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Stationary Solutions to the Stochastic Burgers Equation on the Line

Abstract

We consider invariant measures for the stochastic Burgers equation on \({\mathbb {R}}\), forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each \(a\in {\mathbb {R}}\), there is a unique indecomposable law of a spacetime-stationary solution with mean a, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-a periodic functions converge in law to the extremal space-time stationary solution with mean a as time goes to infinity.

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References

  1. Adler, R.J.: An introduction to continuity, extrema, and related topics for general Gaussian processes, volume 12 of Inst. Math. Sci. Lect. Notes-Monograph Series. Institute of Mathematical Statistics, Hayward, CA (1990)

  2. Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in \(1+1\) dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Bakhtin, Y.: Inviscid Burgers equation with random kick forcing in noncompact setting. Electron. J. Probab. 21(37), 50 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Bakhtin, Y., Khanin, K.: On global solutions of the random Hamilton–Jacobi equations and the KPZ problem. Nonlinearity 31(4), R93–R121 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  6. Bakhtin, Y., Li, L.: Zero temperature limit for directed polymers and inviscid limit for stationary solutions of stochastic Burgers equation. J. Stat. Phys. 172(5), 1358–1397 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Bakhtin, Y., Li, L.: Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation. Commun. Pure Appl. Math. 72(3), 536–619 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Bec, J., Khanin, K.: Burgers turbulence. Phys. Rep. 447(1), 1–66 (2007)

    ADS  MathSciNet  Google Scholar 

  9. Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation. Commun. Math. Phys. 165(2), 211–232 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5–6), 1377–1401 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  11. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  12. Bertini, L., Giacomin, G.: On the long-time behavior of the stochastic heat equation. Probab. Theory Relat. Fields 114(3), 279–289 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Boritchev, A.A.: Turbulence for the generalised Burgers equation. Russ. Math. Surv. 69(6), 957 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Boritchev, A.: Sharp estimates for turbulence in white-forced generalised Burgers equation. Geom. Funct. Anal. 23(6), 1730–1771 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Boritchev, A.: Multidimensional potential Burgers turbulence. Commun. Math. Phys. 342(2), 441–489 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  16. Boritchev, A.: Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing. Stoch. Partial Differ. Equ. Anal. Comput. 6(1), 109–123 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Brzeźniak, Z., Goldys, B., Neklyudov, M.: Multidimensional stochastic Burgers equation. SIAM J. Math. Anal. 46(1), 871–889 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225–236 (1951)

    MathSciNet  MATH  Google Scholar 

  19. Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. NoDEA Nonlinear Differ. Equ. Appl. 1(4), 389–402 (1994)

    MathSciNet  MATH  Google Scholar 

  20. Da Prato, G., Gatarek, D.: Stochastic Burgers equation with correlated noise. Stoch. Stoch. Rep. 52(1–2), 29–41 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems. London Math. Soc. Lecture Note Ser., vol. 229, Cambridge University Press, Cambridge (1996)

  22. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Volume 152 of Encyclopedia Math. Appl., 2nd edn. Cambridge University Press, Cambridge (2014)

  23. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren Math. Wiss., 4th edn. Springer-Verlag, Berlin (2016)

  24. Dawson, D.A., Salehi, H.: Spatially homogeneous random evolutions. J. Multiv. Anal. 10(2), 141–180 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Dunlap, A., Gu, Y., Ryzhik, L., Zeitouni, O.: The random heat equation in dimensions three and higher: the homogenization viewpoint. arXiv:1808.07557v1

  26. Weinan, E., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877–960 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Florescu, I., Viens, F.: Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space. Probab. Theory Relat. Fields 135(4), 603–644 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ. Anal. Comput. 3(2), 159–220 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Gubinelli, M., Perkowski, N.: KPZ reloaded. Commun. Math. Phys. 349(1), 165–269 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Gubinelli, M., Perkowski, N.: The infinitesimal generator of the stochastic Burgers equation. Probab. Theory Relat. Fields 178(3–4), 1067–1124 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Gyöngy, I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73(2), 271–299 (1998)

    MathSciNet  MATH  Google Scholar 

  32. Gyöngy, I., Nualart, D.: On the stochastic Burgers’ equation in the real line. Ann. Probab. 27(2), 782–802 (1999)

    MathSciNet  MATH  Google Scholar 

  33. Hairer, M.: Ergodic theory for stochastic PDEs (2008). http://www.hairer.org/notes/Imperial.pdf

  34. Hairer, M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547–1585 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Hairer, M., Labbé, C.: A simple construction of the continuum parabolic Anderson model on \({ R}^2\). Electron. Commun. Probab. 20(43), 11 (2015)

    MATH  Google Scholar 

  37. Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. Forum Math. Pi 6, e3, 112 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Hairer, M., Weber, H.: Rough Burgers-like equations with multiplicative noise. Probab. Theory Relat. Fields 155(1–2), 71–126 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Hopf, E.: The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230 (1950)

