Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials

We present four types of infinitely many exactly solvable Fokker-Planck equations, which are related to the newly discovered exceptional orthogonal polynomials. They represent the deformed versions of the Rayleigh process and the Jacobi process.

💡 Research Summary

The paper investigates four infinite families of exactly solvable Fokker‑Planck equations that are deformations of the classical Rayleigh (Bessel) and Jacobi diffusion processes. The construction is based on the recent discovery of exceptional orthogonal polynomials (EOP), namely the Xℓ‑Laguerre and Xℓ‑Jacobi families, which extend the ordinary Laguerre and Jacobi polynomial systems by allowing “missing degrees” while preserving orthogonality and completeness.

Starting from the well‑known correspondence between one‑dimensional diffusion processes and Schrödinger‑type quantum Hamiltonians, the authors first recall that a standard Rayleigh process on (0,∞) and a Jacobi process on (0,1) can be written as a similarity‑transformed Hamiltonian H = –∂ₓ² + U(x). The ground‑state wavefunction φ₀(x) of H determines the drift term b(x)=2∂ₓlnφ₀(x) and the stationary density w(x)=φ₀(x)². In the conventional case φ₀ is built from the weight functions of the ordinary Laguerre or Jacobi polynomials, and the eigenfunctions are the corresponding classical orthogonal polynomials with eigenvalues λₙ=n.

The novelty of the work lies in replacing φ₀ by a deformed ground state φ̃₀(x)=φ₀(x)·ξ_ℓ(x), where ξ_ℓ(x) is the non‑polynomial factor that appears in the exceptional families. This modification leads to a new drift term

 b̃(x)=b(x)+2∂ₓlnξ_ℓ(x)

while the diffusion coefficient remains unity. The stationary density becomes w̃(x)=φ̃₀(x)², i.e. the original weight multiplied by ξ_ℓ(x)². Because the exceptional polynomials are eigenfunctions of a shape‑invariant Hamiltonian, the spectrum stays unchanged (λₙ=n) and the eigenfunctions are now the Xℓ‑Laguerre or Xℓ‑Jacobi polynomials Pₙ^{(ℓ)}(x).

Four concrete models are presented:

1. Xℓ‑Laguerre I (L1) – the deformation uses ξ_ℓ^{(L1)}(x)=L_ℓ^{(α)}(−x). The drift acquires an additional term 2∂ₓlnL_ℓ^{(α)}(−x).

2. Xℓ‑Laguerre II (L2) – ξ_ℓ^{(L2)}(x)=L_ℓ^{(−α−1)}(x) yields a different rational correction to the drift.

3. Xℓ‑Jacobi I (J1) – on the interval (0,1) the factor ξ_ℓ^{(J1)}(x)=P_ℓ^{(α,β)}(1−2x) modifies the linear drift of the standard Jacobi process.

4. Xℓ‑Jacobi II (J2) – ξ_ℓ^{(J2)}(x)=P_ℓ^{(−α−1,−β−1)}(2x−1) provides a second independent deformation.

For each case the authors give explicit expressions for the drift, the stationary density, the normalization constants, and they prove orthogonality

 ∫ w̃(x) Pₙ^{(ℓ)}(x) Pₘ^{(ℓ)}(x) dx ∝ δₙₘ.

Because the Fokker‑Planck operator remains self‑adjoint under the inner product weighted by w̃, all spectral properties of the original process are retained. The paper also discusses how the extra rational terms in the drift generate new effective potential barriers, which can dramatically affect first‑passage time statistics and other observables. These features make the deformed processes attractive for modeling phenomena where the standard Rayleigh or Jacobi dynamics are too restrictive, such as population dynamics with Allee effects, stochastic volatility models in finance, or diffusion‑reaction systems with localized traps.

Finally, the authors emphasize that the exceptional‑polynomial framework supplies a systematic method to construct non‑trivial, exactly solvable diffusion processes beyond the classical families. The approach can be extended to multi‑dimensional settings, to time‑dependent drifts, or to other shape‑invariant potentials, opening a broad avenue for future research in stochastic analysis, mathematical physics, and applied probability.