Abstract
We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval [0, 1] in terms of a Markov process Y, which is a Doob’s h transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process \(\mathbb T\): the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of Y and \(\mathbb T\) are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.
This is a preview of subscription content, access via your institution.
Data Availability Statement
No datasets were generated or analysed during the current study.
References
Askey, R.: Beta integrals and the associated orthogonal polynomials. In: Alladi, K. (ed.) Number Theory. Lecture Notes in Mathematics, vol. 1395, pp. 84–121 (1989)
Barraquand, G., Doussal, P.L.: Steady state of the KPZ equation on an interval and Liouville quantum mechanics. Europhys. Lett. (in press, 2021) arXiv:2105.15178
Bertoin, J., Yor, M.: On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Commun. Probab. 6(95), 106 (2001)
Bryc, W.: Quadratic harnesses from generalized beta integrals. In: Bożejko, M., Krystek, A., and Wojakowski, Ł. (eds.) Noncommutative Harmonic Analysis with Applications to Probability III, volume 96 of Banach Center Publications, pp. 67–79. Polish Academy of Sciences. (2012) arXiv:1009.4928
Bryc, W.: On the continuous dual Hahn process. Stoch. Process. Appl. 143, 185–206 (2022). arXiv:2105.06969 [math.PR]
Bryc, W., Kuznetsov, A.: Markov limits of steady states of the KPZ equation on an interval. (2021) arXiv:2109.04462
Bryc, W., Wang, Y.: A dual representation for the Laplace transforms of the Brownian excursion and Brownian meander. Stat. Probab. Lett. 140, 77–83 (2018)
Bryc, W., Wang, Y.: Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries. Ann. l’I.H.P Probab. Stat. 55, 2169–2194 (2019)
Bryc, W., Wesołowski, J.: Askey-Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38(3), 1221–1262 (2010)
Corwin, I., Knizel, A.: Stationary measure for the open KPZ equation. (2021) arXiv:2103.12253
Corwin, I., Shen, H.: Open ASEP in the weakly asymmetric regime. Commun. Pure Appl. Math. 71(10), 2065–2128 (2018)
Craddock, M.: On an integral arising in mathematical finance. In: Nonlinear Economic Dynamics and Financial Modelling, pp. 355–370. Springer (2014)
Dawson, D., Gorostiza, L., Wakolbinger, A.: Schrödinger processes and large deviations. J. Math. Phys. 31(10), 2385–2388 (1990)
de Branges, L.: Tensor product spaces. J. Math. Anal. Appl. 38, 109–148 (1972)
Erdélyi, A.: Higher Transcendental Functions: Vol I: Bateman Manuscript Project. McGraw-Hill, New York (1953)
Erdélyi, A., Magnus, W., Oberhettinger, F.: Tables of Integral Transforms, vol. I. McGraw-Hill, New York (1954)
Farrell, R.H.: Techniques of Multivariate Calculation. Lecture Notes in Mathematics, vol. 520. Springer, New York (1976)
Gerencsér, M., Hairer, M.: Singular SPDEs in domains with boundaries. Probab. Theory Relat. Fields 173(3), 697–758 (2019)
Gradshteyn, I.S., Ryzhik, I.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007)
Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)
Hartman, P., Watson, G.S.: Normal distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2(4), 593–607 (1974)
Jamison, B.: Reciprocal processes. Z. Wahrscheinlichkeit. 30(1), 65–86 (1974)
Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeit. 32(4), 323–331 (1975)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)
Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its qanalogue. http://aw.twi.tudelft.nl/~koekoek/askey.html, report 98-17. Technical University Delft, 2, 20–21 (1998)
Kyprianou, A.E., O’Connell, N.: The Doob–McKean identity for stable Lévy processes (2021) arXiv:2103.12179
Lukacs, E., Szasz, O.: On analytic characteristic functions. Pac. J. Math. 2(4), 615–625 (1952)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, I: Probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, II: some related diffusion processes. Probab. Surv. 2, 348–384 (2005)
Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions Hardback and CD-ROM. Cambridge University Press, Cambridge (2010)
Parekh, S.: The KPZ limit of ASEP with boundary. Commun. Math. Phys. 365(2), 569–649 (2019)
Sharpe, M.: General Theory of Markov Processes. Pure and Applied Mathematics, vol. 133. Academic Press, Boston (1988)
Sousa, R., Yakubovich, S.: The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Commun. Pure Appl. Anal. 17(6), 2351–2378 (2018)
Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second Order Differential Equations, I. Calderon Press, Oxford (1962)
Wilson, J.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)
Yakubovich, S.: The heat kernel and Heisenberg inequalities related to the Kontorovich–Lebedev transform. Commun. Pure Appl. Anal. 10(2), 745–760 (2011)
Yakubovich, S.: On the Yor integral and a system of polynomials related to the Kontorovich–Lebedev transform. Integr. Transf. Spec. Funct. 24(8), 672–683 (2013)
Yakubovich, S.B.: Index Transforms. World Scientific, Singapore (1996)
Yakubovich, S.B.: On the Kontorovich–Lebedev transformation. J. Integr. Equ. Appl. 15(1), 95–112 (2003)
Yakubovich, S.B.: On the least values of \(L_p\)-norms for the Kontorovich–Lebedev transform and its convolution. J. Approx. Theory 131(2), 231–242 (2004)
Acknowledgements
This research project was initiated following WB’s visit in 2018 to Columbia University (whose hospitality is greatly appreciated) and it would not have been possible without Ivan Corwin and Alisa Knizel generously sharing their preliminary results that later appeared in [10]. We extend our thanks to Semyon Yakubovich for helpful comments on the early draft of this paper. We appreciate information from Ofer Zeitouni about the reciprocal processes. We thank Guillaume Barraquand for Ref. [2] and the discussion of its contents. WB’s research was partially supported by Simons Foundation/SFARI Award Number: 703475. AK’s research was partially supported by The Natural Sciences and Engineering Research Council of Canada. YW’s research was partially supported by Army Research Office, US (W911NF-20-1-0139). JW’s research was partially supported by Grant 2016/21/B/ST1/00005 of National Science Centre, Poland.
Author information
Authors and Affiliations
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Additional properties of Kontorovich–Lebedev transform
We collect here facts about \({{\mathcal {K}}}\) and \({{\mathcal {K}}}^{-1}\) that we need in the paper.
Lemma A.1
-
(i)
If F is in \(L_1(K_0)\) then \({{\mathcal {K}}}F\) is bounded.
-
(ii)
If G is in \(L_1(\text {d}\mu )\) and \(\varepsilon >0\), then function \( e^{\varepsilon x} {{\mathcal {K}}}^{-1} G \) is in \(L_1(K_0)\).
-
(iii)
If F is in \(L_1(K_0)\) and G is in \(L_1(\text {d}\mu )\) then we have Parseval’s identity
$$\begin{aligned} \int _\mathbb {R}F(x)\mathcal {K}^{-1}[G](x) \text {d}x = \int _0^\infty G(u) \mathcal {K}[F](u) \mu (\text {d}u). \end{aligned}$$(A1) -
(iv)
If F is in \(L_1(K_0)\) and \(\delta >0\), then we have associativity
$$\begin{aligned} \left( {{\mathcal {K}}}^{-1}e^{-\delta u^2} {{\mathcal {K}}}\right) F={{\mathcal {K}}}^{-1}\left( e^{-\delta u^2}{{\mathcal {K}}}F\right) . \end{aligned}$$(A2) -
(v)
If G is in \(L_1(\text {d}\mu )\) and \(\delta >0\), then we have associativity
$$\begin{aligned} \left( {{\mathcal {K}}}e^{\delta x} {{\mathcal {K}}}^{-1}\right) G={{\mathcal {K}}}\left( e^{\delta x}{{\mathcal {K}}}^{-1} G\right) . \end{aligned}$$(A3)
Proof
(Proof of (A.1)) This follows from \(\vert K_{\text {i}u}(e^x)\vert \le K_0(e^x)\). \(\square \)
Proof
(Proof of (A.1)) This follows from \( \vert {{\mathcal {K}}}^{-1}(G)\vert =\left| \int _0^\infty K_{\text {i}u}(e^x) G(u)\mu (\text {d}u)\right| \le C K_0(e^x)\) and from the well known bounds
for \(x\ge 1\), and
for \(x<0\); the latter is a consequence of the trivial bound \(\cosh t\ge t^{2k}/(2k)!\) applied to (1.11) with \(u=0\) and large enough k.
