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 result in the spirit of the Kolmogorov result.

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.

In spirit it is similar to the deterministic PDEs where one needs to use the regularity of the generator and various embedding theorems. At least in 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, in "Stochastic Equations in Infinite Dimensions" the authors cover a main regularity result. For generator $A$, let $S$ be the semigroup generated by it, then

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

solves the above linear spde. For $W_{A}(t):= \int_{0}^{t} S(t − s)BdW(s)$, they show in theorems 5.14,5.15 a Holder regularity for it

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 result in the spirit of the Kolmogorov result.