The Annals of Probability

Some parabolic PDEs whose drift is an irregular random noise in space

Francesco Russo and Gerald Trutnau

Full-text: Open access


A new class of random partial differential equations of parabolic type is considered, where the stochastic term consists of an irregular noisy drift, not necessarily Gaussian, for which a suitable interpretation is provided. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some appropriate sense) of the associated parabolic equation and a probabilistic interpretation is investigated.

Article information

Ann. Probab., Volume 35, Number 6 (2007), 2213-2262.

First available in Project Euclid: 8 October 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H05: Stochastic integrals 60G48: Generalizations of martingales 60H10: Stochastic ordinary differential equations [See also 34F05]

Singular drifted PDEs Dirichlet processes martingale problem stochastic partial differential equations distributional drift


Russo, Francesco; Trutnau, Gerald. Some parabolic PDEs whose drift is an irregular random noise in space. Ann. Probab. 35 (2007), no. 6, 2213--2262. doi:10.1214/009117906000001178.

Export citation


  • Aronson, D. G. (1967). Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 890--896.
  • Bass, R. F. and Chen, Z.-Q. (2001). Stochastic differential equations for Dirichlet processes. Probab. Theory Related Fields 121 422--446.
  • Bertoin, J. (1986). Les processus de Dirichlet en tant qu'espace de Banach. Stochastics 18 155--168.
  • Bouleau, N. and Yor, M. (1981). Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 292 491--494.
  • Dunford, N. and Schwartz, J. T. (1967). Linear Operators, Part I, General Theory. Wiley, New York.
  • Errami, M. and Russo, F. (2003). $n$-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 259--299.
  • Engelbert, H. J. and Schmidt, W. (1985). On solutions of one-dimensional stochastic differential equations without drift. Z. Wahrsch. Verw. Gebiete 68 287--314.
  • Engelbert, H. J. and Wolf, J. (1998). Strong Markov local Dirichlet processes and stochastic differential equations. Teor. Veroyatnost. i Primenen. 43 331--348.
  • Feyel, D. and De la Pradelle, A. (1999). On fractional Brownian processes. Potential Anal. 18 273--288.
  • Flandoli, F., Russo, F. and Wolf, J. (2003). Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 493--542.
  • Flandoli, F., Russo, F. and Wolf, J. (2004). Some SDEs with distributional drift. II. Lyons--Zheng structure, Itô formula and semimartingale characterization. Random Oper. Stochastic Equations 12 145--184.
  • Föllmer, H. (1981). Dirichlet processes. Stochastic Integrals. Lecture Notes in Math. 851 476--478. Springer, Berlin.
  • Gradinaru, M., Russo, F. and Vallois, P. (2003). Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index $H \ge\frac14$. Ann. Probab. 31 1772--1820.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel.
  • Mastrangelo, M. and Dehen, D. (1992). Opérateurs différentiels paraboliques à coefficients continus par morceaux et admettant un drift génééralisé. Bull. Sci. Math. 116 67--93.
  • Nualart, D. and Viens, F. (2000). Evolution equation of a stochastic semigroup with white-noise drift. Ann. Probab. 28 36--73.
  • Ouknine, Y. (1990). Le ``Skew-Brownian motion'' et les processus qui en dérivent. Theory Probab. Appl. 35 173--179.
  • Portenko, N. I. (1990). Generalized Diffusion Processes. Amer. Math. Soc., Providence, RI.
  • Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403--421.
  • Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to a finite variation process. Stochastics Stochastics Rep. 70 1--40.
  • Russo, F. and Vallois, P. (2007). Elements of stochastic calculus via regularizations. Séminaire de Probabilités XL (C. Donati-Martin, M. Emery, A. Roukeult and C. Stricker, eds.). To appear.
  • Russo, F., Vallois, P. and Wolf, J. (2001). A generalized class of Lyons--Zheng processes. Bernoulli 7 363--379.
  • Stroock, D. W. (1988). Diffusion processes corresponding to uniformly elliptic divergence form operators. Séminare de Probabilitiés XXII. Lecture Notes in Math. 1321 316--347. Springer, Berlin.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
  • Young, L. C. (1936). An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67 251--282.
  • Zvonkin, A. K. (1974). A transformation of the phase space of a diffusion process that removes the drift. Math. Sb. (N.S.) 93(135) 129--149.