Russian)
H. Kunita (1991). Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge
Étienne Pardoux (1975). Equations aux dérivées partielles stochastiques nonlinéaires monotones—Étude de solutions fortes de type Itô, Thése d'État, Univ. Paris XI, Orsay
Étienne Pardoux (1972). Sur des équations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci. Paris Sér. A—B 275, A101–A103
Christopher Preston (1972). Continuity properties of some Gaussian processes,Ann. Math. Statist. 43, 285–292
Michel Talagrand (1985). Régularité des processus gaussiens, C. R. Acad. Sci. Paris Sér. I Math. 301 (7), 379–381 (French, with English summary)
Michel Talagrand (1987). Regularity of Gaussian processes, Acta Math. 159 (1–2), 99–149
G. E. Uhlenbeck and L. S. Ornstein (1930). On the theory of Brownian Motion, Phys. Rev. 36, 823–841
John B. Walsh (1986). An Introduction to Stochastic Partial Differential Equations, In: Lecture Notes in Math. 1180, 265–439, Springer, Berlin
N. Wiener (1923). Differential space, J. Math. Phys. 2, 131–174
Norbert Wiener (1938). The Homogeneous Chaos, Amer. J. Math. 60 (4), 897–936
Editor information
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Khoshnevisan, D. (2009). A Primer on Stochastic Partial Differential Equations. In: Khoshnevisan, D., Rassoul-Agha, F. (eds) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol 1962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85994-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-85994-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85993-2
Online ISBN: 978-3-540-85994-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)