An L (p) -theory for stochastic integral equations

作者：Desch Wolfgang^*; Londen Stig Olof

来源：Journal of Evolution Equations, 2011, 11(2): 287-317.

DOI：10.1007/s00028-010-0092-9

• Stochastic integral equations
• Stochastic fractional differential equation
• Regularity
• Nonnegative operator
• Volterra equation
• Singular kernel

We investigate the stochastic parabolic integral equation of convolution type
u = k(1) * A(p)u + Sigma(infinity)(k=1)k(2) star g(k) + u(0), t >= 0,
and develop an L-p-theory, 2 <= p < infinity, for this equation. The solution u is a function of iota, omega, x with omega in a probability space and x is an element of B, a sigma-finite measure space with positive measure Lambda. The kernels k(1)(t), k(2)(t) are powers of t, i.e., multiples of t(alpha-1), t(beta-1), with alpha is an element of (0, 2), beta is an element of (1/2, 2), respectively. The mapping A(p) is such that - A(p) is a nonnegative linear operator of D(A(p)) subset of L-p(B) into L-p(B). The convolution integrals k(2)star g(k) are stochastic Ito-integrals. By combining an approach due to Krylov with transformation techniques and estimates involving fractional powers of (- A(p)), we obtain existence and uniqueness results. In the case where A(p) is the Laplacian, with B = R-n, sharp regularity results are obtained.

• 出版日期2011-6