    MATH  Google Scholar 

  40. Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems. Commun. Math. Phys. 232(3), 377–428 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  41. Kantorovich, L.V., Rubinshtein, G.S.: On a space of totally additive functions. Vestn. Leningrad. Univ. 13(7), 52–59 (1958)

    MATH  Google Scholar 

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

    ADS  MATH  Google Scholar 

  43. Kim, J.U.: On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete Contin. Dyn. Syst. Ser. B 6(4), 835–866 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Lewis, P., Nualart, D.: Stochastic Burgers’ equation on the real line: regularity and moment estimates. Stochastics 90(7), 1053–1086 (2018)

    MathSciNet  Google Scholar 

  45. Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co. Inc, River Edge (1996)

    MATH  Google Scholar 

  46. Moameni, A.: Invariance properties of the Monge–Kantorovich mass transport problem. Discrete Contin. Dyn. Syst. 36(5), 2653–2671 (2016)

    MathSciNet  MATH  Google Scholar 

  47. Mukherjee, C., Shamov, A., Zeitouni, O.: Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \(d\ge 3\). Electron. Commun. Probab. 21(61), 12 (2016)

    MATH  Google Scholar 

  48. Perkowski, N., Rosati, T.C.: The KPZ equation on the real line. Electron. J. Probab., 24(117), (2019)

  49. Phelps, R.R.: Lectures on Choquet’s Theorem, 2nd edn. Lecture Notes in Mathematics, vol. 1757. Springer, Berlin (2001)

  50. Rosati, T.C.: Synchronization for KPZ. arXiv:1907.06278v1

  51. Sasamoto, T., Spohn, H.: One-dimensional Kardar–Parisi–Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104(23), 230602 (2010)

    ADS  MATH  Google Scholar 

  52. Serre, D.: \(L^1\)-stability of nonlinear waves in scalar conservation laws. In: Evolutionary Equations. Vol. I, Handb. Differ. Equ., pp. 473–553. North-Holland, Amsterdam (2004)

  53. Tessitore, G., Zabczyk, J.: Invariant measures for stochastic heat equations. Probab. Math. Statist., 18(2, Acta Univ. Wratislav. No. 2111), 271–287 (1998)

  54. Tsai, L.-C.: Exact lower tail large deviations of the KPZ equation. arXiv:1809.03410v1

  55. Unterberger, J.: PDE estimates for multi-dimensional KPZ equation. arXiv:1307.1980v4

  56. Unterberger, J.: Global existence and smoothness for solutions of viscous Burgers equation. (2) the unbounded case: a characteristic flow study. arXiv:1510.01539v1

  57. Unterberger, J.: Global existence for strong solutions of viscous Burgers equation. (1) The bounded case. Control Cybern. 46(2), 109–136 (2017)

    MathSciNet  MATH  Google Scholar 

  58. Villani, C.: Optimal Transport. Grundlehren Math. Wiss., vol. 338, Springer, Berlin (2009)

  59. Zhang, X., Zhu, R., Zhu, X.: Singular HJB equations with applications to KPZ on the real line. arXiv:2007.06783v1

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Acknowledgements

We are happy to thank Yuri Bakhtin and Konstantin Khanin for generous explanations of their work, and Kevin Yang for productive discussions. We are also especially grateful to Yu Gu for pointing out a strengthening of Proposition 5.2 that led to an improvement of our main result over the initial version of the paper. AD was partially supported by an NSF Graduate Research Fellowship under grant DGE-1147470, CG by the Fannie and John Hertz Foundation and NSF grant DGE-1656518, and LR by NSF grants DMS-1613603 and DMS-1910023, and ONR grant N00014-17-1-2145.

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Appendices

Proof of Proposition 1.4

We now prove Proposition 1.4. The proof is elementary and independent of the rest of the paper. Let \(\varepsilon >0\). For each \(j=1,\ldots ,N\), define \(w_{j}\in L^{\infty }({\mathbb {R}})\) by

$$\begin{aligned} w_{j}(x)&=\sup _{|y|\ge |x|}|v_{\mathrm {z},j}(y)|, \end{aligned}$$

so \(|v_{\mathrm {z},j}(x)|\le w_{j}(x)\) for all \(x\in {\mathbb {R}}\). Then, \(w_{j}(x)\) is decreasing in |x|, and

$$\begin{aligned} \lim _{|x|\rightarrow \infty }w_{j}(x)=0. \end{aligned}$$

Therefore, we can find a \(K\in {\mathbb {N}}\) so large that

$$\begin{aligned} \frac{1}{KL}\max _{j=1}^{N}\Vert v_{\mathrm {int},j}\Vert _{L^{1}({\mathbb {R}})}&<\varepsilon /2,&\frac{1}{KL}\max _{j=1}^{N}\int _{0}^{KL}w_{j}(x)\,{d}x&<\varepsilon /2. \end{aligned}$$

