The Annals of Probability

Evolution equation of a stochastic semigroup with white-noise drift

David Nualart and Frederi Viens

Full-text: Open access


We study the existence and uniqueness of the solution of a function-valued stochastic evolution equation based on a stochastic semigroup whose kernel $p(s,t,y,x)$ is Brownian in $s$ and $t$.The kernel $p$ is supposed to be measurable with respect to the increments of an underlying Wiener process in the interval $[s, t]$. The evolution equation is then anticipative and, choosing the Skorohod formulation,we establish existence and uniqueness of a continuous solution with values in $L^2(\mathbb{R}^d)$.

As an application we prove the existence of a mild solution of the stochastic parabolic equation

du_t = \Delta_x u dt + v(dt, x) \cdot \nabla u + F(t, x, u) W(dt, x),

where $v$ and $W$ are Brownian in time with respect to a common filtration. In this case, p is the formal backward heat kernel of $\Delta_x + v(dt, x) \cdot \nabla_x$ .

Article information

Ann. Probab., Volume 28, Number 1 (2000), 36-73.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Stochastic parabolic equations anticipating stochastic calculus Skorohod integral stochastic semigroups


Nualart, David; Viens, Frederi. Evolution equation of a stochastic semigroup with white-noise drift. Ann. Probab. 28 (2000), no. 1, 36--73. doi:10.1214/aop/1019160111.

Export citation


  • [1] Al os, E., Nualart, D. and Viens, F. (1998). Stochastic heat equation with white noise drift. Mathematics Preprint Series 21, Univ. Barcelona.
  • [2] Bouleau, N. and Hirsch, F. (1991). Dirichlet Forms and Analysis on Wiener Space. de Gruyter, Berlin.
  • [3] Buckdahn, R. and Ma, J. (1999). Stochastic viscosity solution for non-linear stochastic PDEs. Preprint.
  • [4] DaPrato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • [5] Holden, H., Øksendal, B., Ubøe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Birkh¨auser, Boston.
  • [6] Kotelenez, P. (1995). A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation. Probab. Theory Related Fields 102 159- 188.
  • [7] Kifer, Y. and Kunita, H. (1996). Random positive semigroups and their random infinitesimal generators. In Stochastic Analysis and Applications (I. M. Davies, A. Truman and K. D. Elworthy, eds.) 270-285. World Scientific, Singapore.
  • [8] Krylov, N. V. (1999). An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives. Math. Surveys Monogr. 64 185-242.
  • [9] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
  • [10] Kunita, H. (1994). Generalized solutions of a stochastic partial differential equation. J. Theoret. Probab. 7 279-308.
  • [11] Kurtz, T. and Protter, P. (1966). Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case. Lecture Notes, CIME School in Probability.
  • [12] Leon, J. and Nualart, D. (1999). Stochastic evolution equations with random generators. Unpublished manuscript.
  • [13] Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDEs and Wiener chaos. SIAM J. Math. Anal. 29 452-480.
  • [14] Mikulevicius, R. and Rozovskii, B. L. (1999). Martingale problems for stochastic PDE's. Stochastic partial differential equations: six perspectives. Math. Surveys Monogr. 64 243-325.
  • [15] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [16] Nualart, D. and Pardoux, E. (1988). Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 535-581.
  • [17] Nualart, D. and Zakai, M. (1989). The partial Malliavin calculus. Seminaire de Probabilit´es XXIII. Lecture Notes in Math. 1372 362-381. Springer, Berlin.
  • [18] Pardoux, E. and Peng, S. G. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. Lecture Notes in Control and Inform. Sci. 176 200-217. Springer, Berlin.
  • [19] Pardoux, E. and Protter, P. (1987). Two-sided stochastic integrals and calculus. Probab. Theory Related Fields 76 15-50.
  • [20] Viens, F. A. (1999). Feynman-Kac formula and Lyapunov exponents for anticipative stochastic evolution equations. Unpublished manuscript.
  • [21] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d'Et´e de Probabilit´es de Saint Flour XIV. Lecture Notes in Math. 1180 265-438. Springer, Berlin.
  • [22] Zakai, M. (1967). Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations. Israel J. Math. 5 170-176.