Probabilistic representation formula for the solution of fractional high-order heat-type equations

Abstract

We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order α ∈ (0, 1).

Space-time duality for semi-fractional diffusions

“…2 ) leading to a probabilistic interpretation of a space-fractional 3) and h(x, t 0 ) (solid line) of the time-fractional pde (2.9) for fixed t 0 = 3.5 and α = 1.5 pde of order α ∈ (2, 3) by means of an inverse 1 α -stable subordinator. This stochastic solution is much stronger than the higher order approach in [4].…”

A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders

“…In addition, tree-based methods for differential equations are known to have deep relationships with Hopf algebras [CK99] in the case of Butcher trees [But63], with pseudo processes [BDM19], and they can be used to derive explosion criteria [Tao16].…”

“…Other probabilistic methods that can deal with higher order derivatives usually involve pseudo-processes created as limits of discrete random walks, see [BDM19]. However, the method developed in this paper is different, as instead of creating a specific process whose generator is behaving as a higher order derivative, we use codes that carry information on the branches along the tree.…”

Dimensional Universality of Schauder Estimates Constants for Fourth Order Heat-Type Equations

“…Schauder-type Estimates have been recently considered by many authors in the framework of integro-partial differential equations [12,27,28,38,43,42], however the interest around this kind of estimates for the higher-order parabolic equations is still present [6,13,14,15,48] and it fits in a wider tradition (see, e.g., [1,7,8,24,44]). This latter prompted us to investigate in such a direction.…”