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arxiv: 2606.02036 · v1 · pith:7KVI33BWnew · submitted 2026-06-01 · 🧮 math.PR · math.DS· math.FA

Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction

Olga Izyumtseva , Wasiur R. KhudaBukhsh This is my paper

Pith reviewed 2026-06-28 12:42 UTC · model grok-4.3

classification 🧮 math.PR math.DSmath.FA
keywords self-intersection local timesVolterra processesstochastic flowschange of variable formulaweighted local timesasymptoticsGaussian processesinteracting particles
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The pith

A change of variable formula expresses the self-intersection local times of the flow process x in terms of weighted local times of the Volterra process u.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines self-intersection local times for the process x(u(·),t) arising from a stochastic flow with interaction, where the initial condition is the occupation measure of a Volterra Gaussian process u. It establishes the existence of multiple such local times for x and introduces a change of variable formula to express them using weighted self-intersection local times of u. The work also derives large-time asymptotics for these local times and proves existence of weighted local times for unbounded weights. A sympathetic reader would care because this links geometric self-intersection properties to the coefficients of interacting SDEs in a new way.

Core claim

We prove the existence of multiple self-intersection local times for the process x(u(·),t) and establish a change of variable formula that allows us to describe self-intersection local times for the process x(u(·),t) in terms of the weighted self-intersection local times for the process u. We describe the corresponding asymptotics of the self-intersection local times for x(u(·),t) for large t. Moreover, the existence of weighted self-intersection local times is established for a large class of unbounded weights.

What carries the argument

Change of variable formula relating self-intersection local times of x(u(·),t) to weighted self-intersection local times of u.

If this is right

  • Self-intersection local times of x(u(·),t) are determined by those of u with appropriate weights.
  • Large t asymptotics describe the long-term behavior of these local times for x.
  • Weighted self-intersection local times exist for a large class of unbounded weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This extends previous work on deterministic differential equations with geometric characteristics to the stochastic setting with interaction.
  • The method could be tested on specific Volterra kernels like fractional Brownian motion to check the asymptotics.
  • Such local times might be used to model new types of particle interactions based on self-intersections.

Load-bearing premise

The equation with interaction admits a well-defined stochastic flow when the initial condition is the occupation measure of the Volterra process u.

What would settle it

Numerical simulation of the process x for a simple Volterra kernel where the change of variable formula can be verified directly against the weighted local times of u.

read the original abstract

In this paper, we study self-intersection local times for a stochastic process $x(u(\cdot),t)$, where $u$ is a Gaussian process of the form $u(t)=\int^t_0k(t,s)\mathrm{d}{w(s)}$, $k$ is a deterministic kernel of the Volterra type, $w$ is a Wiener process, and $x$ is a solution to the \emph{equation with interaction}. Equations with interaction are a class of interacting particle system described by stochastic differential equations whose coefficients depend on a random measure (initial distribution of particles) transformed by the flow of solutions. Considering the occupation measure of $u$ as the initial condition for the equation with interaction allows us to define a stochastic flow with interaction driven by self-intersection local times of the process $u$. The study of such stochastic differential equations whose coefficients carry information about the geometric properties of curves is new. They previously appeared only for deterministic differential equations and smooth curves, where the geometric characteristics typically considered are length, curvature, and so on. In this paper, we prove the existence of multiple self-intersection local times for the process $x(u(\cdot),t)$ and establish a ``change of variable formula" that allows us to describe self-intersection local times for the process $x(u(\cdot),t)$ in terms of the weighted self-intersection local times for the process $u.$ We describe the corresponding asymptotics of the self-intersection local times for $x(u(\cdot),t)$ for large $t$. Moreover, the existence of weighted self-intersection local times is established for a large class of unbounded weights, which is of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies self-intersection local times for the process x(u(·),t), where u is a Volterra Gaussian process driven by a Wiener process and x solves an interacting SDE whose initial condition is the occupation measure of u. It claims to prove existence of multiple self-intersection local times for x, a change-of-variable formula relating these to weighted self-intersection local times of u, large-t asymptotics for the local times of x, and existence of weighted self-intersection local times for a broad class of unbounded weights.

