In his 2019 article, Kalinichenko proposed an alternative way of doing stochastic integration in general separable Banach spaces [12].This way circumvents the usual UMD assumption on our separable Banach space X, and instead imposes a strict condition on the integrating process $\Phi : (0,T)\times\Omega\to \Lll(H,X)$. Namely, we require the existence of an $X$-valued Gaussian $g$ such that almost surely for all $x^*\in X^*$,
\[ \int_0^T \|\Phi(t,\omega)^* x^*\|_H^2 \ dt \leq \EE\langle g,x^*\rangle^2. \]
Most notably, this approach works in any separable Banach space. In this thesis we will take a closer look at the proofs used by Kalinichenko, and place his article in the context of the known theory on stochastic analysis in Banach spaces. We will compare the approach both to the UMD and martingale type 2 situation, and discuss the advantages and disadvantages of either strategy.

Moreover, we will compare the conditions imposed in [12] on the stochastic process to the condition of -radonification as assumed in the UMD case [25].

In the divisible sandpile model, we consider a collection of i.i.d. Gaussian heights on a finite graph. It was shown by Levine et al. (2015) that the odometer function in this case equals a discrete, bi-Laplacian field. Subsequently, Cipriani et al. (2016) proved that the scaling limit of the odometer is a continuum bi-Laplacian field, this time on the unit torus. In this thesis, we will determine the scaling limit of a divisible sandpile with an initial configuration of correlated Gaussians, where the covariance is given by a stationary covariance function K(x-y). We show that after appropriate scaling, the odometer still converges to a bi-Laplacian field on the unit torus.