Moment Bounds on Hölder norms of stochastic processes

Question

It is relatively easy to show that a stochastic process is Hölder continuous using Kolmogorov continuity theorem link text. But how does one obtain a bound $\mathbb{E} \left\Vert u\right\Vert _{\gamma}^{a}$ for e.g. an SPDE.

Thanks,

warsaga

Answer 1

In spirit it is similar to the deterministic PDEs where one needs to use the regularity of the generator eg. its spectrum. The following presentation is from "Stochastic Equations in Infinite Dimensions". Let's look at the linear case with additive noise

$$d X(t) = (AX(t) + f (t))dt + BdW(t),$$

where $A: D(A) \subset H \to H$ and $B : U \to H$ are linear operators and $f$ is an $H-$valued stochastic process. For generator $A$, let $S$ be the semigroup generated by it, then

$$X(t) = S(t)X_0 + \int_{0}^{t}S(t − s) f (s)ds + \int_{0}^{t} S(t − s)BdW(s),$$

solve the SPDE. Let $W_{A}(t):= \int_{0}^{t} S(t − s)BdW(s)$.

Removing the White noise Indeed, to estimate moments as the one you mentioned, I would take a look at "4.6 Basic estimates" that return to the deterministic case. For example

So you see here that they transfer moments for the semigroup-formulation to those of the semigroup itself up to some loss in the moments. This is proved for general $\Phi$ and so here you can insert the Holder difference too. Then in "5.3 Continuity of weak solution" they use those estimates to obtain Holder regularity they show in theorems 5.14,5.15 a Holder regularity for them

Using the spectrum Another way is using the spectral information of the operator A eg. "The case when A is self-adjoint". Using the following bounds

they obtain a nice Holder-moments result

and then they apply Kolmogorov result for random fields theorem 3.5.