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Example text

90) Therefore, we will get H¨older-continuity provided that we can establish an estimate of the following type for some γ > 0 and ρ > 0: J(t , x ; s, y) ≤ c(|t − s|γ + |x − y|ρ ). Indeed, this will establish γ21 -H¨older continuity in time, and nuity in space, for all γ1 ∈ ]0 , γ[ and ρ1 ∈ ]0 , ρ[. (91) ρ1 older 2 -H¨ conti- Analysis of J(t , x ; s , y) If there were no absolute values around the increments of G, then we could use the Fourier transform to rewrite J(t , x ; s , y), in the case x = y and s > t, for instance, as s J(t , x ; s , x) = dr R2 0 μ(dη) |F G(t − r)(η) − F G(s − r)(η)|2 (92) t + dr R2 s μ(dη) |F G(t − r)(η)| .

Let L be a partial diﬀerential operator with constant coeﬃcients. A basic example is the wave operator ∂2f (10) Lf = 2 − Δf. ∂t Then there is a (Schwartz) distribution G ∈ S (R+ × Rd ) such that the solution of the PDE (11) L u = ϕ, ϕ ∈ S (Rd ), is ∗ ϕ u = G (t,x) (12) ∗ denotes convolution in the (t , x)-variables. We recall that S (Rd ) where (t,x) denotes the space of smooth test functions with rapid decrease, and S (R+ × Rd ) denotes the space of tempered distributions [15]. The Stochastic Wave Equation 43 When G is a function, this convolution can be written u(t , x) = R+ ×Rd G(t − s , x − y) ϕ(s , y) ds dy.

Moreover, we have by (128) and (134): For all t, t ∈ [0 , T ] and x, x ∈ [0 , L], 2 p E |U (x , t) − U (x , t )| p ≤ A |(x , t) − (x , t )| . 13, page 14) together prove that U has a modiﬁcation which is continuous, in our inhomogeneous norm on (x , t), of any order < 1. Because our norm is equivalent to the usual Euclidean norm, this proves continuity in the ordinary sense. 9. Complete the proof. Be certain that you understand why we have derived H¨ older continuity. For example, prove that there is a modiﬁcation of our solution which is H¨ older continuous in x of any given order < 12 ; and it is H¨ older continuous in t of any given order < 14 .