Let us define

$$\begin{aligned} v_{\pm ,j}(x)&=v_{\mathrm {per},j}(x)\pm w_{j}(x)\pm \sup _{m\in {\mathbb {Z}}}|v_{\mathrm {int},j}(x+mKL)| \end{aligned}$$
(A.1)

and \({\mathbf {v}}_{-}=(v_{-,1},\ldots ,v_{-,N})\) and \({\mathbf {v}}_{+}=(v_{+,1},\ldots ,v_{+,N})\), so \({\mathbf {v}}_{-},{\mathbf {v}}_{+}\in L^{\infty }({\mathbb {R}})^{N}\) and \({\mathbf {v}}_{-}\preceq {\mathbf {v}}\preceq {\mathbf {v}}_{+}\). Then we have

$$\begin{aligned}&\frac{1}{KL}\int _{0}^{KL}{\mathbf {v}}_{-}(x)\,{d}x-{\mathbf {a}} \\&\quad =\frac{1}{KL}\int _{0}^{KL}[{\mathbf {v}}_{-}(x) -{\mathbf {v}}_{\mathrm {per}}(x)]\,{d}x \\&\quad =\frac{1}{KL}\int _{0}^{KL}\left[ -w_{j}(x)+\inf _{i\in {\mathbb {Z}}} v_{\mathrm {int},j}(x+iKL)\right] \,{d}x\\&\quad \succeq -\frac{\varepsilon }{2}(1,\ldots ,1)-\frac{1}{KL}\max _{j=1}^{N}\int _{0}^{KL}\sum _{i\in {\mathbb {Z}}}\left| v_{\mathrm {int},j}(x+iKL)\right| \,{d}x \\&\quad =-\frac{\varepsilon }{2}(1,\ldots ,1)-\frac{1}{KL}\max _{j=1}^{N}\Vert v_{\mathrm {int},j}\Vert _{L^{1}({\mathbb {R}})}\succeq -(\varepsilon ,\ldots ,\varepsilon ). \end{aligned}$$

A similar argument shows that

$$\begin{aligned} \frac{1}{KL}\int _{0}^{KL}{\mathbf {v}}_{+}(x)\,{d}x-{\mathbf {a}}\preceq (\varepsilon ,\ldots ,\varepsilon ). \end{aligned}$$

Since this is possible for every \(\varepsilon >0\), we have \({\mathbf {v}}\in {\mathscr {B}}_{{\mathbf {a}}}\).    \(\square \)

Weighted Spaces

We now record several useful results on weighted spaces. We begin with some basic lemmas that are used throughout the paper. Then we will prove bounds on the heat kernel on weighted spaces.

Basic properties of weighted spaces

Our first lemma allows us to upgrade convergence from one weight to another.

Proposition B.1

Let \({\mathcal {T}}\) be a metric space and \(w_{1},w_{2},w_{3}\) be weights on \({\mathbb {R}}\) so that

$$\begin{aligned} \lim \limits _{|x|\rightarrow \infty }\frac{w_{1}(x)}{w_{2}(x)}=0. \end{aligned}$$

Suppose that \(u_n\in {\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{1}}({\mathbb {R}}))\), and \(u\in {\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{3}}({\mathbb {R}}))\) satisfy

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \Vert u_{n}-u\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{3}} ({\mathbb {R}}))}=0 \quad \text {and} \quad \sup \limits _{n\in {\mathbb {N}}}\Vert u_{n}\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}} ;{\mathcal {C}}_{w_{1}}({\mathbb {R}}))}<\infty . \end{aligned}$$

Then \(u\in {\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{2}}({\mathbb {R}}))\), and \(\lim \limits _{n\rightarrow \infty }\Vert u_{n}-u\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{2}}({\mathbb {R}}))}=0\) as well.

Proof

Fix \(\varepsilon >0\) and define

$$\begin{aligned} K = \sup \limits _{n\in {\mathbb {N}}} \Vert u_{n}\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{1}}({\mathbb {R}}))}<\infty . \end{aligned}$$

Then choose M so that

$$\begin{aligned} \left| \frac{w_{1}(x)}{w_{2}(x)}\right| \le \frac{\varepsilon }{2K}~~~\hbox { if }|x|\ge M, \end{aligned}$$

and N so large that if \(n\ge N\) then

$$\begin{aligned} \Vert u_{n}-u\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{3}}({\mathbb {R}}))} \le \varepsilon \inf \limits _{|x|\le M}\frac{w_{2}(x)}{w_{3}(x)}. \end{aligned}$$

Now, for any \(n\ge N\), if \(|y|\le M\), then for all \(t\in {\mathcal {T}}\) we have

$$\begin{aligned} |(u_{n}-u)(t,y)|\le \varepsilon w_{3}(y)\inf \limits _{x\in [-M,M]}\frac{w_{2}(x)}{w_{3}(x)}\le \varepsilon w_{2}(y) \end{aligned}$$

while if \(|y|\ge M\) we have

$$\begin{aligned} |(u_{n}-u)(t,y)|\le \varepsilon w_{3}(y)\inf \limits _{x\in [-M,M]}\frac{w_{2}(x)}{w_{3}(x)}\le \varepsilon w_{2}(y). \end{aligned}$$

Therefore, \(\Vert u_{n}-u\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}};{\mathcal {C}}_{w_{2}} ({\mathbb {R}}))}<\varepsilon \). This proves that \(\lim \limits _{n\rightarrow \infty }\Vert u_{n}-u\Vert _{{\mathcal {C}}_{\mathrm {b}}({\mathcal {T}}; {\mathcal {C}}_{w_{2}}({\mathbb {R}}))}=0\), as claimed. \(\quad \square \)

We next establish a form of the Arzelà–Ascoli theorem in weighted spaces.

