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arxiv: 2606.27549 · v1 · pith:G5R7YC7Lnew · submitted 2026-06-25 · 🧮 math.PR

A framework for drift transformations of multidimensional diffusion processes with applications to Wiener and Ornstein--Uhlenbeck dynamics

Antonio DiCrescenzo , Verdiana Mustaro , Serena Spina This is my paper

Pith reviewed 2026-06-29 00:36 UTC · model grok-4.3

classification 🧮 math.PR
keywords drift transformationsmultidimensional diffusionstransition probability densityWiener processOrnstein-Uhlenbeck processweight functionstochastic orderingPoissonian resetting
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The pith

A weight function solving a PDE defines drift transformations that produce new multidimensional diffusions with product-form transition densities.

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

This paper develops a framework for transforming the drift of multidimensional diffusion processes using a weight function. The transformation ensures that the transition probability density of the new process is the product of the original density and the weight. Formulated through stochastic differential equations, the method keeps the densities analytically tractable. It is applied to the Wiener and Ornstein-Uhlenbeck processes, deriving explicit expressions and exploring features like stochastic ordering and resetting.

Core claim

The authors establish that for multidimensional diffusions, a drift transformation defined via a weight function w, obtained as the solution to a partial differential equation, yields a new process whose transition p.d.f. can be written as the product of the original transition p.d.f. and the weight function. This holds under general conditions given by SDEs, and leads to closed-form relations for properties such as stochastic ordering and behavior under Poissonian resetting. Explicit constructions are given for transformations of the Wiener process and the Ornstein-Uhlenbeck process in one and two dimensions.

What carries the argument

The weight function w solving a partial differential equation, which determines the drift change and enforces the product-form relation between the transition densities of the original and transformed processes.

Load-bearing premise

The weight function w must exist as the solution of a suitable partial differential equation to ensure the transformed process is a valid diffusion with tractable transition densities.

What would settle it

A calculation showing that the transition density of a transformed two-dimensional Wiener process does not equal the product of the original density and the weight function would disprove the framework.

Figures

Figures reproduced from arXiv: 2606.27549 by Antonio DiCrescenzo, Serena Spina, Verdiana Mustaro.

Figure 1
Figure 1. Figure 1: The transition p.d.f. of the transformed Wiener process [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The drifts of 𝐗̂ 𝑡 and 𝐗𝑡 represented as vector field (on top) and the potential of 𝐗̂ 𝑡 in blue and of 𝐗𝑡 in yellow (bottom). The choices of the parameters are 𝜌 = 0.5, 𝜎1 = 1, 𝜎2 = 2, 𝑚1 = −1, 𝑚2 = 2, (𝑟1 , 𝑟2 ) = (−2.1094, 0.6386) and 𝑐 = 1 (left) or 𝑐 = 30 (right). mixture (2.15) with mixing parameter 𝜃𝑐 (𝐲) (see Eq. (3.7) for 𝑛 = 2). In particular, in the first case of [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the density functions of the transformed OU process [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The drifts of the OU process 𝐗𝑡 and of the transformed process 𝐗̂ 𝑡 represented as vector fields (on top), and the potentials of 𝐗𝑡 in yellow and of 𝐗̂ 𝑡 in blue (bottom). The choices of the parameters are specified in [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
read the original abstract

We investigate a class of drift-based transformations between multidimensional diffusion processes. The approach allows to construct a new process whose transition probability density function (p.d.f.)\ can be expressed in a product form involving the p.d.f.\ of the original diffusion. The framework is formulated in terms of stochastic differential equations providing general conditions under which the transformed p.d.f.\ remains analytically tractable in the multidimensional setting. The transformation is defined through a weight function $w$, derived as the solution of a suitable partial differential equation. Also, specific forms of $w$ yield certain mixture representations of the transformed p.d.f., which also leads to identify a bimodal feature. We establish closed-form relations between the original and transformed diffusions focusing on features such as stochastic ordering, Poissonian resetting, and diffusions in potential fields. The analysis of the usual stochastic order highlights how the transformation can substantially alter the probabilistic behavior of the process. Furthermore, the product-form relation is shown to persist under Poisson-paced resets, allowing for explicit stationary distributions in certain cases. Two fundamental case studies are presented, based on transformations of the Wiener and Ornstein-Uhlenbeck processes, for which explicit expressions of the weight function, potential function, and transition densities are derived. Special attention is given to the two-dimensional setting, where symmetry properties and absorbing boundaries are explored, providing further insight into the structure and behavior of the transformed diffusions.