Thus \(e^{\varepsilon x} K_0^2(x)\le C_\varepsilon ^2\exp (\varepsilon x/2)1_{x<0}+ K_0(0)+C^2\exp (-2\exp (x))1_{x>0} \) is integrable, i.e. \(e^{\varepsilon x} K_0(x)\in L_1(K_0)\). \(\square \)
Proof
(Proof of (A.1)) (See [38, Lemma 2.4] and the discussion therein.) Since
we can apply Fubini’s theorem to interchange the order of integrals:
which is (A1). \(\square \)
Proof
(Proof of (A.1)) Identity (A2) in the integral form consists of change of order in the integrals:
The integrand is dominated by \(K_0(e^x)K_0(e^y)\vert F(y)\vert e^{-\delta u^2}\). From (1.10) we see that \(e^{-\delta u^2}\in L_1(\text {d}\mu )\), so with \(\vert F\vert \in L_1(K_0)\), this justifies the use of Fubini’s theorem. \(\square \)
Proof
(Proof of (A.1)) Identity (A3) in the integral form is
The use of Fubini’s theorem is again justified by the fact that \(e^{\delta x} K_0(e^x)\) is in \(L_1(K_0)\); the latter follows from the bounds stated in the proof of (A.1). \(\square \)
Appendix B: Semigroup property for \({{\mathcal {Q}}}_{s,t}\)
In this section we prove Proposition 6.1. The fact that
is a consequence of Beta integral [1, (7.i)] or [25, Section 1.3] for the weight function of the continuous dual Hahn polynomials
We use (B1) with
The Chapman–Kolmogorov equations
are a consequence of de Branges–Wilson integral [14, 35]
after we note that
with
Related algebraic identity is used for similar purposes in [4, Section 3] and in [10, Section 7.3].
Appendix C: Continuous dual Hahn process
To define the continuous dual Hahn process, Corwin and Knizel [10] specify its marginal distributions as a \(\sigma \)-finite measure on \(\mathbb {R}\) and transition probability distributions, and then show that they are consistent. For technical reasons, they define the process only on the interval [0, S), where \(S=2+( {\mathsf {c}} -2)1_{(0,2)}( {\mathsf {c}} )\). For Theorem 1.3, we need this process to be extended to \(s\in [0, {\mathsf {c}} )\) with \( {\mathsf {c}} >0\). For Theorem 1.2, we need this process for \(s\in (- {\mathsf {a}} , {\mathsf {c}} )\).
The following summarizes Ref. [5], restricted to the time domain, \((-\infty , {\mathsf {c}} ]\), where \( {\mathsf {c}} \in \mathbb {R}\). To define the family of transition probabilities we use the orthogonality probability measure for the continuous dual Hahn orthogonal polynomials. These are monic polynomials which depend on three parameters traditionally denoted by a, b, c, but to avoid confusion with the parameters \( {\mathsf {a}} , {\mathsf {c}} \) above, we will denote them by \(\alpha ,\beta ,\gamma \).
We take \(\alpha \in \mathbb {R}\) and \(\beta ,\gamma \) are either real or complex conjugates. Let
The continuous dual Hahn orthogonal polynomials in real variable x are defined by the three step recurrence relation
with the usual initialization \(p_{-1}(x)=0\), \(p_0(x)=1\). Compare [25, (1.3.5)].
Recursion (C1) defines a family of monic polynomials. Favard’s theorem, as stated in [9, Theorem A.1], allows us to recognize orthogonality from the condition that
If parameters \(\alpha ,\beta ,\gamma \) are such that (C2) holds, then there exists a unique (see [5, Lemma 2.1]) probability measure \(\rho (\text {d}x\vert \alpha ,\beta ,\gamma )\) such that for \(m\ne n\) we have
(Notice the factor x/4 to match the scaling in [10]. This factor is not present in [5], so we need to recalculate some of the results that we quote from there.)
