Articles on Stochastic partial 
differential equations 

Subject index:

• Introductory texts:

 

 

Walsh, J.B. An introduction to stochastic partial differential equations, Ecole d'Eté de Prob. de St. Flour XIV, 1984, Lect. Notes in Math 1180, Springer Verlag (1986).  Math Review link

          Etienne Pardoux: Stochastic partial differential equations, a review, Bull. Sc. Math.117, 29-47, 1993.

 

 

          Robert C Dalang:
           Extending the Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E.s"
            Electronic Journal of Probability, 1999.

• General results on SPDEs:

           Istvan Gyöngy

Existence and uniqueness results for semilinear stochastic partial differential equations

Stochastic Processes and their Applications, 73 (2) (1998) pp. 271-299

[Abstract][Full text]

 

 

           E Pardoux & D. Nualart: White noise driven quasilinear SPDEs with reflection, Prob. Theory and Rel. Fields 93, 77-89, 1992.

• Parabolic SPDEs and stochastic heat equation:

 

 

Bally, V., Gyongy, I., Pardoux, E., 1994. White noise driven parabolic SPDEs with measurable drift.
J. Funct. Anal. 120, 484 - 510.

Gyongy  I., Pardoux, E., Weak and strong solutions of white noise driven parabolic SPDEs. preprint.

Etienne Pardoux & Zhang Tusheng:
Absolute continuity of the law of the solution of a parabolic SPDE, J. of Funct. Anal. 112, 447-458, 1993.

Carl Mueller: Long-time existence for signed solutions of the heat equation with a noise term
Probab Theory Relat Fields 110 (1998) 1, 51-68.

 

Gianmario Tessitore and Jerzy Zabczyk

Strict positivity for stochastic heat equations

Stochastic Processes and their Applications, 77 (1) (1998) pp. 83-98 [Abstract][Full text]

Carl Mueller : The heat equation with Lévy noise
Stochastic Processes and their Applications, 74 (1) (1998) pp. 67-82 [Abstract][Full text]

           Jan Ubøe and Tusheng Zhang

A stability property of the stochastic heat equation

Stochastic Processes and their Applications, 60 (2) (1995) pp. 247-260 [Abstract]

 

 
 
 

        Lorenzo Bertini, Giambattista Giacomin:
       On the long time behavior of the stochastic heat equation
        Probab Theory Relat Fields 114 (1999) 3, 279-289

        Elisa Alòs, Jorge A. León, David Nualart:
       Stochastic heat equation with random coefficients
        Probab Theory Relat Fields 115 (1999) 1, 41-94.

          Anna A. Kwiecioska

Stabilization of partial differential equations by noise

Stochastic Processes and their Applications, 79 (2) (1999) pp. 179-184 [Abstract][Full text]

• Stochastic wave equation:



Dalang, R.C. and Frangos, N.E. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26-1 (1998), 187-212.   Math Review link

Albeverio, S., Haba, Z. and Russo, F. Trivial solutions for a non-linear two space-dimensional wave equation perturbed by a space-time white noise.  Stochastics and Stochastics Rep. 56 (1996), 127-160.  Math Review link

Robert C Dalang: "Extending the Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E.'s" Electronic Journal of Probability, 1999.

Annie Millet and Pierre-Luc Morien

On a stochastic wave equation in two space dimensions: regularity of the solution and its density.
Stochastic Processes and their Applications, 86 (1) (2000) pp. 141-162[Abstract][Full text]

           David Márquez-Carreras and Marta Sanz-Solé

Small perturbations in a hyperbolic stochastic partial differential equation

Stochastic Processes and their Applications, 68 (1) (1997) pp. 133-154

[Abstract]

         Mueller, C. : Long time existence for the wave equation with a noise term.

        Ann. Probab. 25-1 (1997), 133-151. Math Review link.

D. Nualart & S. Tindel: Quasilinear stochastic hyperbolic differential equations with nondecreasing coefficient. Potential analysis 7 (1997).

• Nonlinear SPDEs (see also: Viscosity solutions for SPDEs )
  

Alain Bensoussan, Roger Temam (1972):  Equations aux dérivées partielles stochastiques nonlinéaires,
Israel Journal of Mathematics, Vol. 11, No. 1, 95-129.

Alain Bensoussan (1992): Some existence results for stochastic PDEs. in:
Da Prato & Tubaro (Editors): Stochastic partial differential equations and applications,
Pitman Research Notes in Mathematics, Vol. 268.

Szymon Peszat, Jerzy Zabczyk:
Nonlinear stochastic wave and heat equations
Probab Theory Relat Fields 116 (2000) 3, 421-443.

Thomas G. Kurtz and Jie Xiong
Particle representations for a class of nonlinear SPDEs
Stochastic Processes and their Applications, 83 (1) (1999) pp. 103-126 [Abstract][Full text]

Douglas Blount

Fourier analysis applied to SPDEs

Stochastic Processes and their Applications, 62 (2) (1996) pp. 223-242 [Abstract]

• Relation with non-linear filtering:

Etienne Pardoux:
Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3, 127-167, 1979.

Kunita (1981):  Stochastic partial differential equations connected with nonlinear filtering, pp 100-168
in: S.K. Mitter & A. Moro (Editos): Nonlinear filtering and stochastic control, Lecture notes in mathematics, Vol. 972.

Etienne Pardoux :
Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, in Ecole d'été de Probabilités de St Flour XIX, LNM 1464, Springer, 67-163, 1991.

Etienne Pardoux: Equations du filtrage non linéaire, de la prédiction et du lissage, Stochastics 6, 193-231, 1982.

E Pardoux, M. Chaleyat-Maurel, D. Michel: Un théorème d'unicité pour l'équation de Zakai, Stochastics 29, 1-13, 1990.