Significance. If the well-posedness of the interacting flow holds, the change-of-variable formula and the extension to unbounded weights would constitute a substantive advance in connecting geometric properties of Gaussian paths to stochastic flows with interaction, moving beyond deterministic curves. The large-t asymptotics could also be of independent interest in the theory of local times for Volterra processes.

major comments (1)
  1. [Abstract / Setup of x(u(·),t)] The construction of x(u(·),t) as the solution to the equation with interaction starting from the occupation measure of u is invoked throughout (abstract and the paragraph introducing the process) without an existence/uniqueness statement, without a reference to a theorem covering the given Volterra kernel class, and without verification that the interaction term (driven by local times of u) preserves the required regularity. All subsequent results on existence of local times for x, the change-of-variable formula, and the asymptotics rest on this assumption.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this foundational issue. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Setup of x(u(·),t)] The construction of x(u(·),t) as the solution to the equation with interaction starting from the occupation measure of u is invoked throughout (abstract and the paragraph introducing the process) without an existence/uniqueness statement, without a reference to a theorem covering the given Volterra kernel class, and without verification that the interaction term (driven by local times of u) preserves the required regularity. All subsequent results on existence of local times for x, the change-of-variable formula, and the asymptotics rest on this assumption.

    Authors: We agree that the current version assumes without explicit justification the existence and uniqueness of the solution x(u(·),t) to the interacting SDE with initial occupation measure of u. No dedicated statement, reference to a well-posedness theorem for the relevant Volterra kernels, or verification that the local-time-driven interaction preserves regularity appears in the manuscript. In the revised version we will add a short subsection (or appendix) that either cites an appropriate existence/uniqueness result for equations with interaction under the given kernel assumptions or supplies a brief argument confirming that the interaction term maintains the necessary path regularity. This will directly support the subsequent claims on local times, the change-of-variable formula, and the asymptotics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external assumption of flow well-posedness

full rationale

The paper defines x(u(·),t) as the solution to the interacting SDE with initial occupation measure of the Volterra process u, then proves existence of its multiple self-intersection local times, a change-of-variable formula relating them to weighted local times of u, and large-t asymptotics. No quoted step reduces any claimed result to its inputs by construction: the local-time existence and change-of-variable are presented as consequences of standard stochastic calculus once the flow is assumed to exist. The well-posedness assumption is invoked but not derived from the local-time claims themselves, nor is it a self-citation, fitted parameter, or ansatz smuggled via prior work. No self-citations appear in the provided text. The derivation is therefore self-contained against the stated assumptions, warranting score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard existence theory for SDEs with measure-dependent coefficients and on properties of Volterra kernels; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The equation with interaction admits a stochastic flow when the initial measure is the occupation measure of u.
    Invoked when defining x(u(·),t) whose local times are studied.
  • standard math Standard regularity conditions on the Volterra kernel k and the Wiener process w hold.
    Required for u to be a well-defined Gaussian process.

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Works this paper leans on

39 extracted references · 32 canonical work pages

  1. [1]

    volume 125 of Applied Mathematical Sciences

    Topological methods in hydrodynamics. volume 125 of Applied Mathematical Sciences . Second ed., Springer, Cham. doi: 10.1007/978-3-030-74278-2 . Baxendale, P., Harris, T.E.,

    work page doi:10.1007/978-3-030-74278-2

  2. [2]

    Isotropic stochastic flows. Ann. Probab. 14, 1155–1179. URL: http://links.jstor.org/sici?sici= 0091-1798(198610)14:4<1155:ISF>2.0.CO;2-C&origin=MSN . Berman, S.M., 1973/74. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69–94. doi: 10.1512/iumj. 1973.23.23006. Carverhill, A.,

    work page doi:10.1512/iumj 1973

  3. [3]

    Stochastics 14, 273–317

    Flows of stochastic dynamical systems: ergodic theory. Stochastics 14, 273–317. doi: 10.1080/17442508508833343. Chaintron, L.P., Diez, A., 2022a. Propagation of chaos: a review of models, methods and applications. I. Models and methods. Kinet. Relat. Models 15, 895–1015. doi: 10.3934/krm.2022017. Chaintron, L.P., Diez, A., 2022b. Propagation of chaos: a r...

    work page doi:10.1080/17442508508833343

  4. [4]

    Chen, X.,

    doi: 10.1016/j.spl.2024.110168. Chen, X.,

    work page doi:10.1016/j.spl.2024.110168 2024

  5. [5]

    Parabolic Anderson model with rough or critical Gaussian noise. Ann. Inst. Henri Poincaré Probab. Stat. 55, 941–976. doi:10.1214/18-aihp904. Chen, X.,

    work page doi:10.1214/18-aihp904

  6. [6]

    Izyumtseva and KhudaBukhsh Page 19 of 21 Self-intersection local times in stochastic flows Chen, X., Li, W.V .,

    doi: 10.1214/23-ejp985. Izyumtseva and KhudaBukhsh Page 19 of 21 Self-intersection local times in stochastic flows Chen, X., Li, W.V .,

    work page doi:10.1214/23-ejp985

  7. [7]

