The Dynkin-type isomorphism theorem of Le Jan, Marcus and Rosen (Ann. Probab. 2015) relates the continuous additive functionals LtνL_t^\nu of a transient Borel right process — indexed by bounded measures ν\nu and constructed through a loop soup — to permanental fields, and proves that {Ltν,(t,ν)∈R+×V}\{L_t^\nu,(t,\nu)\in\mathbb R_+\times\mathcal V\} is jointly continuous under sufficient hypotheses: abstractly, uniform continuity of the potential-convolutions in ν\nu plus a tightness condition for general Borel right processes, and concretely, for exponentially killed Lévy processes, L2L^2-control of the potential density, radial regularity of its Fourier transform, and a Fourier integrability condition on ν^\hat\nu, with ν\nu restricted to Revuz measures R+(X)\mathcal R^+(X). The open question the source itself poses (Remark 6.1) about the exact domain of validity of this theorem concerns the measure-side restriction: ν∈R+(X)\nu\in\mathcal R^+(X) would be automatic if Getoor's conjecture held — that, apart from extremely nonsymmetric degenerate exceptions such as uniform motions (deterministic drift), every Lévy process satisfies Hunt's hypothesis (H) (every semipolar set is polar) — a problem the 2021 Hu–Sun survey still calls a major unsolved problem of Markov-process potential theory, open even for general pure-jump subordinators (nondecreasing Lévy processes).

Background

A continuous additive functional (CAF) of a Markov process generalizes Brownian local time: it is an adapted, almost surely continuous and nondecreasing process AtA_t with A0=0A_0=0 satisfying the flow identity At+s=At+As∘θtA_{t+s}=A_t+A_s\circ\theta_t, where θt\theta_t is the path-shift operator. For a measure ν\nu on the state space one wants a CAF LtνL_t^\nu behaving like an integral of pointwise local times against ν\nu; working directly with measure-indexed functionals avoids requiring pointwise local times, which need not exist. The setting of the source paper is a Borel right process (a Markov process with right-continuous paths and the strong Markov property, in the sense of the general theory of Markov processes) that is transient, so that its potential density u(x,y)=∫0∞pt(x,y) dtu(x,y)=\int_0^\infty p_t(x,y)\,dt, the expected total occupation kernel, is finite off a small set; crucially, uu is not assumed symmetric (the process need not be reversible) and may be infinite on the diagonal. Each bounded positive measure ν\nu that arises as the Revuz measure of some CAF (Revuz duality associates a measure to every additive functional through the resolvent; the class is denoted R+(X)\mathcal R^+(X)) determines a CAF LtνL_t^\nu. A classical obstruction: because a CAF can only increase while the process sits on the charged set, the Revuz measure of a CAF cannot charge semipolar sets — countable unions of thin sets, where a set is thin if the process cannot hit it instantaneously. A set is polar if the process, started anywhere, never hits it. Hunt's hypothesis (H) asserts that every semipolar set is polar. When excessive functions (the Markov-process analogue of nonnegative superharmonic functions) are lower semicontinuous, (H) is equivalent to the bounded positivity, energy, maximum, and regularity principles of potential theory, to the dichotomy of capacity, and to the statement that every natural additive functional is continuous; in particular, for a Lévy process (a process with stationary independent increments on Rd\mathbb R^d) satisfying (H), every bounded measure charging no polar set is the Revuz measure of a CAF, so the restriction ν∈R+(X)\nu\in\mathcal R^+(X) becomes vacuous.

Le Jan–Marcus–Rosen connect these functionals to permanental fields: processes ψ={ψ(ν),ν∈V}\psi=\{\psi(\nu),\nu\in\mathcal V\} indexed by measures whose joint moments are α\alpha-permanents of the matrix (∬u dνi dνj)\big(\iint u\,d\nu_i\,d\nu_j\big), a generalization of second-order Gaussian chaoses (squares of Gaussian processes correspond to the symmetric case), realized as renormalized sums of continuous additive functionals over a loop soup, a Poisson ensemble of Markov loops. A Dynkin-type isomorphism theorem relates ψ\psi to the terminal functionals L∞νL^\nu_\infty under a change of measure, and continuity of ψ\psi transfers to {L∞ν}\{L^\nu_\infty\} in the index ν\nu; the index space is metrized by a proper norm, a norm on measures under which the nn-fold integrals governing permanental moments obey product bounds. The hard step is joint continuity of {Ltν,(t,ν)∈R+×V}\{L_t^\nu,(t,\nu)\in\mathbb R_+\times\mathcal V\}. Theorem 5.1 of the paper (via Lemma 5.1) proves it for general transient Borel right processes under two abstract hypotheses: (i) the first and second uu-convolutions ν↦∫u(y,z) dν(z)\nu\mapsto\int u(y,z)\,d\nu(z) and ν↦∬u(y,z)u(z,w) dφ(w) dν(z)\nu\mapsto\iint u(y,z)u(z,w)\,d\varphi(w)\,d\nu(z) are continuous in ν\nu, uniformly in the auxiliary variables, and (ii) a tightness condition on {L∞ν}\{L^\nu_\infty\} under the loop measures Qρ,φQ^{\rho,\varphi}. Theorem 1.4 verifies concrete hypotheses when XX is a Lévy process killed at an independent exponential time with characteristic exponent κ\kappa (the logarithm of its characteristic function, so that u^=1/κ\hat u=1/\kappa): ∥u∥2<∞\|u\|_2<\infty together with integrability of e−Re κe^{-\mathrm{Re}\,\kappa}, radial regularity of u^\hat u at infinity (two-sided comparison with a regularly varying function, that is, one with asymptotic power-law behaviour), and the Fourier condition ∫1∞(∫∣ξ∣≥x∣ν^(ξ)∣2γ(ξ) dξ)1/2dxx<∞\int_1^\infty\big(\int_{|\xi|\ge x}|\hat\nu(\xi)|^2\gamma(\xi)\,d\xi\big)^{1/2}\frac{dx}{x}<\infty with γ=∣u^∣∗∣u^∣\gamma=|\hat u|*|\hat u|; then {Ltνx}\{L_t^{\nu_x}\} is jointly continuous with an explicit modulus of continuity.

