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January 2000 Evolution equation of a stochastic semigroup with white-noise drift
David Nualart, Frederi Viens
Ann. Probab. 28(1): 36-73 (January 2000). DOI: 10.1214/aop/1019160111


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 $\Delta_x + v(dt, x) \cdot \nabla_x$ .


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David Nualart. Frederi Viens. "Evolution equation of a stochastic semigroup with white-noise drift." Ann. Probab. 28 (1) 36 - 73, January 2000.


Published: January 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60052
MathSciNet: MR1755997
Digital Object Identifier: 10.1214/aop/1019160111

Primary: 60H15
Secondary: 60H07

Keywords: anticipating stochastic calculus , Skorohod integral , Stochastic parabolic equations , stochastic semigroups

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 1 • January 2000
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