Quasiregular stochastic convolutions

Convolution-like Structures, Differential Operators and Diffusion Processes

“…Condition III requires the existence of an integral transform with bounded kernels which determines uniquely a given measure 𝜇 ∈ M C (𝐸) (in the sense that if 𝜇(𝜗) = 𝜈(𝜗) for all 𝜗 ∈ Θ, then 𝜇 = 𝜈) and trivializes the convolution in the same way as the Fourier transform trivializes the ordinary convolution. As noted in [184], it is possible, in principle, to study infinite divisibility of probability measures on measure algebras not satisfying Condition III; however, it is natural to require Condition III to hold, not only because, to the best of our knowledge, all known examples of convolution-like structures are constructed from a product formula of the form 𝜗(𝑥)𝜗(𝑦) = (𝛿 𝑥 𝛿 𝑦 ) (𝜗) (and therefore possess such a trivializing family of functions) but also because this trivialization property leads to a richer theory. Lastly, condition IV expresses the motivating goal discussed in the Introduction: the Feller semigroup {𝑇 𝑡 } should have the convolution semigroup property with respect to the operator or, in other words, the Feller process {𝑋 𝑡 } 𝑡 ≥0 determined by {𝑇 𝑡 } is a Lévy process with respect to in the sense that we have…”

On the construction of convolution-like operators associated with multidimensional diffusion processes

“…Condition III requires the existence of an integral transform with bounded kernels which determines uniquely a given measure µ ∈ M C (E) (in the sense that if µ(ϑ) = ν(ϑ) for all ϑ ∈ Θ, then µ = ν) and trivializes the convolution in the same way as the Fourier transform trivializes the ordinary convolution. As noted in [59], it is possible, in principle, to study infinite divisibility of probability measures on measure algebras not satisfying Condition III; however, it is natural to require Condition III to hold, not only because, to the best of our knowledge, all known examples of convolution-like structures are constructed from a product formula of the form ϑ(x)ϑ(y) = (δ x δ y )(ϑ) (and therefore possess such a trivializing family of functions) but also because this trivialization property leads to a richer theory. Lastly, condition IV expresses the probabilistic motivation mentioned above: the Feller semigroup {T t } is conservative and has the convolution semigroup property with respect to the operator ; in other words, a Feller process {X t } t≥0 associated to {T t } is a Lévy process with respect to , in the sense that we have P X t ∈ •|X s = x = γ t−s δ x for every 0 ≤ s ≤ t and x ∈ E.…”