Earlier frameworks for continuity of local times are more restrictive: the symmetric theory via Gaussian isomorphisms developed in the Marcus–Rosen monograph, point-indexed local times of Borel right processes (Eisenbaum–Kaspi 2007, who connect continuity conditions to a necessary-and-sufficient condition for continuity of an associated Gaussian process obtained as a central-limit limit), and Marcus–Rosen's sufficient condition for joint continuity via permanental processes (2013), where necessity is known only in the square-of-Gaussian subcase. Necessity of none of the sufficient conditions of the source paper — abstract or Fourier-analytic — is known, and the Lévy-specific machinery has no analogue on a general state space.

Problem Statement

Let XX be a transient Borel right process with state space SS and potential density u(x,y)u(x,y), not assumed symmetric and allowed to be infinite on the diagonal. A norm ∥⋅∥\|\cdot\| on the space M(S)\mathcal M(S) of finite signed Radon measures on SS is called proper for uu (Le Jan–Marcus–Rosen, Eqs. (1.4)–(1.5)) if for every n≥2n\ge2 and all ν1,…,νn∈M(S)\nu_1,\dots,\nu_n\in\mathcal M(S), ∣∫∏j=1nu(xj,xj+1)∏j=1ndνj(xj)∣  ≤  Cn∏j=1n∥νj∥\Big|\int\prod_{j=1}^{n}u(x_j,x_{j+1})\prod_{j=1}^{n}d\nu_j(x_j)\Big|\;\le\;C^{n}\prod_{j=1}^{n}\|\nu_j\| for a universal constant C<∞C<\infty, with the cyclic convention xn+1=x1x_{n+1}=x_1. Fix such a proper norm and let V⊆R∥⋅∥+\mathcal V\subseteq\mathcal R^{+}_{\|\cdot\|} — the positive Revuz measures of finite ∥⋅∥\|\cdot\|-norm — be a separable locally compact metric space with metric dd inducing some topology (as in their Lemma 5.1); continuity in the index ν\nu below is meant with respect to this topology. In the Lévy case of their Theorem 1.4, the family V={νx,x∈Rd}\mathcal V=\{\nu_x,x\in\mathbb R^d\} is metrized by the proper norm ∥ν∥γ,2=(∫∣ν^(ξ)∣2γ(ξ) dξ)1/2\|\nu\|_{\gamma,2}=\big(\int|\hat\nu(\xi)|^2\gamma(\xi)\,d\xi\big)^{1/2} with γ=∣u^∣∗∣u^∣\gamma=|\hat u|*|\hat u|. Say that joint continuity holds for V\mathcal V if the continuous additive functionals {Ltν, (t,ν)∈R+×V}\{L_t^\nu,\,(t,\nu)\in\mathbb R_+\times\mathcal V\} admit a version that is jointly continuous PxP^x-almost surely for every x∈Sx\in S.

The joint-continuity theorems of Le Jan–Marcus–Rosen impose the measure-side hypothesis ν∈R+(X)\nu\in\mathcal R^+(X), and Remark 6.1 of the paper notes that, in all cases where Hunt's hypothesis (H) is known for a Lévy process, this condition is superfluous. The research objective is to settle the status of this hypothesis for Lévy processes, which is Getoor's conjecture in its standard formulation: apart from the degenerate exceptions already known to violate (H) — the uniform motions (deterministic drift) and, more generally, the processes admitting a uniform-motion projection, such as the drift-plus-degenerate-Gaussian process of Example 3.5 of the Hu–Sun survey — the recognized obstruction being a nonzero drift, so that a subordinator satisfying (H) must have zero drift (Hu–Sun 2012, Prop. 1.6; survey Prop. 2.24) — does every Lévy process on Rd\mathbb R^d satisfy Hunt's hypothesis (H), that every semipolar set is polar? The sources state this exception class only informally (Remark 6.1 of Le Jan–Marcus–Rosen speaks of 'essentially all' Lévy processes; the Hu–Sun survey, of 'essentially all Lévy processes ... except for some extremely nonsymmetric cases like uniform motions'); for this task the scope is fixed operationally by the description above, so that a process lies in scope exactly when it is not one of these recognized exceptions. Under (H), every bounded measure charging no polar set is the Revuz measure of a continuous additive functional, so the restriction ν∈R+(X)\nu\in\mathcal R^+(X) in the joint-continuity theorems would be automatic. The conjecture is currently open even for pure-jump subordinators (nondecreasing Lévy processes without drift or Gaussian component) with infinite Lévy measure, for one-dimensional Lévy processes with ∫(1∧∣x∣) μ(dx)=∞\int(1\wedge|x|)\,\mu(dx)=\infty, and for sums of independent Lévy processes that individually satisfy (H) (Questions 5–7 of the Hu–Sun survey). The deliverable is the single question of the status of Hunt's hypothesis (H) — Getoor's conjecture — for these Lévy processes; the proper norm, the index space V\mathcal V and the property 'joint continuity holds for V\mathcal V' introduced in the preceding paragraph are background that explains why the measure-side restriction ν∈R+(X)\nu\in\mathcal R^+(X) matters, and are not themselves a separate deliverable.

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