Fractional stochastic partial differential equations in space and time

Abstract

In this thesis, we study deterministic and stochastic abstract evolution equations which are of fractional order, either spatially or spatiotemporally.

The preliminary concepts necessary to interpret such equations are collected in the first chapter.

The second chapter is devoted to analyzing a class of equations which generalizes the spatial Whittle–Matérn stochastic partial differential equations (SPDEs) to space–time. We define solution concepts, establish their well-posedness and equivalence, and study their spatiotemporal regularity and covariance structure. The abstract spatial operator in these equations is assumed to be the negative generator of a strongly continuous semigroup on a separable Hilbert space; for certain spatial regularity results, it is moreover assumed to be analytic.

In the third chapter, we define various higher-order Markov properties for Hilbert-space-valued stochastic processes and investigate the relations between them. For solutions to linear abstract SPDEs, we identify two sets of additional conditions under which locality of the precision operator is either necessary or sufficient to satisfy the weakest Markov property. We show that the mild solutions from Chapter 2 satisfy a higher-order Markov property if the orders of the equations are non-fractional (i.e., integer) and that, conversely, a necessary condition for the weakest Markov property is, in general, not satisfied if this parameter is fractional. We moreover establish that an infinite-dimensional analog of the fractional Brownian motion can be obtained as a limiting case of this class of equations.

The fourth chapter concerns the deterministic natural Dirichlet problem for nonlocal abstract space–time operators posed on an arbitrary Banach space. We impose that the whole "past" of the solution equals a given function, i.e., its values are prescribed at all times preceding some (non-trivial) initial time. If the semigroup associated to the spatial operator is exponentially stable, we show that the problem is well-posed in an L^p-sense with p in [1, infinity]. Whenever such solutions are continuous, they satisfy a mild solution formula which expresses them in terms of the initial data and the semigroup, thus generalizing the well-known variation of constants formula for the first-order abstract Cauchy problem. A comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems is included.

Finally, in the fifth chapter, we study the convergence of a sequence of semilinear parabolic stochastic evolution equations posed on a sequence of Banach spaces approximating a limiting space. The abstract "discrete-to-continuum approximation" setting is encoded using projection and lifting operators. These allow us to define and compare the discretized equations, and to formulate conditions under which their solutions are well posed and convergent when lifted to a common state space. Our framework is applied to the case where the limiting problem is an SPDE whose linear part is a generalized Whittle–Matérn operator on a manifold, discretized by a sequence of geometric graphs constructed from a (random) point cloud.