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Markov processes related to the stationary measure for the open KPZ equation

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.

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Data Availability Statement

No datasets were generated or analysed during the current study.

References

  1. 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)

  2. 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

  3. Bertoin, J., Yor, M.: On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Commun. Probab. 6(95), 106 (2001)

    MathSciNet  MATH  Google Scholar 

  4. 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

  5. Bryc, W.: On the continuous dual Hahn process. Stoch. Process. Appl. 143, 185–206 (2022). arXiv:2105.06969 [math.PR]

    Article  MathSciNet  MATH  Google Scholar 

  6. Bryc, W., Kuznetsov, A.: Markov limits of steady states of the KPZ equation on an interval. (2021) arXiv:2109.04462

  7. 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)

    Article  MATH  Google Scholar 

  8. 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)

    MathSciNet  MATH  Google Scholar 

  9. Bryc, W., Wesołowski, J.: Askey-Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38(3), 1221–1262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Corwin, I., Knizel, A.: Stationary measure for the open KPZ equation. (2021) arXiv:2103.12253

  11. Corwin, I., Shen, H.: Open ASEP in the weakly asymmetric regime. Commun. Pure Appl. Math. 71(10), 2065–2128 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Craddock, M.: On an integral arising in mathematical finance. In: Nonlinear Economic Dynamics and Financial Modelling, pp. 355–370. Springer (2014)

  13. Dawson, D., Gorostiza, L., Wakolbinger, A.: Schrödinger processes and large deviations. J. Math. Phys. 31(10), 2385–2388 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Branges, L.: Tensor product spaces. J. Math. Anal. Appl. 38, 109–148 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Erdélyi, A.: Higher Transcendental Functions: Vol I: Bateman Manuscript Project. McGraw-Hill, New York (1953)

    Google Scholar 

  16. Erdélyi, A., Magnus, W., Oberhettinger, F.: Tables of Integral Transforms, vol. I. McGraw-Hill, New York (1954)

    MATH  Google Scholar 

  17. Farrell, R.H.: Techniques of Multivariate Calculation. Lecture Notes in Mathematics, vol. 520. Springer, New York (1976)

    Book  Google Scholar 

  18. Gerencsér, M., Hairer, M.: Singular SPDEs in domains with boundaries. Probab. Theory Relat. Fields 173(3), 697–758 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gradshteyn, I.S., Ryzhik, I.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  20. Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hartman, P., Watson, G.S.: Normal distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2(4), 593–607 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jamison, B.: Reciprocal processes. Z. Wahrscheinlichkeit. 30(1), 65–86 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeit. 32(4), 323–331 (1975)

    Article  MATH  Google Scholar 

  24. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)

    Article  MATH  Google Scholar 

  25. 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)

  26. Kyprianou, A.E., O’Connell, N.: The Doob–McKean identity for stable Lévy processes (2021) arXiv:2103.12179

  27. Lukacs, E., Szasz, O.: On analytic characteristic functions. Pac. J. Math. 2(4), 615–625 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, I: Probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, II: some related diffusion processes. Probab. Surv. 2, 348–384 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    MATH  Google Scholar 

  31. Parekh, S.: The KPZ limit of ASEP with boundary. Commun. Math. Phys. 365(2), 569–649 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sharpe, M.: General Theory of Markov Processes. Pure and Applied Mathematics, vol. 133. Academic Press, Boston (1988)

    MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second Order Differential Equations, I. Calderon Press, Oxford (1962)

    Book  MATH  Google Scholar 

  35. Wilson, J.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yakubovich, S.: The heat kernel and Heisenberg inequalities related to the Kontorovich–Lebedev transform. Commun. Pure Appl. Anal. 10(2), 745–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yakubovich, S.B.: Index Transforms. World Scientific, Singapore (1996)

    Book  MATH  Google Scholar 

  39. Yakubovich, S.B.: On the Kontorovich–Lebedev transformation. J. Integr. Equ. Appl. 15(1), 95–112 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

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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

  1. (i)

    If F is in \(L_1(K_0)\) then \({{\mathcal {K}}}F\) is bounded.

  2. (ii)

    If G is in \(L_1(\text {d}\mu )\) and \(\varepsilon >0\), then function \( e^{\varepsilon x} {{\mathcal {K}}}^{-1} G \) is in \(L_1(K_0)\).

