A maximal L-p-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Abstract

Let Z = ( Z(t))(t >= 0) be an additive process with a bounded triplet (0, 0,Lambda(t))(t >= 0). Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: A(Z) (t)u(t, x) = lim(h down arrow 0) E[u(t, x + Z(t+ h) - Zt) - u(t, x)]/h = integral(d)(R) (u(t, x + y) - u(t, x) - y center dot del(x)u(t, x)1(|y|<= 1))Lambda(t) (dy). Suppose that for any Schwartz function. on R-d whose Fourier transform is in C-c(infinity)(B-cs\B-cs-1), there exist positive constants N-0, N-1, and N-2 such that integral(d)(R) |E[phi(x + r(-1)Z(t))]|dx <= N(0)e(-N1t/s(r)), for all(r, t)is an element of (0, 1) x [0, T], and ||phi(mu) (r(-1)D)phi||L-1(R-d) <= N-2/s(r), for all r is an element of (0, 1). where s is a scaling function (Definition 2.4), c(s) is a positive constant related to s, mu is a symmetric Levy measure on R-d,phi(mu)(r(-1)D)phi(x) = F-1 [psi(mu)(r(-1)xi)F[phi]] (x) and psi(mu)(xi):= integral(d)(R) (eiy(center dot xi) - 1- iy center dot xi 1(|y|<= 1))mu(dy). In particular, above assumptions hold for Levy measures.t having a nice lower bound and mu satisfying aweak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Levy measures Lambda(t) and they do not have to be symmetric. In this paper, we establish the L p-solvability to the initial value problem partial derivative u/partial derivative t (t, x) = AZ(t)u(t, x), u(0, center dot) = u(0), (t, x). (0, T) x R-d, (0.2) where u(0) is contained in a scaled Besov space B-p,q(s;gamma-2/q)(R-d) (see Definition 2.8) with a scaling function s, exponent p is an element of(1,infinity), q is an element of [1,8), and order gamma is an element of [0,infinity). We show that equation (0.2) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z. In other words, there exists a unique solution u to equation (0.2) in L-q ((0, T); H-p(mu;gamma) (R-d)), where H-p(mu;gamma) (Rd) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies ||u||(Lq) ((0, T); H-p(mu;gamma) (R-d)) <= N ||u(0)|| B-p,q(s;gamma- 2/q)(R-d), where N is independent of u and u(0). We finally remark that our operators AZ (t) include logarithmic operators such as -a(t) log(1-Delta) (Corollary 3.2) and operators whose symbols are non-smooth such as - Sigma(d)(j =1) c(j) (t)(-Delta)(xj)(alpha/2) (Corollary 3.9).