• Elliptic SPDEs:

           David Nualart and Samy Tindel

Quasilinear stochastic elliptic equations with reflection

Stochastic Processes and their Applications, 57 (1) (1995) pp. 73-82 [Abstract]

• SPDEs and turbulence:



T. Deck, J. Potthoff:
On a class of stochastic partial differential equations related to turbulent transport Probab Theory Relat Fields 111 (1998) 1, 101-122.


• SPDEs and backward stochastic differential equations:
  Rainer Buckdahn and Shige Peng : Stationary backward stochastic differential equations and associated partial differential equations,
  
  Probability Theory and Related Fields  Volume 115, Number 3, 1999, 383 - 399.
  
  

• SPDEs with Poisson noise / jumps:
   


Erwan Saint Loubert Bié:
Étude d'une EDPS conduite par un bruit poissonnien Probab Theory Relat Fields 111 (1998) 2, 287-321

Nicolas Fournier, Malliavin calculus for parabolic SPDEs with jumps, Probability theory and related fields, Vol 111, 1998.



Leonid Mytnik Stochastic partial differential equation driven by stable noise. Probab. Theory Related Fields 123 (2002), no. 2, 157--201.



Parabolic SPDEs driven by Poisson white noise

Stochastic Processes and their Applications, 74 (1) (1998) pp. 21-36 [Abstract][Full text]

• Malliavin calculus for SPDEs
 
Malliavin calculus is a stochastic version of variational calculus, initially developed on the Wiener space by P. Malliavin, D Stroock and J.M. Bismut.


E Pardoux & V. Bally: Malliavin calculus for white noise driven parabolic SPDEs, Potential Analysis, 9, 27-64, 1998.

Nicolas Fournier, Malliavin calculus for parabolic SPDEs with jumps,
Probability theory and related fields, Vol 111, 1998.

• Viscosity solutions for nonlinear SPDEs
 
The notion of viscosity solution, developed by Crandall, Ishii and PL Lions, is a notion of generalized solution which is appropriate for many nonlinear PDEs with nonlinearities appearing in the highedt derivative and which verify a generalized maximum principle. This notion has been recently extended to the case of nonlinear SPDEs.




P L Lions and P Souganidis (1998) Fully nonlinear SPDEs  Comptes rendus de l Academie des Sciences , Vol. 326 (1998).

P L Lions and P Souganidis (1998) EDP stochastiques nonlineaires et solutions de viscosite,
Seminaire X-EDP , (1998).

Rainer Buckdahn and Jin Ma: Stochastic viscosity solutions for nonlinear SPDEs I
Stochastic processes and applications , .93 (2001) 181-204

Rainer Buckdahn and Jin Ma: Stochastic viscosity solutions for nonlinear SPDEs II
Stochastic processes and applications , .93 (2001) 205-228

• Large deviations for SPDEs

 

C. Rovira, S. Tindel:
Sharp large deviation estimates for the stochastic heat equation.
Prépublication de mathématiques de l'Université Paris 13, no 98-32 (1998).

C. Rovira, S. Tindel
Sharp Laplace asymptotics for SPDEs: the parabolic and elliptic cases.
Prépublication de mathématiques de l'Université de Barcelone, no 243 (1997).

C. Rovira, S. Tindel
Sharp Laplace asymptotics for an hyperbolic SPDE.
Prépublication de mathématiques de l'Université Autonome de Barcelone, no 26 (1996).

Kallianpur, Gopinath; Xiong, Jie
Large deviations for a class of stochastic partial differential equations.
Ann. Probab. 24 (1996), no. 1, 320--345.

Caroline Cardon-Weber
Large deviations for a Burgers'-type SPDE.
Stochastic Processes and their Applications, 84 (1) (1999) pp. 53-70.


 
• Numerical methods for SPDEs

          István Gyöngy and David Nualart

Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise

Stochastic Processes and their Applications, 58 (1) (1995) pp. 57-72 [Abstract]

• Parameter estimation for SPDEs

 

 

Huebner, M., Parameter estimation for SPDE's. Ph.D. Thesis, University of Southern California,  June 1993.

Huebner, M., Khasminskii, R., Rozovskii, B.L., 1992. Two examples of parameter estimation. In: Cambanis, S., Ghosh, J.K., Karandikar, R.L., Sen, P.K. (Eds.), Stochastic Processes, Springer, New York.

Huebner, M., Rozovskii, B., 1995. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's. Probab. Theory Related Fields 103, 143 {163.

Ibragimov, I.A., Khasminskii, R.Z., 1997a. Some estimation problems in infnite dimensional white noise, in Pollard, D., Torgersen, E., Yang, G. (Eds.), Festschrift for Lucien Le Cam, Springer, Berlin, pp. 259 - 274.

Ibragimov, I.A., Khasminskii, R.Z., 1997b. Some nonparametric estimation problems for parabolic SPDE.
Technical Report #31, Wayne State University, Department of Mathematics.

Sergey V. Lototsky and Boris L. Rosovskii
Spectral asymptotics of some functionals arising in statistical inference for SPDEs
Stochastic Processes and their Applications, 79 (1) (1999) pp. 69-94 [Abstract][Full text]

Piterbarg, L., Rozovskii, B., 1996. Maximum likelihood estimators in the equations of physical oceanography.
In: Adler, R.J. et al. (Eds.), Stochastic Modeling in Physical Oceanography, Birkh. auser, Boston,pp. 397 - 421.

Piterbarg, L., Rozovskii, B., 1997. On asymptotic problems of parameter estimation in stochastic PDE's: discrete time sampling. Mathematical Methods of Statistics 6(2), 200 { 223.