On the solution of fractional differential equations: a probabilistic approach

In the last decades, the theory of fractional differential equations (FDE's) has been actively studied due to its successful application at providing more accurate models to describe a variety of physical phenomena. To solve FDE's various numerical and analytical approaches have been investigated (e.g., the Laplace transform, the Mellin transform, the Fourier transform techniques, the operational calculus method). As for a probabilistic framework, some connections between probability and fractional differential equations are also known in the literature; for instance, the probabilistic interpretation of the Green (or fundamental) solution to the time-space fractional diffusion equation.

In this talk we aim to give a general overview of a probabilistic approach to study the
well-posedness for boundary value problems for fractional ordinary differential equations such as

Dβa+*u(x) = λ(x)u(x) − g(x), x є (a, b]

and, more generally, fractional partial differential equations of the type

tDβ a+*u(t, x) = Au(t,x) − g(t, x), t є (a, b], x є Rd

u(a,x) = φa(x), x є Rd

where Dβa+* stands for the fractional Caputo derivative of order β є (0, 1) and A is the infinitesimal generator of a Markov process. The approach presented here is based on the probabilistic interpretation of fractional Caputo derivatives as the generators of β−stable processes interrupted on the first attempt to cross certain boundary point.