Let X = {X(t), t epsilon R+} be a real-valued symmetric Levy process with local times {L-t(x), (t, X) epsilon R+ x R} and characteristic function e(i lambda X(t)) = et(-t psi(lambda)). Let sigma(2)(0)(x-y) = 4/pi integral(infinity)(0) sin(2)(lambda(x-y)/2)/psi(lambda) d lambda. If sigma(2)(0)(h) is concave, and satisfies some additional very weak regularity conditions, then for any p >= t and all t epsilon R+, lim(h down arrow 0)integral(b)(a)vertical bar L-t(x+h) - L-t(x)/sigma(0)(h)vertical bar(p) dx = 2(p/2) E vertical bar eta vertical bar(p) integral(b)(a) vertical bar L-t(x)vertical bar(p/2) dx for all a, b in the extended real line almost surely, and also in L-m, m >= 1. (Here eta is a normal random variable with mean zero and variance one.) This result is obtained via the Eisenbaum Isomorphism Theorem and depends on the related result for Gaussian processes with stationary increments, {G(x), x epsilon R-1}, for which E(G(x) - G(Y))(2) = sigma(2)(0)(x-y); lim(h down arrow 0)integral(b)(a)vertical bar G(x+h) - G(x)/sigma(0)(h)vertical bar(p) dx = E vertical bar eta vertical bar(p) (b-a) for all a,b epsilon R-1, almost surely.
