pith. sign in

arxiv: 2205.04162 · v7 · pith:V6XOIYQMnew · submitted 2022-05-09 · 🧮 math.PR · math.AP

Fractional Boundary Value Problems and elastic sticky Brownian motions, II: Non-local dynamic boundary conditions on smooth domains

Mirko D'Ovidio This is my paper

Pith reviewed 2026-05-24 12:19 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords fractional boundary value problemssticky diffusionsdynamic boundary conditionstime changesright-inversessmooth domainstrap effectBrownian motions
0
0 comments X

The pith

Fractional dynamic boundary conditions produce sticky diffusions that spend infinite mean time on the boundary.

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

The paper shows how fractional versions of dynamic boundary conditions lead to sticky diffusion processes on bounded smooth domains. These processes spend only finite time on the boundary but an infinite mean time there. This is achieved by constructing the processes through time changes using right-inverses of suitable processes. A reader would care because it provides a probabilistic interpretation for solutions to fractional boundary value problems and models trap effects at the boundary. The approach applies to bounded domains with smooth boundaries.

Core claim

Fractional boundary value problems involving fractional dynamic boundary conditions lead to sticky diffusions spending an infinite mean time and finite time on a lower-dimensional boundary. Such a behaviour can be associated with a trap effect in the macroscopic point of view. The processes are characterized by using time changes obtained by right-inverses of suitable processes to describe the fractional sticky conditions and the associated boundary behaviours on smooth domains.

What carries the argument

Time changes obtained by right-inverses of suitable processes, used to describe fractional sticky conditions on the boundary.

If this is right

  • The diffusions spend finite time but infinite mean time on the boundary.
  • The behaviour corresponds to a macroscopic trap effect.
  • The processes can remain or move on the boundary according to dynamics different from the interior.
  • The governing equations include a fractional time derivative in the boundary condition.

Where Pith is reading between the lines

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

  • The construction might extend to model memory effects at boundaries in physical systems with fractional time derivatives.
  • Numerical path simulations of the resulting processes could be compared against solutions of the fractional PDE to test the infinite mean time prediction.
  • Similar time-change techniques could apply to elastic sticky motions in higher-dimensional or non-smooth settings.
  • The infinite mean time suggests connections to anomalous diffusion models where boundaries act as long-term traps.

Load-bearing premise

That time changes obtained by right-inverses of suitable processes correctly describe the fractional sticky conditions and the associated boundary behaviours on smooth domains.

What would settle it

A direct calculation of the expected occupation time on the boundary for the solution of the fractional boundary value problem, checking whether it diverges while the actual occupation time remains finite.

Figures

Figures reproduced from arXiv: 2205.04162 by Mirko D'Ovidio.

Figure 1
Figure 1. Figure 1: The Koch domain (pre-fractal, first step) on the left and the modified Koch domain [PITH_FULL_IMAGE:figures/full_fig_p034_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Koch domain (pre-fractal) with inward curves and outward curves. The construction [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
read the original abstract

Sticky diffusion processes on bounded domains spend finite time (and finite mean time) on the lower-dimensional space given by the boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on the boundary it can stay or move according to dynamics that are different from those in the interior. Such processes may be characterized by a time-derivative appearing in the boundary condition for the governing problem. We use time changes obtained by right-inverses of suitable processes in order to describe fractional sticky conditions and the associated boundary behaviours. We obtain that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions spending an infinite mean time (and finite time) on a lower-dimensional boundary. Such a behaviour can be associated with a trap effect in the macroscopic point of view.

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

0 major / 2 minor

Summary. The manuscript claims that time changes constructed via right-inverses of suitable processes realize fractional dynamic (non-local) boundary conditions for diffusions on smooth bounded domains; the resulting processes are sticky, spending finite time but infinite mean time on the boundary, thereby furnishing a probabilistic interpretation of the associated fractional boundary-value problems and a macroscopic trap effect.