When defined, probability measures \(\rho (\text {d}x\vert \alpha ,\beta ,\gamma )\) may be discrete, continuous, or of mixed type, depending on the parameters. We use them to define transition probabilities for \(-\infty<s<t\le {\mathsf {c}} \) as follows
By [5, Proposition 3.1], these measures are well defined. A version of [5, Theorem 3.5(i)] says that Chapman–Kolmogorov equations hold: for a Borel subset \(U\subset \mathbb {R}\), \( x\in \mathbb {R}\), and \(-\infty<s<t<u\le {\mathsf {c}} \) we have
Definition C.1
We denote by \((\mathbb {T}_t)_{-\infty <t\le {\mathsf {c}} }\) the Markov process on state space \(\mathbb {R}\), with transition probabilities (C3).
Next we follow [10, Definition 7.8] and introduce a family of \(\sigma \)-finite positive measures that are the entrance law [32, pg. 5] for the Markov process \((\mathbb {T}_t)\). For \( {\mathsf {a}} >- {\mathsf {c}} \) and \(s\le {\mathsf {c}} \), let
where for \(j\in \mathbb {Z}\cap [0, -( {\mathsf {a}} +s)/2) \) the discrete masses are given by
This is recalculated from [10, Definition 7.8]; the parameters in [10, (1.12)] are \(u= {\mathsf {c}} /2\), \(v= {\mathsf {a}} /2\).
Remark C.1
We note that there are no atoms when \(- {\mathsf {a}}<s< {\mathsf {c}} \), so in this case \({\mathfrak {p}}_s(\text {d}x)\) is absolutely continuous with the density supported on \((0,\infty )\).
The following summarizes properties of measures \({\mathfrak {p}}_s\) and transition probabilities \({\mathfrak {p}}_{s,t}(x,\text {d}y)\) that we will need.
Theorem C.2
Suppose that \( {\mathsf {a}} + {\mathsf {c}} >0\), and \(U\subset \mathbb {R}\) is a Borel set.
-
(i)
Measures \(\{{\mathfrak {p}}_s: -\infty< s< {\mathsf {c}} \}\) are the entrance law for the continuous dual Hahn process, i.e., they satisfy
$$\begin{aligned} {\mathfrak {p}}_t(U)=\int _\mathbb {R}{\mathfrak {p}}_{s,t}(x,U) {\mathfrak {p}}_s(\text {d}x), -\infty< s<t\le {\mathsf {c}} . \end{aligned}$$(C7) -
(ii)
For \(-\infty<s<t< {\mathsf {c}} \) and \(x>0\), transition probabilities for the continuous dual Hahn process are absolutely continuous with density (1.14).
Proof
-
(i)
is a version of [5, Theorem 4.1 ]
-
(ii)
This is the case of measure \(\rho (\text {d}x\vert \alpha ,\beta ,\gamma )\) with parameters \(\alpha >0\), \(\beta ={\bar{\gamma }}\), \(\text {Re}(\beta )>0\), so the density can be read out from [25, Section 1.3], see also [4, Section 3] and [10, Section 1.3].
\(\square \)
Remark C.3
By Remark C.1 and Theorem C.2(ii), transition probabilities (C3) with the entrance law (C5) define an absolutely continuous Markov process \((\mathbb {T}_t)_{- {\mathsf {a}}<t< {\mathsf {c}} }\) on the state space \((0,\infty )\). This is the process that appears in Theorem 1.2. However, if the state space is enlarged to include points \(x<0\), then the transition probabilities may have a discrete component; for example, at \(x=-( {\mathsf {c}} -s)^2\), transition probability \({\mathfrak {p}}_{s,t}(x,\text {d}y)\) is a degenerate measure concentrated at \(-( {\mathsf {c}} -t)^2\), see [5]. In particular, if \(s<- {\mathsf {a}} \) then the entrance law (C5) may put some mass at points in \((-\infty ,0)\).
Rights and permissions
About this article
Cite this article
Bryc, W., Kuznetsov, A., Wang, Y. et al. Markov processes related to the stationary measure for the open KPZ equation. Probab. Theory Relat. Fields 185, 353–389 (2023). https://doi.org/10.1007/s00440-022-01110-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-022-01110-7
Keywords
- Kontorovich–Lebedev transform
- Open KPZ equation
- Yakubovich’s heat kernel
- Killed Brownian motion
- Dual representation for Laplace transform
- Liouville potential
- Dual semigroups
Mathematics Subject Classification
- 60J25
- 44A15
- 34L10