Proposition B.2

Suppose that \(w_{1},w_{2},w_{3}\) are weights so that \(\lim \limits _{|x|\rightarrow \infty }\frac{w_{1}(x)}{w_{2}(x)}=0\) and fix \(\alpha > 0\). Then the embedding

$$\begin{aligned} {\mathcal {C}}_{w_{1}}({\mathbb {R}})\cap {\mathcal {C}}_{w_{3}}^{\alpha } ({\mathbb {R}})\hookrightarrow {\mathcal {C}}_{w_{2}}({\mathbb {R}}) \end{aligned}$$

is compact, where \({\mathcal {C}}_{w_{1}}({\mathbb {R}})\cap {\mathcal {C}}_{w_{3}}^{\alpha }({\mathbb {R}})\) is equipped with the norm \(\Vert u\Vert _{{\mathcal {C}}_{w_{1}}({\mathbb {R}})\cap {\mathcal {C}}_{w_{3}}^{\alpha } ({\mathbb {R}})}=\Vert u\Vert _{{\mathcal {C}}_{w_{1}}({\mathbb {R}})} +\Vert u\Vert _{{\mathcal {C}}_{\mathrm {p}_{w_{3}}}^{\alpha }({\mathbb {R}})}\).

Proof

It suffices to show that the unit ball of \({\mathcal {C}}_{w_{1}}({\mathbb {R}})\cap {\mathcal {C}}_{w_{3}}^{\alpha }({\mathbb {R}})\) is precompact in \({\mathcal {C}}_{w_{2}}({\mathbb {R}})\). Fix a sequence \((v_{n})_{n}\) of elements of this unit ball. On any compact subset of \({\mathbb {R}}\), \((v_{n})\) is uniformly bounded and Hölder. Thus by Arzelà–Ascoli and diagonalization, there exists a subsequence \((v_{n_{k}})_{k}\) which converges locally uniformly to some \(v\in {\mathcal {C}}({\mathbb {R}})\). As noted in the proof of Lemma B.4, this is equivalent to convergence in some weighted space. Since \((v_{n})\) is uniformly bounded in \({\mathcal {C}}_{w_{1}}({\mathbb {R}})\), Proposition B.1 implies that \(v\in {\mathcal {C}}_{w_{2}}({\mathbb {R}})\) and \(\lim \limits _{k\rightarrow \infty }v_{n_{k}}=v\) in \({\mathcal {C}}_{w_{2}}({\mathbb {R}})\). \(\quad \square \)

Finally, we record a compactness criterion in \({\mathcal {X}}_m\).

Lemma B.3

If \(K\subset {\mathcal {X}}_{m}\) is such that K is compact in the topology of \({\mathcal {C}}_{\mathrm {p}_{\ell }}({\mathbb {R}})\) for each \(\ell >m\), then K is compact in the topology of \({\mathcal {X}}_{m}\) as well.

Proof

Let \((v_{n})_{n}\) be a sequence of elements in K. By a diagonal argument, there is a subsequence \((v_{n_k})_k\) which converges in the topology of \({\mathcal {C}}_{\mathrm {p}_{\ell }}({\mathbb {R}})\) for all \(\ell >m\), and hence in the topology of \({\mathcal {X}}_{m}\). \(\quad \square \)

Heat kernel bounds in weighted spaces

Here, we prove some weighted estimates for the heat kernel.

Lemma B.4

Fix a weight \(w\in \{(\log \langle \cdot \rangle )^{3/4}\}\cup \{\mathrm {p}_{\ell }{\ :\ }\ell \in {\mathbb {R}}\}\), \(\beta \ge \alpha \ge 0\), and \(T<\infty \). There is a constant \(C=C(w,\alpha ,\beta ,T)<\infty \) so that for all \(t\in (0,T]\) and \(f\in {\mathcal {C}}_{w}^{\alpha }({\mathbb {R}})\) we have

$$\begin{aligned} \Vert G_{t}*f\Vert _{{\mathcal {C}}_{w}^{\beta }({\mathbb {R}})}\le Ct^{-\frac{\beta -\alpha }{2}}\Vert f\Vert _{{\mathcal {C}}_{w}^{\alpha }({\mathbb {R}})} \end{aligned}$$
(B.1)

In particular,

$$\begin{aligned} \Vert \partial _{x}G_{t}*f\Vert _{{\mathcal {C}}_{w}^{\beta }({\mathbb {R}})}\le Ct^{-\frac{\beta -\alpha +1}{2}}\Vert f\Vert _{{\mathcal {C}}_{w}^{\alpha }({\mathbb {R}})}. \end{aligned}$$
(B.2)

In the case \(\alpha =0\), it is only necessary to assume that \(f\in L_{w}^{\infty }({\mathbb {R}})\) and the norm \(\Vert f\Vert _{{\mathcal {C}}_{w}^{\alpha }({\mathbb {R}})}\) can be replaced by \(\Vert f\Vert _{L_{w}^{\infty }({\mathbb {R}})}\) on the right-hand sides of (B.1) and (B.2).