  3. (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)
  4. (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)
  5. (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

$$\begin{aligned} K_0(x)\le K_{1/2}(x)= e^{-x}\frac{\sqrt{\pi }}{\sqrt{2x}}\le C e^{-x} \end{aligned}$$
(A4)

for \(x\ge 1\), and

$$\begin{aligned} K_0(e^x)\le C_\varepsilon e^{-\varepsilon x/4} \end{aligned}$$
(A5)

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

$$\begin{aligned} \int _\mathbb {R}\int _0^\infty \vert F(x)K_{\text {i}u}(e^x) G(u)\vert \text {d}x \mu (\text {d}u)\le \int _\mathbb {R}\vert F(x)\vert K_{0}(e^x) \text {d}x \int _0^\infty \vert G(u)\vert \mu (\text {d}u)<\infty , \end{aligned}$$

we can apply Fubini’s theorem to interchange the order of integrals:

$$\begin{aligned} \int _\mathbb {R}F(x) \left( \int _0^\infty \,K_{\text {i}u}(e^x) G(u) \mu (\text {d}u)\right) \text {d}x = \int _0^\infty \left( \int _\mathbb {R}F(x) K_{\text {i}u}(e^x) \text {d}x\right) G(u) \mu (\text {d}u), \end{aligned}$$

which is (A1). \(\square \)

Proof

(Proof of (A.1)) Identity (A2) in the integral form consists of change of order in the integrals:

$$\begin{aligned}&\int _\mathbb {R}\left( \int _0^\infty e^{-\delta u^2} K_{\text {i}u}(e^x)K_{\text {i}u}(e^y)\mu (\text {d}u)\right) F(y)\text {d}y \\&\quad = \int _0^\infty K_{\text {i}u}(e^x) e^{-\delta u^2}\left( \int _\mathbb {R}K_{\text {i}u}(e^y)F(y)\text {d}y \right) \mu (\text {d}u). \end{aligned}$$

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

$$\begin{aligned}&\int _0^\infty G(v) \left( \int _\mathbb {R}e^{sx}K_{\text {i}u}(e^x)K_{\text {i}v} (e^x) \text {d}x\right) \mu (\text {d}v) \\&\quad =\int _\mathbb {R}K_{\text {i}u}(e^x)e^{\delta x} \left( \int _0^\infty K_{\text {i}v}(e^x)G(v)\mu (\text {d}v)\right) \text {d}x. \end{aligned}$$

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

$$\begin{aligned} \int _0^\infty q_{s,t}(u,v)\text {d}v=1 \end{aligned}$$

is a consequence of Beta integral [1, (7.i)] or [25, Section 1.3] for the weight function of the continuous dual Hahn polynomials

$$\begin{aligned} \int _0^\infty \frac{\prod _{j=1}^3 \left( \Gamma ( {\mathsf {c}} _j+\text {i}{x})\Gamma ( {\mathsf {c}} _j-\text {i}{x})\right) }{\left| \Gamma (2\text {i}{x})\right| ^2} dx =2 \pi \prod _{1\le k<j\le 3}\Gamma ( {\mathsf {c}} _k+ {\mathsf {c}} _j). \end{aligned}$$
(B1)

We use (B1) with

$$\begin{aligned} x=v/2, {\mathsf {c}} _1=( {\mathsf {c}} -t)/2, {\mathsf {c}} _2=(t-s+\text {i}u)/2, {\mathsf {c}} _3=(t-s-\text {i}u)/2. \end{aligned}$$

The Chapman–Kolmogorov equations

$$\begin{aligned} \int _0^\infty q_{r,s}(u,v) q_{s,t}(v,w)\text {d}v= q_{r,t}(u,w), \quad r<s<t, \end{aligned}$$

are a consequence of de Branges–Wilson integral [14, 35]

$$\begin{aligned} \int _0^\infty \frac{\prod _{j=1}^4 \left( \Gamma ( {\mathsf {c}} _j+\text {i}{x})\Gamma ( {\mathsf {c}} _j-\text {i}{x})\right) }{\left| \Gamma (2\text {i}{x})\right| ^2} dx =\frac{2 \pi \prod _{1\le k<j\le 4}\Gamma ( {\mathsf {c}} _k+ {\mathsf {c}} _j)}{\Gamma ( {\mathsf {c}} _1+ {\mathsf {c}} _2+ {\mathsf {c}} _3+ {\mathsf {c}} _4)}, \end{aligned}$$

after we note that

$$\begin{aligned}&\frac{q_{r,s}(u,x) q_{s,t}(x,w)}{q_{r,t}(u,w)}\\&\quad = \frac{\Gamma ( {\mathsf {c}} _1+ {\mathsf {c}} _2+ {\mathsf {c}} _3+ {\mathsf {c}} _4)}{4 \pi \prod _{1\le k<j\le 4}\Gamma ( {\mathsf {c}} _k+ {\mathsf {c}} _j)}\frac{\prod _{j=1}^4 \left( \Gamma ( {\mathsf {c}} _j+\text {i}{x}/2)\Gamma ( {\mathsf {c}} _j-\text {i}{x}/2)\right) }{\left| \Gamma (\text {i}{x})\right| ^2}. \end{aligned}$$

with

$$\begin{aligned} {\mathsf {c}} _1= & {} (s-r+\text {i}u)/2,\; {\mathsf {c}} _2=(s-r-\text {i}u)/2,\; {\mathsf {c}} _3=(t-s+\text {i}w)/2,\; \\ {\mathsf {c}} _4= & {} (t-s-\text {i}w)/2. \end{aligned}$$