Significance. If the correspondence is rigorously established, the work supplies a probabilistic representation for a class of fractional boundary conditions that extends the theory of elastic sticky Brownian motions to non-local settings on domains; this could connect fractional PDE theory with sticky-process constructions and provide a concrete mechanism for anomalous boundary residence times.

minor comments (2)
  1. The abstract states the main correspondence but the provided text supplies no explicit derivation or error estimate for the time-change construction; the full manuscript should contain a self-contained verification that the right-inverse time change indeed produces the claimed fractional dynamic boundary condition (e.g., via generator calculation or resolvent identity).
  2. Notation for the fractional-order operators and the precise definition of the right-inverse process should be introduced with a short preliminary section or appendix so that the boundary-condition statement can be checked without external references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The summary accurately captures the main contribution regarding the probabilistic interpretation of fractional dynamic boundary conditions via time-changed sticky diffusions. Since the major comments section contains no specific points or requests for clarification, we have no individual items to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on time-change constructions using right-inverses of suitable processes to realize fractional dynamic boundary conditions, leading to the claimed sticky behavior with infinite mean time on the boundary. This is presented as a direct consequence of the method applied to smooth domains rather than any fitted parameter renamed as a prediction, self-definitional loop, or load-bearing self-citation chain. The abstract and description show an independent construction without reduction to inputs by construction. No steps matching the enumerated circularity patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract introduces no explicit free parameters, new axioms, or invented entities beyond standard notions of sticky processes and fractional derivatives already in the literature.

pith-pipeline@v0.9.0 · 5668 in / 943 out tokens · 22718 ms · 2026-05-24T12:19:29.127025+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Time-dependent Robin heat equation via Markovian switching

    math.PR 2026-04 unverdicted novelty 5.0

    The Robin heat equation with Markovian switching boundary reactivity is solved via contraction semigroups in the annealed case and non-autonomous evolution families in the quenched case, converging to a deterministic ...

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    Amir, Sticky Brownian motion as the strong limit of a sequence of random walks

    M. Amir, Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Processes and their Applications 39, (1991) 221–237. 35

    work page 1991

  2. [2]

    Arendt, G

    W. Arendt, G. Metafune, D. Pallara, S. Romanelli, The Laplacian with Wentzell-Robin Boundary Conditions on Spaces of Continuous Functions. Semigroup Forum 67, (2003), 247 – 261

    work page 2003

  3. [3]

    Arendt, M

    W. Arendt, M. Sauter. The Wentzell Laplacian via forms and the approximative trace. Discrete and Continuous Dynamical Systems - S. doi: 10.3934/dcdss.2022148

    work page doi:10.3934/dcdss.2022148

  4. [4]

    Baeumer, M

    B. Baeumer, M. M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, No 4 (2001), 481–500

    work page 2001

  5. [5]

    Bandle, J

    C. Bandle, J. von Below, W. Reichel, Parabolic problems with dynamical boundary condi- tions: eigenvalue expansions and blow up, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl., 17 (1) (2006), 35 – 67

    work page 2006

  6. [6]

    E. G. Bazhlekova, Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal. 3, No 3 (2000), 213–230

    work page 2000

  7. [7]

    Bertoin, L´ evy Processes

    J. Bertoin, L´ evy Processes. Cambridge University Press (1996)

    work page 1996

  8. [8]

    N. H. Bingham, Limit theorems for occupation times of markov processes. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 17 (1071), 1–22

    work page

  9. [9]

    Blumenthal, R

    R. Blumenthal, R. Getoor, Markov Processes and Potential Theory. Academic Press (2007). Republication of 1968

    work page 2007

  10. [10]

    Bouchaud, P

    J. Bouchaud, P. Kr¨ uger, A. Landier, D. Thesmar, Sticky Expectations and the Profitability Anomaly. Journal of Finance 74 (2018), 639 – 674

    work page 2018

  11. [11]

    Bou-Rabee, M

    N. Bou-Rabee, M. Holmes-Cerfon, Sticky Brownian Motion and its Numerical Solution, SIAM Review 62, (2020) Iss. 1

    work page 2020

  12. [12]

    Burdzy, ZQ

    K. Burdzy, ZQ. Chen, D. Marshall, Traps for reflected Brownian motion. Math. Z. 252, (2006) 103 – 132

    work page 2006

  13. [13]

    Brezis, Analisi Funzionale: Teoria e applicazioni

    H. Brezis, Analisi Funzionale: Teoria e applicazioni . Liguori, 1986. Napoli, Italia

    work page 1986

  14. [14]

    Capitanelli, M

    R. Capitanelli, M. D’Ovidio, Fractional equations via convergence of forms. Fractional Calculus and Applied Analysis 22 (2019), 844 - 870

    work page 2019

  15. [15]

    Capitanelli, M

    R. Capitanelli, M. D’Ovidio, Delayed and Rushed motions through time change. ALEA, Lat. Am. J. Probab. Math. Stat. 17 (2020), 183–204

    work page 2020

  16. [16]