The proof of this lemma is word-for-word the same as that of [36, Lemma 2.8]. There, only exponential weights (since a uniformity statement in the weight is needed) and continuous functions are considered, but there is no difference in the treatment given the Gaussian decay of the heat kernel. The essence of the argument is that only singularity in the heat kernel is at \(t=0\), \(x=0\), so the part of the heat kernel that is exposed to the growth of f at infinity is smooth, and moreover decays quickly enough not to pose any difficulty for these estimates.

Lemma B.5

Fix \(m \in {\mathbb {R}}\). If \(p \in [1, \infty )\) and \(f \in L_{\mathrm {p}_m}^p({\mathbb {R}})\), then \(G_t *f \rightarrow f\) in \(L_{\mathrm {p}_m}^p({\mathbb {R}})\) as \(t \downarrow 0\). If \(f \in L_{\mathrm {p}_m}^\infty ({\mathbb {R}})\), then \(G_t *f \overset{\mathrm {w}^*}{\longrightarrow } f\) in \(L_{\mathrm {p}_m}^\infty ({\mathbb {R}})\) as \(t \downarrow 0\). Finally, if \(f \in {\mathcal {C}}_{\mathrm {p}_{m}}({\mathbb {R}})\), then \(G_t *f \rightarrow f\) in \({\mathcal {X}}_m\) as \(t \downarrow 0\).

Proof

First fix \(p \in [1, \infty )\) and \(f \in L_{\mathrm {p}_m}^p({\mathbb {R}})\). We provide a simple variant of a standard “approximation of the identity” argument to deal with the weighted spaces. Using the scaling symmetry of G, we can write

$$\begin{aligned} G_{t}*f(x)-f(x)=\int _{{\mathbb {R}}}[f(x-\sqrt{t}y)-f(x)]G_{1}(y)\,{d}y, \end{aligned}$$

so by the triangle inequality,

$$\begin{aligned} \Vert G_{t}*f-f\Vert _{L_{\mathrm {p}_m}^{p}({\mathbb {R}})}\le \int _{{\mathbb {R}}}\Vert \tau _{\sqrt{t}y}f-f\Vert _{L_{\mathrm {p}_m}^{p}({\mathbb {R}})}G_{1}(y)\,{d}y. \end{aligned}$$
(B.3)

We will use the dominated convergence theorem, so we first establish an integrable majorant. Assume \(t\in (0,1]\). We can easily verify that there exists \(C = C(m) <\infty \) such that

$$\begin{aligned} \mathrm {p}_{-m}(a + b) \le C \mathrm {p}_{-m}(a) \mathrm {p}_{|m|}(b) \end{aligned}$$
(B.4)

for all \(a, b \in {\mathbb {R}}\). In the sequel, we permit C to change from line to line. Then

$$\begin{aligned} \Vert \tau _{\sqrt{t}y}f\Vert _{L_{\mathrm {p}_m}^{p}({\mathbb {R}})} = \left( \int _{{\mathbb {R}}} |f(x)|^p \mathrm {p}_{m}(x+\sqrt{t}y)^{-p}\,{d}x\right) ^{\frac{1}{p}} \le C\mathrm {p}_{|m|}(y)\Vert f\Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})}. \end{aligned}$$

Since \(\mathrm {p}_{|m|}\) is integrable against the Gaussian \(G_{1}\), this is a suitable majorant.

By the dominated convergence theorem, it now suffices to prove pointwise (in y) convergence to 0 as \(t\downarrow 0\) in (B.3). We can therefore fix \(y \in {\mathbb {R}}\) and consider \(t > 0\) such that \(\sqrt{t} y \le 1\). For fixed \(\varepsilon >0\), we can find a compactly-supported continuous function \(\zeta \) on \({\mathbb {R}}\) so that \(\Vert \zeta -f\Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})} < \varepsilon \). Then we have

$$\begin{aligned} \Vert \tau _{\sqrt{t}y}f-f\Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})}\le \Vert \tau _{\sqrt{t}y}f-\tau _{\sqrt{t}y}\zeta \Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})} +\Vert \tau _{\sqrt{t}y}\zeta -\zeta \Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})} +\Vert \zeta -f\Vert _{L_{\mathrm {p}_{m}}^{p}({\mathbb {R}})}. \end{aligned}$$

By (B.4) and \(\sqrt{t} y \le 1\), the first and third terms are each less than a constant times \(\varepsilon \) and the second term goes to 0 as \(t \rightarrow 0\). Therefore \(G_t *f \rightarrow f\) in \(L_{\mathrm {p}_m}^p({\mathbb {R}})\) as \(t \downarrow 0\).

