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Evolution equation of a stochastic semigroup with white-noise drift
Nualart, David Viens, Frederi
Nualart, David
Viens, Frederi
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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 L2(Rd).
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 Δx+v(dt,x)⋅∇x .
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This is the published version, also available here: http://dx.doi.org/10.1214/aop/1019160111.
Date
2000-09-20
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Institute of Mathematical Statistics (IMS)
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Nualart, David; Viens, Frederi. Evolution equation of a stochastic semigroup with white-noise drift. Ann. Probab. 28 (2000), no. 1, 36--73. http://dx.doi.org/10.1214/aop/1019160111.