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 / 3 minor

Summary. The manuscript develops a drift transformation framework for multidimensional diffusion processes. A weight function w is defined as the solution to a PDE such that the transition density of the transformed process factors as a product involving the original density. General conditions are formulated via SDEs; explicit closed-form expressions for w, the potential, and the densities are derived for the Wiener and Ornstein-Uhlenbeck processes (including symmetric 2D cases with absorbing boundaries). The paper also treats stochastic ordering, Poisson resets (with persistence of the product form and explicit stationary distributions), and mixture representations that can produce bimodality.

Significance. If the derivations hold, the framework supplies a systematic, checkable route to new analytically tractable multidimensional diffusions from standard ones. The parameter-free explicit expressions for the Wiener and OU cases, together with direct verification against the Kolmogorov equations and the preservation of the product form under resets, are concrete strengths that make the results falsifiable and potentially useful for applications.

minor comments (3)
  1. [§2] The PDE satisfied by w is central yet introduced only after the SDE formulation; moving its statement to the beginning of §2 would improve readability.
  2. [§4.2] In the 2D OU example the symmetry assumptions and boundary conditions are stated but the explicit form of the transformed density is not written out; adding the formula would make the product representation immediate to check.
  3. [§3.1] The claim that the transformation 'substantially alters the probabilistic behavior' is illustrated only for the usual stochastic order; a brief numerical check or plot for the Wiener case would strengthen the point without lengthening the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment, including the recommendation for minor revision. The referee's summary correctly identifies the core elements of the drift transformation framework, the explicit constructions for Wiener and Ornstein-Uhlenbeck processes, and the preservation of the product form under Poisson resets.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a drift transformation via a weight function w solving a PDE, yielding product-form transition densities for the transformed diffusion. Explicit closed-form solutions are derived for Wiener and Ornstein-Uhlenbeck cases (including 2D with boundaries) and verified to satisfy the Kolmogorov forward/backward equations directly. These steps are parameter-free, self-contained within the SDE/PDE framework, and do not reduce to fitted quantities or self-citations. No load-bearing premise relies on prior author work as an unverified uniqueness theorem, nor does any 'prediction' equate to its input by construction. The framework remains independently checkable against the original diffusion's density.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, axioms, or invented entities; the central construction rests on the existence of a weight function solving an unspecified PDE and on standard properties of multidimensional diffusions.

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Works this paper leans on

45 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    Boege, M

    T. Boege, M. Drton, B. Hollering, S. Lumpp, P. Misra, D. Schkoda, Conditional independence in stationary distributions of diffusions. Stoch. Proc. Appl. 184(C) (2025). https://doi.org/10.1016/j.spa.2025.104604

    work page doi:10.1016/j.spa.2025.104604 2025

  2. [2]

    Markovian Maximal Coupling of Markov Processes

    B. Böttcher, Markovian maximal coupling of Markov processes (2017). https://doi.org/10.48550/arXiv.1710.09654

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1710.09654 2017

  3. [3]

    Celletti, From perturbative methods to machine learning techniques in space science, Boll

    A. Celletti, From perturbative methods to machine learning techniques in space science, Boll. Unione Mat. Ital. 18 (2025) 149–166. https://doi.org/10.1007/s40574-024-00422-x

    work page doi:10.1007/s40574-024-00422-x 2025

  4. [4]

    Chen, S.F

    M.F. Chen, S.F. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab. 17 (1989) 151-177. https://doi.org/10.1214/aop/1176991501

    work page doi:10.1214/aop/1176991501 1989

  5. [5]

    https://doi.org/10.1007/978-3-031-85160-5

    S.Chewi,J.Niles-Weed,P.Rigollet,StatisticalOptimalTransport,Lect.NotesMath.,SpringerCham,2025. https://doi.org/10.1007/978-3-031-85160-5

    work page doi:10.1007/978-3-031-85160-5 2025

  6. [6]