Now suppose \(f \in L_{\mathrm {p}_m}^\infty ({\mathbb {R}})\), and fix \(\phi \) in the dual space \(L_{\mathrm {p}_{-m}}^{1}({\mathbb {R}})\). We must show that

$$\begin{aligned} \langle \phi ,G_{t}*f\rangle \rightarrow \langle \phi ,f\rangle \quad \text {as } t \downarrow 0. \end{aligned}$$
(B.5)

Since \(G_{t}\) is symmetric, we have \(\langle \phi ,G_{t}*f\rangle =\langle G_{t}*\phi ,f\rangle \). But we have just shown that \(G_{t}*\phi \rightarrow \phi \) in \(L_{\mathrm {p}_{-m}}^1({\mathbb {R}})\), so (B.5) follows.

Finally, suppose that \(f \in {\mathcal {C}}_{\mathrm {p}_{m}}({\mathbb {R}})\). Fix \(\varepsilon > 0\) and \(x \in {\mathbb {R}}\). We write

$$\begin{aligned} G_t *f(x) - f(x) = \int _{{\mathbb {R}}}[f(y)-f(x)]G_t(x-y)\,{d}y. \end{aligned}$$

Now \(|f(y)|\le \Vert f\Vert _{{\mathcal {C}}_{\mathrm {p}_{m}}({\mathbb {R}})}\mathrm {p}_{m}(y)\) and \(\mathrm {p}_{m} G_t(x - \cdot )\in L^{1}({\mathbb {R}})\). When \(t \in (0, 1]\), \(G_t\) is decreasing in t outside a compact set. Thus there exists a compact set \(K\subset {\mathbb {R}}\) containing x so that

$$\begin{aligned} \int _{{\mathbb {R}}\setminus K}|f(y)-f(x)|G_t(x-y)\,{d}y<\varepsilon \end{aligned}$$

for all \(n\ge 0\). On K, f is uniformly continuous and bounded. Thus there exists a \(\delta >0\) such that \(|f(y)-f(x)|<\varepsilon \) when \(|y-x|<\delta \). Since \(G_t\) has unit mass, this implies

$$\begin{aligned} \int _{B_{\delta }(x)}|f(y)-f(x)|G_t(x-y)\,{d}y<\varepsilon \end{aligned}$$

for all \(t \in (0, 1]\). Finally, \(G_t(x-y)\rightarrow 0\) uniformly on \(K\setminus B_{\delta }(x)\) as \(t \downarrow 0\), so there exists \(\delta > 0\) such that

$$\begin{aligned} \int _{K\setminus B_{\delta }(x)}|f(y)-f(x)|G_t(x-y)\,{d}y<\varepsilon \end{aligned}$$

for all \(t < \delta \). Together, these bounds show that \(|G_t *f(x) - f(x)|\rightarrow 0\) as \(n\rightarrow 0\).

In fact, the convergence is locally uniform in x. But locally uniform convergence is equivalent to the existence of a weight w such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert G_t *f - f\Vert _{{\mathcal {C}}_{w}({\mathbb {R}})}=0. \end{aligned}$$

Combining this with the uniform bound (B.1) with \(\alpha = \beta = 0\) and \(w = \mathrm {p}_m\), Proposition B.1 implies

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert G_t *f - f\Vert _{{\mathcal {C}}_{\mathrm {p}_{\ell }}({\mathbb {R}})}=0 \end{aligned}$$

for any \(\ell >m\). That is, \(G_t *f \rightarrow f\) in \({\mathcal {X}}_m\). \(\quad \square \)

Next, we show an estimate with super-exponential weights. The restriction \(\beta <2\) is needed in the following lemma simply because the heat equation is not well-posed for initial conditions growing like \(\exp (cx^{2})\) with \(c>0\).

Lemma B.6

Fix \(\beta \in [3/2,2)\) and define, for \(\lambda \ge 0\), \(\mathrm {q}_{\lambda }(x)=\mathrm {e}^{\lambda \langle x\rangle ^{\beta }}\). For any \(\Lambda >0\) and \(T>0\), there exists a \(C<\infty \) so that for all \(\lambda \in [0,\Lambda ]\), \(t\in (0,T]\), \(f\in {\mathcal {C}}_{q_{\lambda }}({\mathbb {R}})\), and \(x\in {\mathbb {R}}\), we have

$$\begin{aligned} |\partial _{x}G_{t}*f(x)|\le Ct^{-\frac{1}{2}}\mathrm {e}^{Ct\langle x\rangle ^{2(\beta -1)}}q_{\lambda }(x)\Vert f\Vert _{{\mathcal {C}}_{\mathrm {q}_{\lambda }} ({\mathbb {R}})}. \end{aligned}$$
(B.6)

Remark B.7

This also holds for \(\beta \in (1,3/2)\). The argument is similar but not identical, and is not needed in this paper, so we omit it.