    Critical Brownian sheet does not have double points. Ann. Probab. 40, 1829–1859. doi: 10.1214/11-AOP665. Dalang, R.C., Lee, C.Y., Mueller, C., Xiao, Y.,

    work page doi:10.1214/11-aop665

  8. [8]

    Dalang, R.C., Mueller, C.,

    doi:10.1214/21-EJP589. Dalang, R.C., Mueller, C.,

    work page doi:10.1214/21-ejp589

  9. [9]

    II (Charlotte, NC, 1990)

    Isotropic stochastic flows: a survey, in: Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990). Birkhäuser Boston, Boston, MA. volume 27 of Progr. Probab., pp. 75–94. Dimitroff, G.,

    1990

  10. [10]

    Some properties of isotropic Brownian and Ornstein-Uhlenbeck flows. Ph.D. thesis. URL: https://depositonce. tu-berlin.de/handle/11303/1627, doi:10.14279/DEPOSITONCE-1330. Dimitroff, G., Scheutzow, M.,

    work page doi:10.14279/depositonce-1330

  11. [11]

    Stochastic Process

    Dispersion of volume under the action of isotropic Brownian flows. Stochastic Process. Appl. 119, 588–601. doi:10.1016/j.spa.2008.03.005. Dorogovtsev, A., Gnedin, A., Izyumtseva, O.,

    work page doi:10.1016/j.spa.2008.03.005 2008

  12. [12]

    arXiv preprint arXiv:1910.09492

    Self-intersection local times of random fields in stochastic flows. arXiv preprint arXiv:1910.09492 . Dorogovtsev, A., Hlyniana, K., Liu, S.,

    arXiv 1910

  13. [13]

    arXiv preprint arXiv:2506.13347

    Ergodic theorem for the differential equations with interaction. arXiv preprint arXiv:2506.13347 . Dorogovtsev, A., Izyumtseva, O.,

    arXiv

  14. [14]

    Stochastics 91, 143–154

    Hilbert-valued self-intersection local times for planar Brownian motion. Stochastics 91, 143–154. doi:10.1080/17442508.2018.1521412. Dorogovtsev, A., Weiß, A.,

    work page doi:10.1080/17442508.2018.1521412 2018

  15. [15]

    volume 3 ofDe Gruyter Series in Probability and Stochastics

    Measure-valued processes and stochastic flows. volume 3 ofDe Gruyter Series in Probability and Stochastics. De Gruyter, Berlin. Translated from the 2007 Russian [2375817]. Dorogovtsev, A.A., Weiss, A.,

    2007

  16. [16]

    Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 53, 856–862. doi:10.1017/s0305004100032989. Dynkin, E.B.,

    work page doi:10.1017/s0305004100032989

  17. [17]

    Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10, 192–217. doi: 10.1214/ aoap/1019737669. Goldman, A.,

    arXiv

  18. [18]

    Points multiples des trajectoires de processus gaussiens. Z. Wahrsch. Verw. Gebiete 57, 481–494. doi: 10.1007/BF01025870. Harang, F.A., Ling, C.,

    work page doi:10.1007/bf01025870

  19. [19]

    Regularity of local times associated with Volterra-Lévy processes and path-wise regularization of stochastic differential equations. J. Theoret. Probab. 35, 1706–1735. doi: 10.1007/s10959-021-01114-4 . van der Hofstad, R., den Hollander, F., König, W.,

    work page doi:10.1007/s10959-021-01114-4

  20. [20]

    Large deviations for the one-dimensional Edwards model. Ann. Probab. 31, 2003–2039. doi:10.1214/aop/1068646376. van der Hofstad, R., Klenke, A.,

    work page doi:10.1214/aop/1068646376 2003

  21. [21]

    Self-attractive random polymers. Ann. Appl. Probab. 11, 1079–1115. doi: 10.1214/aoap/1015345396. den Hollander, F.,

    work page doi:10.1214/aoap/1015345396

  22. [22]

    volume 1974 of Lecture Notes in Mathematics

    Random polymers. volume 1974 of Lecture Notes in Mathematics . Springer-Verlag, Berlin. doi: 10.1007/ 978-3-642-00333-2 . lectures from the 37th Probability Summer School held in Saint-Flour,

    1974

  23. [23]

    doi:10.48550/ARXIV.2409.04377

    Local times of self-intersection and sample path properties of Volterra Gaussian processes. doi:10.48550/ARXIV.2409.04377. Izyumtseva, O.L.,

    work page doi:10.48550/arxiv.2409.04377

  24. [24]

