Lorenzo Toniazzi

Consider the Caputo evolution equation (EE) \(\partial^\beta_t u = \Delta u\) with initial condition \(\phi\) on \(\{0\}\times\mathbb R^d\). As it is well known, the solution reads \(u(t, x) = \mathbf{E}_x\left[\phi (B_{E_t})\right]\). Here \(B_t\) is a Brownian motion and the independent time change \(E_t\) is an inverse \(\beta\)-stable subordinator. This is a popular model for subdiffusion [6], with remarkable universality properties [1]. We substitute the Caputo derivative \(\partial^\beta_t\) with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a time-nonlocal initial condition \(\phi\) on \((-\infty, 0] \times \mathbb{R}^d\). We derive the new stochastic representation for the solution, namely \(u(t, x) = \mathbf{E}_x\left[\phi(-W_t, B_{E_t})\right]\). This stochastic representation has a pleasing interpretation due to the non-obvious presence of \(W_t\). Here \(W_t\) denotes the waiting/trapping time of the subdiffusion \(B_{E_t}\). We discuss classical-wellposedness for the space-fractional case, following [7]. Additionally, following [3, 4], we discuss weak-wellposedness for the respective extensions of Caputo-type EEs (such as in [2, 5]).

References
[1] Barlow, Martin T.; Cerny, Jiri (2011). Probability theory and related fields, 149.3-4: 639-673.
[2] Chen Z-Q., Kim P., Kumagai T., Wang J. (2017). arXiv:1708.05863.
[3] Du Q., Yang V., Zhou Z. (2017). Discrete and
continuous dynamical systems series B, Vol 22, n. 2.
[4] Du Q., Toniazzi L., Zhou Z. (2018).
Preprint. Expected submission date: Sept. 2018.
[5] Hernández-Hernández, M.E., Kolokoltsov, V.N., Toniazzi, L. (2017). Chaos, Solitons & Fractals, 102, 184-196.
[6] Meerschaert, M.M., Sikorskii, A. (2012). Stochastic Models for Fractional Calculus, De Gruyter
Studies in Mathematics, Book 43.
[7] Toniazzi L. (2018). To appear in J Math Anal Appl. arXiv:1805.02464.