Proof

Throughout the proof, C denotes a positive constant that depends only on \(\Lambda \) and T. It may change from line to line. We may assume without loss of generality that \(\Vert f\Vert _{{\mathcal {C}}_{\mathrm {q}_{\lambda }}({\mathbb {R}})}=1\). We begin by noting that \(|y|\le \exp (y^{2}/4)\) for all \(y\in {\mathbb {R}}\), so

$$\begin{aligned} |\partial _{x}G_{t}|\le \sqrt{2}t^{-\frac{1}{2}}G_{2t}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} |\partial _{x}G_{t}*f(x)|\le \sqrt{2}t^{-\frac{1}{2}}G_{2t}*\mathrm {q}_{\lambda }(x), \end{aligned}$$
(B.7)

so it remains to bound \(|G_{2t}*\mathrm {q}_{\lambda }(x)|\). The function \(w(s,\cdot ) = G_{s}*q_{\lambda }\) solves the heat equation with initial condition \(\mathrm {q}_{\lambda }\), so we can bound it from above by constructing a supersolution v with the same initial condition. Set

$$\begin{aligned} v(s,x)=\exp \left( As\langle x\rangle ^{2(\beta -1)}+Bs^{\frac{\beta }{2-\beta }}\right) q_{\lambda }(x) \end{aligned}$$
(B.8)

for constants \(A,B>0\) to be determined. Then we have

$$\begin{aligned} \partial _{s}v\ge \left( A\langle x\rangle ^{2(\beta -1)}+Bs^{\frac{2(\beta -1)}{2-\beta }}\right) v \end{aligned}$$

and

$$\begin{aligned} \partial _{xx}v\le 8\left( \lambda ^{2}+\lambda +A(\lambda +1)s\langle x \rangle ^{\beta -2}+A^{2}s^{2}\langle x\rangle ^{2(\beta -2)}\right) \langle x\rangle ^{2(\beta -1)}v. \end{aligned}$$

Comparing these, in order for v to be a supersolution for the heat equation, i.e. to satisfy

$$\begin{aligned} \partial _{s}v\ge \frac{1}{2}\partial _{xx}v, \end{aligned}$$
(B.9)

it suffices to choose A and B so that

$$\begin{aligned} 4\left( \lambda ^{2}+\lambda +A(\lambda +1)s\langle x\rangle ^{\beta -2}+A^{2}s^{2}\langle x\rangle ^{2(\beta -2)}\right) \le A+Bs^{\frac{2(\beta -1)}{2-\beta }}\langle x\rangle ^{-2(\beta -1)}.\nonumber \\ \end{aligned}$$
(B.10)

To accomplish this, let \(A=4(\lambda ^{2}+\lambda )+2\) and \(\xi =s\langle x\rangle ^{\beta -2}\). Then for (B.10) to hold, it suffices to choose B so that

$$\begin{aligned} 4(\lambda +1)A\xi +4A^{2}\xi ^{2}\le 2+B\xi ^{\frac{2(\beta -1)}{2-\beta }} \end{aligned}$$
(B.11)

for all \(\xi \ge 0\). When \(\xi \le 1/(4A(\lambda +1))\), the left side of (B.11) is bounded by 2, and the inequality holds regardless of B. Moreover, \(2(\beta -1)/(2-\beta )\ge 2\) since \(\beta \ge 3/2\), so the right side of (B.11) grows at least as fast as the left as \(\xi \rightarrow +\infty \). Thus, there exists \(B=B(\Lambda )\) sufficiently large that (B.11) holds also when \(\xi \ge 1/(4A(\lambda +1))\). With these values of A and B, v satisfies (B.9) and \(v(0,\cdot )\equiv \mathrm {q}_{\lambda }\equiv w(0,\cdot )\). Thus \(w(s,x)\le v(s,x)\) for all \(s\ge 0\) and \(x\in {\mathbb {R}}\) by the comparison principle for the heat equation. The bound (B.6) then follows from (B.7) and (B.8). \(\quad \square \)

Classical Solutions are Mild

In this appendix we prove a converse to Lemma 2.8.

Lemma C.1

(Classical solutions are mild). Suppose that \(m\in (0,1)\), \(L\in (0,\infty ]\), \(T>0\) and \(\theta ^{[L]}\in \tilde{{\mathcal {Z}}}_{m,T}\) is a classical solution to (2.14). Then \(\theta ^{[L]}\) satisfies (2.21).

Proof

Let \(f=-\partial _{x}(\theta ^{[L]}+\psi ^{[L]})^{2}\) denote the nonlinear forcing in (2.14). Fix \(t\in (0,T]\) and define, for \(s\in [0,t]\),

$$\begin{aligned} \Theta (s,x)&=G_{s}*\theta ^{[L]}(t-s,x),&F(s,x)&=G_{s}*f(t-s,x). \end{aligned}$$

It is clear that \(\Theta \) is twice-differentiable in space. We claim that it is also continuous in s, pointwise in x. Fix \(\ell >m\), \((s,x)\in [0,t]\times {\mathbb {R}}\), and a sequence \((s_{n})_{n \in {\mathbb {N}}}\subset (0,t)\) converging to s. If \(s > 0\), we use

$$\begin{aligned} \begin{aligned} |\Theta (s_{n},x)-\Theta (s,x)|&\le |(G_{s_{n}}-G_{s})*\theta ^{[L]}(t-s_{n},\cdot )(x)|\\&\quad +|G_{s}*[\theta ^{[L]}(t-s_{n},\cdot )-\theta ^{[L]}(t-s,\cdot )](x)|. \end{aligned} \end{aligned}$$
(C.1)

