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Generalized Kac's moment formula for positive continuous additive functionals of symmetric Markov processes

Naotaka Kajino, Ryoichiro Noda

TL;DR

The paper addresses extending Kac's moment formula from Brownian motion to general positive continuous additive functionals (PCAFs) of symmetric Hunt processes tied to regular Dirichlet forms. It develops a concise, self-contained approach based on the Revuz correspondence and the strong Markov property to derive a Lebesgue–Stieltjes integral formula for expectations against PCAFs, and then obtains a generalized Kac formula that expresses $E_x[A_t^k]$ and mixed moments $E_x[f(X_t) \prod_i A^{(i)}_t]$ in terms of the process's heat kernel $p_t(x,y)$ and the Revuz measures $\mu_i$. The results recover the classical Kac formula when $\mu$ corresponds to a density $f$, and extend to corollaries including the extended Kato class $\mathcal{S}_{\mathrm{EK}}$ with uniformly bounded exponential moments and to part processes on domains. These contributions facilitate analysis of PCAFs in Feynman–Kac-type settings and occupation-time problems for a broad class of symmetric Markov processes.

Abstract

We establish a formula for moments of certain random variables involving positive continuous additive functionals of symmetric Hunt processes whose Dirichlet forms are regular, generalizing the classical Kac's moment formula.

Generalized Kac's moment formula for positive continuous additive functionals of symmetric Markov processes

TL;DR

The paper addresses extending Kac's moment formula from Brownian motion to general positive continuous additive functionals (PCAFs) of symmetric Hunt processes tied to regular Dirichlet forms. It develops a concise, self-contained approach based on the Revuz correspondence and the strong Markov property to derive a Lebesgue–Stieltjes integral formula for expectations against PCAFs, and then obtains a generalized Kac formula that expresses and mixed moments in terms of the process's heat kernel and the Revuz measures . The results recover the classical Kac formula when corresponds to a density , and extend to corollaries including the extended Kato class with uniformly bounded exponential moments and to part processes on domains. These contributions facilitate analysis of PCAFs in Feynman–Kac-type settings and occupation-time problems for a broad class of symmetric Markov processes.

Abstract

We establish a formula for moments of certain random variables involving positive continuous additive functionals of symmetric Hunt processes whose Dirichlet forms are regular, generalizing the classical Kac's moment formula.

Paper Structure

This paper contains 5 sections, 8 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Setting, PCAFs and smooth measures in the strict sense
  4. Main results
  5. Proofs of the main theorems

Key Result

Theorem 2.4

Let $A = (A_{t})_{t \geq 0} \in \mathbf{A}_{c, 1}^{+}$, $\mu \in \mathcal{S}_{1}$, and consider the following condition 2. item: Revuz correspondence. Then, $\mu$ is the Revuz measure of $A$ if and only if 2. item: Revuz correspondence holds.

Theorems & Definitions (21)

  • Definition 2.1: PCAF, Fukushima_Oshima_Takeda_11_Dirichlet
  • Definition 2.2: PCAF in the strict sense, Fukushima_Oshima_Takeda_11_Dirichlet
  • Definition 2.3: Smooth measure in the strict sense, Fukushima_Oshima_Takeda_11_Dirichlet
  • Theorem 2.4: Revuz correspondence
  • proof
  • Remark 2.5
  • Lemma 2.6: Chen_Fukushima_12_Symmetric
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 11 more