Abstract
We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.
Highlights
We study the space-time coupled evolution equation (CEE)
That −Hν = (∂t − L)ν is the subordination of the heat operator (∂t − L) by an infinite Lévy measure ν
We prove the stochastic representation for the solution u(t, x) to (1.1) to be
Summary
We could not treat this case, as our method relies on Dynkin formula, and we could not recover a suitable version for the left continuous process Ssν− This case is treated in the general setting of space-time Feller semigroups in [4], as discussed below. Besides the introduction of initial conditions on the past for CEEs, this work appears to be the first one that formulates and solves the governing equation of Y in differential form, without relying on Fourier-Laplace transform techniques This was part of the contribution of [4], which treats different CEEs, as mentioned above. The article is organised as follows; Section 2 introduces general notation, our assumptions and the main semigroup results used to treat the operator Hν ; Section 3 proves Theorem 3.5 and presents some concrete fundamental solutions to (1.1); Section 4 proves Theorem 4.8
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