The Annals of Probability

Evolution equation of a stochastic semigroup with white-noise drift

David Nualart and Frederi Viens

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Abstract

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

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

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160111

Digital Object Identifier
doi:10.1214/aop/1019160111

Mathematical Reviews number (MathSciNet)
MR1755997

Zentralblatt MATH identifier
1044.60052

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

Keywords
Stochastic parabolic equations anticipating stochastic calculus Skorohod integral stochastic semigroups

Citation

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. https://projecteuclid.org/euclid.aop/1019160111


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