    The constant of renormalization for self-intersection local time of diffusion process in the plane. Ukraïn. Mat. Zh. 60, 1489–1498. doi: 10.1007/s11253-009-0166-4 . Jung, P., Markowsky, G.,

    work page doi:10.1007/s11253-009-0166-4

  25. [25]

    Stochastic Process

    On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion. Stochastic Process. Appl. 124, 3846–3868. doi: 10.1016/j.spa.2014.07.001. Kardar, M., Parisi, G., Zhang, Y.C.,

    work page doi:10.1016/j.spa.2014.07.001 2014

  26. [26]

    doi: 10.1103/physrevlett.56

    work page doi:10.1103/physrevlett.56

  27. [27]

    Double points of a Gaussian sample path. Z. Wahrsch. Verw. Gebiete 45, 175–180. doi: 10.1007/BF00715191. Kunita, H.,

    work page doi:10.1007/bf00715191

  28. [28]

    Wiener sausage and self-intersection local times. J. Funct. Anal. 88, 299–341. doi: 10.1016/0022-1236(90)90108-W . Le Jan, Y.,

    work page doi:10.1016/0022-1236(90)90108-w

  29. [29]

    On isotropic Brownian motions. Z. Wahrsch. Verw. Gebiete 70, 609–620. doi: 10.1007/BF00531870. Lifshits, M.,

    work page doi:10.1007/bf00531870

  30. [30]

    SpringerBriefs in Mathematics, Springer, Heidelberg

    Lectures on Gaussian processes. SpringerBriefs in Mathematics, Springer, Heidelberg. doi: 10.1007/978-3-642-24939-6 . Lyu, Y., Zhang, R.,

    work page doi:10.1007/978-3-642-24939-6

  31. [31]

    Exponential asymptotics for 𝑝-multiple self-intersection local time of 1- 𝑑 diffusion process. Ann. Appl. Probab. 35, 4067–4105. doi: 10.1214/25-AAP2214. Revuz, D., Yor, M.,

    work page doi:10.1214/25-aap2214

  32. [32]

    volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Continuous martingales and Brownian motion. volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Third ed., Springer-Verlag, Berlin. doi: 10.1007/978-3-662-06400-9 . Izyumtseva and KhudaBukhsh Page 20 of 21 Self-intersection local times in stochastic flows Rosen, J.,

    work page doi:10.1007/978-3-662-06400-9

  33. [33]

    Springer, Berlin

    A renormalized local time for multiple intersections of planar Brownian motion, in: Séminaire de Probabilités, XX, 1984/85. Springer, Berlin. volume 1204 of Lecture Notes in Math., pp. 515–531. doi: 10.1007/BFb0075738. Sznitman, A.S.,

    work page doi:10.1007/bfb0075738 1984

  34. [34]

    Topics in Propagation of Chaos

    Topics in propagation of chaos, in: École d’Été de Probabilités de Saint-Flour XIX—1989. Springer, Berlin. volume 1464 of Lecture Notes in Math., pp. 165–251. doi: 10.1007/BFb0085169. Talagrand, M.,

    work page doi:10.1007/bfb0085169 1989

  35. [35]

    Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theory Related Fields 112, 545–563. doi:10.1007/s004400050200. Vadlamani, S., Adler, R.J.,

    work page doi:10.1007/s004400050200

  36. [36]

    Electron

    Global geometry under isotropic Brownian flows. Electron. Comm. Probab. 11, 182–192. doi: 10.1214/ECP. v11-1212. Varadhan, S.S.,

    work page doi:10.1214/ecp

  37. [37]

    On Edwards’ model for polymer chains. III. Borel summability. Comm. Math. Phys. 84, 459–470. URL: http: //projecteuclid.org/euclid.cmp/1103921283. Xiao, Y., Zhang, T.,

    arXiv

  38. [38]

    Local times of fractional Brownian sheets. Probab. Theory Related Fields 124, 204–226. URL: https://doi.org/ 10.1007/s004400200210, doi:10.1007/s004400200210. Zirbel, C.L.,

    work page doi:10.1007/s004400200210

  39. [39]

    Springer, New York

    Mass transport by Brownian flows, in: Stochastic models in geosystems (Minneapolis, MN, 1994). Springer, New York. volume 85 of IMA Vol. Math. Appl., pp. 459–492. doi: 10.1007/978-1-4613-8500-4\_22 . Izyumtseva and KhudaBukhsh Page 21 of 21

    work page doi:10.1007/978-1-4613-8500-4 1994