engleski

On a class of stochastic evolution equations - Pathwise approach

Let X be a Banach space, L(X) the Banach algebra of bounded, linear operators on X with uniform topology, and A_j in C(R,L(X)), i=1,...,d (noncommutative) families of linear operators. Let w=(w_j)_{j=1}^d be a real, d-dimensional, Wiener white noise. For u_0 in X, the following evolution equation is solved: partial u(t) / partial t = sum_{j=1}^d w_j(omega,t)A_j(t)u(t), (1) u(0) = u_0. (2) White noise, itself, is given as a distribution-valued stochastic process, w from Omega to D'(R)^d with characteristic functional: C_w(phi_1,...,phi_d)= E(exp {i\sum_j (w_j(.),\phi_j )}) := exp {-1/2 {sum_j ||phi_j||^2} }, with ||\phi||^2:= int |phi(t)|^2 dt. It is well known that almost all trajectories of white noise are distributional derivatives of continuous functions (i.e. of Brownian motion trajectories). Thus, equation (1) may be interpreted pathwise, in the space D'(R,X), provided, multiplication of distributions is suitably defined. To that end, problem (1)-(2) is shifted into the framework of Colombeau algebra of random generalized functions with values in X. It is an associative, differential algebra, containing distributions in a canonical way, preserving differentiation of distributions and multiplication of C^infinity-functions. Then, generalized random evolution family U(t,s), t,s in R is constructed, from which the unique solution reads: u(t)=U(t,0)u_0. Next, it is shown that solution have generalized moments of all orders, and finally, the expectation, i.e. deterministic generalized function Eu, is proved to admit an associated distribution v in C^1(R,X), which satisfies the equation: partial v(t) / partial t = 1/2 sum_{j=1}^d A_j(t)^2 v(t).

generalized random functions; white noise; expectation of solution

nije evidentirano