    A.Comtet,F.Cornu,G.Schehr,Last-passagetimeforlineardiffusionsandapplicationtotheemptyingtime of a box, J. Stat. Phys. 181 (2020) 1565–1602. https://doi.org/10.1007/s10955-020-02637-6

    work page doi:10.1007/s10955-020-02637-6 2020

  7. [7]

    Del Moral, D

    P. Del Moral, D. Villemonais, Exponential mixing properties for time inhomogeneous diffusion processes with killing, Bernoulli 24 (2018) 1010–1032. https://doi.org/10.3150/16-BEJ845

    work page doi:10.3150/16-bej845 2018

  8. [8]

    Dharmaraja, A

    S. Dharmaraja, A. Di Crescenzo, V. Giorno, A.G. Nobile, A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation. J. Stat. Phys. 161 (2015) 326–345. https://doi.org/10.1007/s10955-015-1336-4

    work page doi:10.1007/s10955-015-1336-4 2015

  9. [9]

    Di Crescenzo, V

    A. Di Crescenzo, V. Giorno, A.G. Nobile, Analysis of reflected diffusions via an exponential time-based transformation, J. Stat. Phys. 163 (2016) 1425–1453. https://doi.org/10.1007/s10955-016-1525-9

    work page doi:10.1007/s10955-016-1525-9 2016

  10. [10]

    Di Crescenzo, V

    A. Di Crescenzo, V. Giorno, A.G. Nobile, L.M. Ricciardi, On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes, J. Appl. Probab. 32 (1995) 316-336. https://doi.org/10.2307/3215291

    work page doi:10.2307/3215291 1995

  11. [11]

    Di Crescenzo, V

    A. Di Crescenzo, V. Giorno, A.G. Nobile, S. Spina, First-exit-time problems for two-dimensional Wiener and Ornstein-Uhlenbeck processes through time-varying ellipses, Stochastics 96 (2024) 696-727. https://doi.org/10.1080/17442508.2024.2315274

    work page doi:10.1080/17442508.2024.2315274 2024

  12. [12]

    Di Crescenzo, B

    A. Di Crescenzo, B. Martinucci, On a symmetric, nonlinear birth-death process with bimodal transition probabilities, Symmetry 1 (2009) 201-214. https://doi.org/10.3390/sym1020201

    work page doi:10.3390/sym1020201 2009

  13. [13]

    Evans, S

    M. Evans, S. Majumdar. Diffusion with stochastic resetting. Physical Review Letters 106 (2011) 160601. 10.1103/PhysRevLett.106.160601

    work page doi:10.1103/physrevlett.106.160601 2011

  14. [14]

    Forman, M

    J.L. Forman, M. Sørensen, A transformation approach to modelling multi-modal diffusions, J. Stat. Plan. Inference 146 (2014) 56–69. https://doi.org/10.1016/j.jspi.2013.09.013

    work page doi:10.1016/j.jspi.2013.09.013 2014

  15. [15]

    Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Vol

    C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Vol. 13, Springer-Verlag, Berlin, 2004

    2004

  16. [16]

    Giorno, A.G

    V. Giorno, A.G. Nobile, On the construction of a special class of time-inhomogeneous diffusion processes, J. Stat. Phys. 177 (2019) 299–323. https://doi.org/10.1007/s10955-019-02369-2 36

    work page doi:10.1007/s10955-019-02369-2 2019

  17. [17]

    Giorno, A.G

    V. Giorno, A.G. Nobile, E. Pirozzi, L. Ricciardi, On the construction of first-passage-time densities for diffusion processes, Sci. Math. Jpn. 64 (2006)

    2006

  18. [18]

    Giorno, A

    V. Giorno, A. G. Nobile, S. Spina, On some time non-homogeneous queueing systems with catastrophes. Appl. Math. Comput. 245 (2014) 220–234. https://doi.org/10.1016/j.amc.2014.07.076

    work page doi:10.1016/j.amc.2014.07.076 2014

  19. [19]

    https://doi.org/10.1214/20-EJS1708

    E.Gobet,U.Stazhynski,Parametricinferencefordiffusionsobservedatstoppingtimes,Electron.J.Stat.14 (2020) 2098–2122. https://doi.org/10.1214/20-EJS1708

    work page doi:10.1214/20-ejs1708 2020

  20. [20]

