# Filtering of Gaussian processes in Hilbert spaces

## TL;DR

This paper develops filtering techniques for infinite-dimensional Gaussian processes, deriving integral equations for the filter and error covariance, and applies these results to linear SPDEs driven by Gauss-Volterra processes observed at finite points.

## Contribution

It introduces a new filtering framework for infinite-dimensional Gaussian processes and derives integral equations, with applications to SPDEs driven by Gauss-Volterra processes.

## Key findings

- Derived integral equations for the filter and error covariance.
- Applied results to linear SPDEs with finite-point observations.
- Extended filtering theory to infinite-dimensional Gaussian processes.

## Abstract

Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.