Stochastic integrals for spde’s: A comparison

Abstract

We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $$k \ge 1$$

“…where we have used elementary properties of the Fourier transform (see also Dalang [3], Nualart and Quer-Sardanyons [18], and Dalang and Quer-Sardanyons [8] for properties of the stochastic integral). This last formula is convenient since…”

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

“…The two different approaches to solving SPDEs that we have presented above, using on the one hand, two-parameter "space-time white noise", and on the other hand, infinite-dimensional processes, have each generated a considerable literature; nonetheless there are important cases where they both give rise to the same solution, see e.g. [16].…”

Moderate deviations for a fractional stochastic heat equation with spatially correlated noise

“…In order to prove our main result, we are going to give other equivalent approach to the solution of Eq α δ,ε pd, b, σq, see [11]. To start with, let us denote by H the Hilbert space obtained by the completion of SpR d q with the inner production xφ, ψy H :"…”

“…In the following, we can define the stochastic integral with respect to cylindrical Wiener process pB t phqq tě0 (see e.g. [9] or [11]) of any predictable square-integrable process with values in H as follows…”