    Gradshteyn, I.M

    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007

    2007

  21. [21]

    Grigorescu, M

    I. Grigorescu, M. Kang, Ergodic properties of multidimensional Brownian Motion with rebirth, Electron. J. Probab. 12 (2007) 1299–1322. https://doi.org/10.1214/EJP.v12-450

    work page doi:10.1214/ejp.v12-450 2007

  22. [22]

    Herrmann, P

    S. Herrmann, P. Imkeller, D. Peithmann, Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: a large deviations approach, Ann. Appl. Probab. 16 (2006) 1851–1892. https://doi.org/10.1214/105051606000000385

    work page doi:10.1214/105051606000000385 2006

  23. [23]

    M.O.Hongler,R.Filliger,P.Blanchard,Solublemodelsfordynamicsdrivenbysuper-diffusivenoise.Phys. A: Stat. Mech. Appl. 370 (2006) 301-315. https://doi.org/10.1016/j.physa.2006.02.036

    work page doi:10.1016/j.physa.2006.02.036 2006

  24. [24]

    Iyengar, Hitting lines with two-dimensional Brownian motion, SIAM J

    S. Iyengar, Hitting lines with two-dimensional Brownian motion, SIAM J. Appl. Math. 45 (1985) 983–989. https://doi.org/10.1137/0145060

    work page doi:10.1137/0145060 1985

  25. [25]

    Jacod, Parametric inference for discretely observed non-ergodic diffusions, Bernoulli 12 (2006) 383–401

    J. Jacod, Parametric inference for discretely observed non-ergodic diffusions, Bernoulli 12 (2006) 383–401. https://doi.org/10.3150/bj/1151525127

    work page doi:10.3150/bj/1151525127 2006

  26. [26]

    Johnson, J.L

    P. Johnson, J.L. Pedersen, G. Peskir, C. Zucca (2022) Detecting the presence of a random drift in Brownian motion, Stoch. Proc. Appl. 150 (2022) 1068–1090. https://doi.org/10.1016/j.spa.2021.05.006

    work page doi:10.1016/j.spa.2021.05.006 2022

  27. [27]

    Kusuoka, H

    S. Kusuoka, H. Takahashi, Y. Tamura, Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments, Electron. Commun. Probab. 22 (2017) 1–11. https://doi.org/10.1214/16-ECP36

    work page doi:10.1214/16-ecp36 2017

  28. [28]

    Lavenant, S

    H. Lavenant, S. Zhang, Y.H. Kim, G. Schiebinger, Toward a mathematical theory of trajectory inference, Ann. Appl. Probab. 34(1A) (2024), 428-500. https://doi.org/10.1214/23-AAP1969

    work page doi:10.1214/23-aap1969 2024

  29. [30]

    41 (2013) 1350-1380

    C.Li,Maximum-likelihoodestimationfordiffusionprocessesviaclosed-formdensityexpansions,Ann.Stat. 41 (2013) 1350-1380. https://doi.org/10.1214/13-AOS1118

    work page doi:10.1214/13-aos1118 2013

  30. [31]

    Y. Li, M. Tao, S. Wang, Essential barrier height and a probabilistic approach in characterizing potential landscape, Stoch. Process. Appl. 190 (2025) 104763. https://doi.org/10.1016/j.spa.2025.104763

    work page doi:10.1016/j.spa.2025.104763 2025

  31. [32]

    In: Cools, R., Nuyens, D

    H.Maatouk,X.Bay.AnewrejectionsamplingmethodfortruncatedmultivariateGaussianrandomvariables restricted to convex sets. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proc. Math. Stat. (2016), vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0

    work page doi:10.1007/978-3-319-33507-0 2016

  32. [33]

    Magdziarz, K

    M. Magdziarz, K. Taźbierski Stochastic representation of processes with resetting. Physical Review E. 106 (2022), https://doi.org/10.1103/PhysRevE.106.014147

    work page doi:10.1103/physreve.106.014147 2022

  33. [34]