    Capitanelli, M

    R. Capitanelli, M. D’Ovidio, Fractional Cauchy problem on random snowflakes. Journal of Evolution Equations 21 (2021), 2123–2140

    work page 2021

  17. [17]

    Caputo, Elasticit` a e Dissipazione

    M. Caputo, Elasticit` a e Dissipazione. Zanichelli, Bologna, (1969)

    work page 1969

  18. [18]

    Caputo, F Mainardi, Linear models of dissipation in anelastic solids

    M. Caputo, F Mainardi, Linear models of dissipation in anelastic solids. La Rivista del Nuovo Cimento 1, (1971), 161–198

    work page 1971

  19. [19]

    Caputo, F

    M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism.PAGEOPH 91, (1971), 134–147

    work page 1971

  20. [20]

    Chen, Time fractional equations and probabilistic representation, Chaos, Solitons & Fractals 102 (2017) 168-174

    Z.-Q. Chen, Time fractional equations and probabilistic representation, Chaos, Solitons & Fractals 102 (2017) 168-174

    work page 2017

  21. [21]

    Cl´ ement, C

    P. Cl´ ement, C. A. Timmermans, On C0-semigroups generated by differential operators satisfying Ventcel’s boundary conditions. Indag. Math. 89 (1986), 379 – 387. 36

    work page 1986

  22. [22]

    Colantoni, M

    F. Colantoni, M. D’Ovidio, Non-local Boundary Value Problems for Brownian motions on the half line. Submitted, arXiv:2209.14135

    work page arXiv

  23. [23]

    D’Ovidio, From Sturm-Liouville problems to fractional and anomalous diffusions

    M. D’Ovidio, From Sturm-Liouville problems to fractional and anomalous diffusions. Stochastic Processes and their Applications 122, No 10 (2012), 3513–3544

    work page 2012

  24. [24]

    D’Ovidio, Fractional Boundary Value Problems

    M. D’Ovidio, Fractional Boundary Value Problems. Fract. Calc. Appl. Anal. 1 25, (2022) 29 – 59

    work page 2022

  25. [25]

    D’Ovidio Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, I: The half line

    M. D’Ovidio Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, I: The half line. Submitted, arXiv:2402.12982

    work page arXiv

  26. [26]

    M. M. Dzherbashian, Integral Transforms and Representations of Functions in the Complex Plane. Nauka, Moscow, (1966) (in Russian)

    work page 1966

  27. [27]

    M. M. Dzherbashian, A. B. Nersessian, Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 3 (1968), No 1, 1–29 (in Russian)

    work page 1968

  28. [28]

    Engel, G

    K.-J. Engel, G. Frangnelli, Analyticity of semigroups generated by operators with general- ized Wentzell boundary conditions. Advances in Differential Equations 10 (2005), No 11, 1301–1320

    work page 2005

  29. [29]

    Fattler, M

    T. Fattler, M. Grothaus, R. Voßhall, Construction and analysis of a sticky reflected distorted Brownian motion. Annales de l’Institut Henri Poincar´ e - Probabilit´ es et Statistiques52, No. 2, (2016), 735–762

    work page 2016

  30. [30]

    Feller, The parabolic differential equations and the associated semi-groups of transfor- mations

    W. Feller, The parabolic differential equations and the associated semi-groups of transfor- mations. Ann. of Math. 55, No 2 (1952), 468–519

    work page 1952

  31. [31]

    Gerali, S

    A. Gerali, S. Neri, L. Sessa, F. M.Signoretti, Credit and Banking in a DSGE Model of the Euro Area, Journal of Money, Credit and Banking 42, Supplement 1 (September 2010), 107 – 141

    work page 2010

  32. [32]

    G. R. Goldstein, Derivation and physical interpretation of general boundary conditions. Adv. Differential Equations 11, No 4 (2006), 457–480

    work page 2006

  33. [33]

    Goldstein, C.-Y

    J.A. Goldstein, C.-Y. Lin, Highly degenerate parabolic boundary value problems. Diff. Int. Eqns. 2 (1989), 216 – 227

    work page 1989

  34. [34]

    Gorenflo, Y

    R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18, No 3 (2015), 799–820

    work page 2015

  35. [35]

    C. Graham, The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary Annales de l’Institut Henri Poincar´ e - Probabilit´ es et Statistiques 24, No 1 (1988), 45–72

    work page 1988

  36. [36]

    Grothaus, R

    M. Grothaus, R. Voßhall, Stochastic differential equations with sticky reflection and bound- ary diffusion, Electron. J. Probab. 22 (2017), 1 – 37

    work page 2017

  37. [37]

