an:01453271
Zbl 0970.60068
Al??s, E.; Nualart, D.; Viens, F.
Stochastic heat equation with white-noise drift
EN
Ann. Inst. Henri Poincar??, Probab. Stat. 36, No. 2, 181-218 (2000).
00064343
2000
j
60H15 35K05 60H25 60H05
anticipative stochastic evolution equation; Skorokhod integral; stochastic semigroup; white noise drift; real line; backward heat kernel
The authors study the existence and uniqueness of solutions for a one-dimensional anticipative stochastic evolution equation on the real line
\[
u(t,x)=\int_{\mathbb R} p(0,t,y,x)u_0(y) dy + \int_{\mathbb R} \int_0^t p(s,t,y,x)F(s,y,u(s,y)) dW_{s,y}
\]
driven by a two-parameter Wiener process \(W_{t,x}\) and based on a stochastic semigroup defined by the kernel \(p(s,t,y,x)\). This kernel is supposed to be measurable w.r.t. the increments of the Wiener process on \([s,t]\times\mathbb R\). The results are based on \(L^p\)-estimates for the Skorokhod integral. As an application they establish the existence of a weak solution for the following heat equation on the real line subject to white noise drift,
\[
\partial_t u(t,x)=\partial_x^2 u(t,x)+\dot{v}(t,x)\partial_x u(t,x)+F(t,x,u)\partial_t\partial_x W(t,x),
\]
where \(\dot{v}\) is a white noise in time.
Dirk Bl??mker (Augsburg)