In his comment [1] on our paper [2], Pikovsky contrasts two definitions for the phase of a stochastic oscillator by way of an analytically solvable model system. In [3] the phase is defined in terms of a system of isochrons Σθ, analogous to Poincaré sections, with the property that for any initial condition on one isochron Σa, the mean first passage time (MFPT) to a second isochron Σb, b > a, w...

We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ α t u(x, t) = Lu(x, t), where 0 < α ≤ 2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of ths weak solutions and the asymptotic behaviour as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. ...

We prove an uncertainty principle for certain eigenfunction expansions on L2(R+,w(r)dr) and use it to analogues of theorems Chernoff Ingham Laplace-Beltrami operators compact symmetric spaces, special Hermite operator Cn Rn.

is completely integrable and hence L(k) is in a commuting system of differential operators with n algebraically independent operators. Then we have the following fundamental result (cf. [1]). Theorem [Heckman, Opdam]. When kα are generic, the function F (λ, k;x) has an analytic extension on R and defines a unique simultaneous eigenfunction of the commuting system of differential operators with ...