    Maller, G

    R.A. Maller, G. Müller, A. Szimayer, Ornstein–Uhlenbeck processes and extensions, in: T. Mikosch, et al. (Eds.), Handbook of Financial Time Series, Springer, Berlin, Heidelberg, 2009, pp. 493–512. https://doi.org/10.1007/978-3-540-71297-8

    work page doi:10.1007/978-3-540-71297-8 2009

  34. [35]

    https://doi.org/10.1007/BF00340014 37

    M.Nagasawa,TransformationsofdiffusionandSchrödingerprocesses.Probab.Th.Rel.Fields82,109–136 (1989). https://doi.org/10.1007/BF00340014 37

    work page doi:10.1007/bf00340014 1989

  35. [36]

    Nielsen, K

    F. Nielsen, K. Okamura, On the𝑓-divergences between densities of a multivariate location or scale family, Stat. Comput. 34 (2024) 60. https://doi.org/10.1007/s11222-023-10373-6

    work page doi:10.1007/s11222-023-10373-6 2024

  36. [37]

    A. Oga, Y. Koike, Drift estimation for a multi-dimensional diffusion process using deep neural networks, Stoch. Processes Appl. 170 (2024) 104240. https://doi.org/10.48550/arXiv.2112.13332

    work page doi:10.48550/arxiv.2112.13332 2024

  37. [38]

    https://link.springer.com/book/10.1007/978-1-4939-1323-7

    G.A.Pavliotis,StochasticProcessesandApplications.DiffusionProcesses,theFokker-PlanckandLangevin Equations, Springer, 2014. https://link.springer.com/book/10.1007/978-1-4939-1323-7

    work page doi:10.1007/978-1-4939-1323-7 2014

  38. [39]

    Pitt, Positively correlated normal variables are associated, Ann

    L.D. Pitt, Positively correlated normal variables are associated, Ann. Probab. 10(2) (1982), 496-499. https://doi.org/10.1214/aop/1176993872

    work page doi:10.1214/aop/1176993872 1982

  39. [40]

    Ricciardi, On the transformation of diffusion processes into the Wiener process, J

    L.M. Ricciardi, On the transformation of diffusion processes into the Wiener process, J. Math. Anal. Appl. 54 (1976) 185–199. https://doi.org/10.1016/0022-247X(76)90244-4

    work page doi:10.1016/0022-247x(76)90244-4 1976

  40. [41]

    Ricciardi, A

    L.M. Ricciardi, A. Di Crescenzo, V. Giorno, A.G. Nobile, An outline of theoretical and algorithmic ap- proaches to first passage time problems with applications to biological models, Math. Japon. 50 (1999) 247–322. https://www.researchgate.net/publication/268054855

    work page arXiv 1999

  41. [42]

    Nocedal, S

    M. Shaked, J.G. Shanthikumar, Stochastic Orders. Springer, New York, 2007. https://doi.org/10.1007/978- 0-387-34675-5

    work page doi:10.1007/978- 2007

  42. [43]

    D.Trevisan,Well-posednessofmultidimensionaldiffusionprocesseswithweaklydifferentiablecoefficients, Electron. J. Probab. 21 (2016) 1–41. https://doi.org/10.1214/16-EJP4453

    work page doi:10.1214/16-ejp4453 2016

  43. [44]

    Vuong, S

    Q.N. Vuong, S. Bedbur, U. Kamps, Distances between models of generalized order statistics, J. Multiv. Analysis 118 (2013) 24–36. http://dx.doi.org/10.1016/j.jmva.2013.03.010

    work page doi:10.1016/j.jmva.2013.03.010 2013

  44. [45]

    J. Wang, C. Lin, Y. Liu, R. Xu, Z. Dou, X. Long, H. Guo, T. Komura, W. Wang, and X. Li, PDT: point distribution transformation with diffusion models. In Proceedings of SIGGRAPH 78 (2025) 1–11. https://doi.org/10.1145/3721238.3730717

    work page doi:10.1145/3721238.3730717 2025

  45. [46]

    F.Xiao,Z.Su,H.Zheng,S.Wang,Modelingofouterradiationbeltelectronsbymultidimensionaldiffusion process, J. Geophys. Res. 114 (2009) A03201. https:/doi.org/10.1029/2008JA013580 38

    work page doi:10.1029/2008ja013580 2009