Abstract
We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.
Highlights
The study of the evolution equations with complex spatial variables is a quite recent research topic in the theory of the partial differential equations and complex analysis
We complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable
This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator
Summary
The study of the evolution equations with complex spatial variables is a quite recent research topic in the theory of the partial differential equations and complex analysis. ), u(x, 0) = f (x), where t > 0, x ∈ R and f ∈ BUC(R) This approach is interesting because, by exploiting the theory of semigroups of linear operators, it is possible to obtain a new complex version of the heat equation (see (2.2) below) and study the properties of its analytic solution (see [8,9]). We study the stochastic solution of the complex heat equation, when the time-derivative is replaced by a convolution-type operator, which generalizes the Caputo fractional derivative. The generalized fractional setting and the complex time-fractional heat equation are introduced, where the stochastic solution related to the related complex Cauchy problem is obtained. K=1 k=1 as in the previous point n (B(tk ))αk ∼ W N
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