    P. Hsu. On the excursions of reflecting Brownian motion. Transactions of the American Mathematical Society 296, (1986) 239 - 264

    work page 1986

  38. [38]

    Kiryakova, Generalized Fractional Calculus and Applications, vol

    V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Re- search Notes in Mathematics Series , Longman Scientific & Technical, Harlow, UK (1994)

    work page 1994

  39. [39]

    A. N. Kochubei, The Cauchy problem for evolution equations of fractional order.Differential Equations 25, No 8 (1989), 967–974. 37

    work page 1989

  40. [40]

    A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory 71 (2011), 583–600

    work page 2011

  41. [41]

    V. N. Kolokoltsov, The probabilistic point of view on the generalized fractional partial differential equations, Fract. Calc. Appl. Anal. 22, No 3 (2019), 543–600

    work page 2019

  42. [42]

    A. M. Kr¨ ageloh, Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups. J. Math. Anal. Appl. 283, (2003), 459 – 467

    work page 2003

  43. [43]

    Kumar, S

    D. Kumar, S. Bhattacharya, G. Shankar, Weak adhesion at the mesoscale: particles at an interface Soft Matter 29, 2013

    work page 2013

  44. [44]

    Itˆ o, H

    K. Itˆ o, H. P. JR. McKean. Brownian motions on a half line. Illinois J. Math. 7, (1963), 181-231

    work page 1963

  45. [45]

    Jerison, C

    D. Jerison, C. Kenig, The Neumann problem on Lipschitz domains,Bulletin of the American Mathematical Society 4, (1981), 203 - 207

    work page 1981

  46. [46]

    J. L. Lions , E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Grundlehren der mathematischen Wissenschaften (GL, volume 181). Springer Berlin, Hei- delberg, 1972

    work page 1972

  47. [47]

    Luchko, Fractional derivatives and the fundamental theorem of Fractional Calculus

    Y. Luchko, Fractional derivatives and the fundamental theorem of Fractional Calculus. Fract. Calc. Appl. Anal. 23, No 4 (2020), 939–966

    work page 2020

  48. [48]

    Mainardi, Y

    F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153–192

    work page 2001

  49. [49]

    M. M. Meerschaert, E. Nane, P. Vellaisamy, Fractional Cauchy problems on bounded do- mains. Ann. Probab. 37, No 3 (2009), 979–1007

    work page 2009

  50. [50]

    Orsingher, L

    E. Orsingher, L. Beghin, Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37 (2009), 206–249

    work page 2009

  51. [51]

    V. G. Papanicolaou, The probabilistic solution of the third boundary value problem for second order elliptic equations. Probab. Th. Rel. Fields 87 (1990), 27-77

    work page 1990

  52. [52]

    Peskir, Sticky Bessel diffusions

    G. Peskir, Sticky Bessel diffusions. Stochastic Processes and their Applications , (2021) in press

    work page 2021

  53. [53]

    Peskir, D

    G. Peskir, D. Roodman, Sticky Feller Diffusions, Research Report No. 1, 2020, Probab. Statist. Group Manchester (32 pp)

    work page 2020

  54. [54]

    R. Song, Z. Vondrˇ acek, Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields 125, (2003), 578 – 592

    work page 2003

  55. [55]

    Taira, A

    K. Taira, A. Favini, S. Romanelli, Feller semigroups generated by degenerate elliptic oper- ators. Semigroup Forum 60 (2000), 296 – 309

    work page 2000

  56. [56]

    B. Toaldo. Convolution-type derivatives, hitting-times of subordinators and time-changed C0-semigroups. Potential Analysis 42 (2015), 115–140

    work page 2015

  57. [57]

    von Below, G

    J. von Below, G. Fran¸ cois, Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition. Bull. Belg. Math. Soc. 12 (2005), 505 – 519

    work page 2005

  58. [58]

    Warren, Branching processes, the Ray-Knight theorem, and sticky brownian motion S´ eminaire de probabilit´ es(Strasbourg), tome 31 (1997), 1–15

    J. Warren, Branching processes, the Ray-Knight theorem, and sticky brownian motion S´ eminaire de probabilit´ es(Strasbourg), tome 31 (1997), 1–15

    work page 1997

  59. [59]

    A. D. Wentzell, On boundary conditions for multidimensional diffusion processes, Theor. Probab. Appl. IV (1959), No 2, 164 – 177. 38

    work page 1959