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 abc, 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

$$\begin{aligned} A_n=(n+\alpha +\beta )(n+\alpha +\gamma ),\quad C_n=n(n-1+\beta +\gamma ). \end{aligned}$$

The continuous dual Hahn orthogonal polynomials in real variable x are defined by the three step recurrence relation

$$\begin{aligned} x p_n(x)= p_{n+1}(x)+(A_n+C_n-\alpha ^2)p_n(x)+A_{n-1}C_n p_{n-1}(x), \end{aligned}$$
(C1)

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

$$\begin{aligned} \prod _{k=0}^nA_kC_{k+1}\ge 0 \hbox { for all}\ n. \end{aligned}$$
(C2)

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

$$\begin{aligned} \int _\mathbb {R}p_n(\tfrac{x}{4}\vert \alpha ,\beta ,\gamma ) p_m(\tfrac{x}{4}\vert \alpha ,\beta ,\gamma )\rho (\text {d}x\vert \alpha ,\beta ,\gamma )=0. \end{aligned}$$

(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

$$\begin{aligned} {\mathfrak {p}}_{s,t}(x,\text {d}y)={\left\{ \begin{array}{ll} \rho (\text {d}y\vert \tfrac{ {\mathsf {c}} -t}{2},\tfrac{t-s - \text {i}\sqrt{x}}{2},\tfrac{t-s + \text {i}\sqrt{x}}{2}) &{} x\ge 0, \\ \rho (\text {d}y\vert \tfrac{ {\mathsf {c}} -t}{2},\tfrac{t-s - \sqrt{-x}}{2},\tfrac{t-s + \sqrt{-x}}{2}) &{} -( {\mathsf {c}} -s)^2\le x<0, \\ \delta _{-( {\mathsf {c}} -t)^2} &{} x<-( {\mathsf {c}} -s)^2. \end{array}\right. } \end{aligned}$$
(C3)

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

$$\begin{aligned} {\mathfrak {p}}_{s,u}(x,U)=\int _\mathbb {R}{\mathfrak {p}}_{t,u}(y,U) {\mathfrak {p}}_{s,t}(x,\text {d}y). \end{aligned}$$
(C4)

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

$$\begin{aligned}&{\mathfrak {p}}_{s}(\text {d}x)= \frac{( {\mathsf {a}} + {\mathsf {c}} )( {\mathsf {a}} + {\mathsf {c}} +2)}{16\pi } \frac{\vert \Gamma (\frac{s+ {\mathsf {a}} +\text {i}\sqrt{x}}{2},\frac{ {\mathsf {c}} -s+\text {i}\sqrt{x}}{2} )\vert ^2}{2\sqrt{x}\vert \Gamma (\text {i}\sqrt{x})\vert ^2}1_{x>0}\;\text {d}x \nonumber \\&\quad + \sum _{\{j:\; 2j+ {\mathsf {a}} +s<0\}} p_j(s) \delta _{-(2j+ {\mathsf {a}} +s)^2}(\text {d}x), \end{aligned}$$
(C5)

where for \(j\in \mathbb {Z}\cap [0, -( {\mathsf {a}} +s)/2) \) the discrete masses are given by

$$\begin{aligned} p_j(s)&= \frac{( {\mathsf {a}} + {\mathsf {c}} )( {\mathsf {a}} + {\mathsf {c}} +2)}{4}\frac{\Gamma (\frac{ {\mathsf {c}} - {\mathsf {a}} -2s}{2},\frac{ {\mathsf {a}} + {\mathsf {c}} }{2})}{\Gamma (- {\mathsf {a}} -s)}\cdot \frac{( {\mathsf {a}} +2j+s)( {\mathsf {a}} +s,\frac{ {\mathsf {a}} + {\mathsf {c}} }{2})_j}{( {\mathsf {a}} +s)j!(1+\frac{2s+ {\mathsf {a}} - {\mathsf {c}} }{2})_j}. \end{aligned}$$
(C6)

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.

  1. (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)
  2. (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

  1. (i)

    is a version of [5, Theorem 4.1 ]

  2. (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 \((-\infty ,0)\).

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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

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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