Then \(\Vert G_{s_{n}}-G_{s}\Vert _{L_{\mathrm {p}_{-\ell }}^{1}({\mathbb {R}})}\rightarrow 0\). Since \(\theta ^{[L]}\) is uniformly bounded in \(L_{\mathrm {p}_{\ell }}^{\infty }({\mathbb {R}})\), we have

$$\begin{aligned} |(G_{s_{n}}-G_{S})*\theta ^{[L]}(t-s_{n},\cdot )(x)|\le \Vert G_{s_{n}}-G_{s} \Vert _{L_{\mathrm {p}_{-\ell }}^{1}({\mathbb {R}})}\Vert \theta ^{[L]}(t-s_{n},\cdot ) \Vert _{L_{\mathrm {p}_{\ell }}^{\infty }({\mathbb {R}})}\rightarrow 0. \end{aligned}$$

On the other hand,

$$\begin{aligned}&G_{s}*[\theta ^{[L]}(t-s_{n},\cdot )-\theta ^{[L]}(t-s,\cdot )](x)\\&\quad =\int _{{\mathbb {R}}}G_{s}(x-y)[\theta ^{[L]}(t-s_{n},y)-\theta ^{[L]}(t-s,y)]\,{d}y\rightarrow 0 \end{aligned}$$

by the weak-\(*\) continuity of \(\theta ^{[L]}\), since \(G_{s}\in L_{\mathrm {p}_{-\ell }}^{1}({\mathbb {R}})\). By (C.1), \(|\Theta (s_{n},x)-\Theta (s,x)|\rightarrow 0\) as \(n\rightarrow \infty \).

Now suppose \(s=0\). Then \(G_{0}=\delta _{0}\) is singular, so we must argue differently. In this case, we are considering \(\theta ^{[L]}\) near a time \(t>0\), where it is continuous. Fix \(\varepsilon >0\), and consider the opposite decomposition

$$\begin{aligned} \begin{aligned} |\Theta (s_{n},x)-\Theta (s,x)|&\le |G_{s_{n}}*[\theta ^{[L]}(t-s_{n},\cdot )- \theta ^{[L]}(t,\cdot )](x)|\\&\qquad +|(G_{s_{n}}-\delta _{0})*\theta ^{[L]}(t,\cdot )(x)|. \end{aligned} \end{aligned}$$
(C.2)

Since \(\theta ^{[L]}\in {\mathcal {C}}_{\mathrm {b}}((0,S];{\mathcal {C}}_{\mathrm {p}_{\ell }} ({\mathbb {R}}))\), there exists a \(\delta >0\) such that

$$\begin{aligned} \Vert \theta ^{[L]}(t-s_{n},\cdot )-\theta ^{[L]}(t,\cdot )\Vert _{{\mathcal {C}}_{\mathrm {p}_{\ell }} ({\mathbb {R}})}<\varepsilon \end{aligned}$$

when \(s_{n}<\delta \). Now \(G_{s_{n}}*\mathrm {p}_{\ell }\le C_\ell \mathrm {p}_{\ell }\), so

$$\begin{aligned} |G_{s_{n}}*[\theta ^{[L]}(t-s_{n},\cdot )-\theta ^{[L]}(t,\cdot )](x)|\le C_\ell \varepsilon \mathrm {p}_{\ell }(x). \end{aligned}$$

For the second term of (C.2), we use the fact that \(\theta ^{[L]}(t,\cdot )\in {\mathcal {C}}_{\mathrm {p}_{\ell }}({\mathbb {R}})\), so by Lemma B.5, we have pointwise convergence and \(|(G_{s_{n}}-\delta _{0})*\theta ^{[L]}(t,\cdot )(x)|\rightarrow 0\). Thus by (C.2), \(|\Theta (s_{n},x)-\Theta (0,x)|\rightarrow 0\) as \(n\rightarrow \infty \), as desired.

Next, (2.9) implies that \(\Theta \) is differentiable in s on (0, t) and that

$$\begin{aligned} \partial _s \Theta (s, x)&= (\partial _s G_s) *\theta ^{[L]}(t - s, x) - G_s *\partial _t \theta ^{[L]}(t - s, x)\\&= \frac{1}{2} \partial _{xx} G_s *\theta ^{[L]}(t - s, x) - \frac{1}{2} G_s *\partial _{xx}\theta ^{[L]}(t - s, x) - F(s, x). \end{aligned}$$

Now \(\theta ^{[L]}(t - s, \cdot )\) is a tempered distribution, so we may exchange differentiation and convolution to find

$$\begin{aligned} \partial _s \Theta (s, x) = -F. \end{aligned}$$

The continuity of \(\Theta \) in time ensures that

$$\begin{aligned} \Theta (t,x)-\Theta (0,x)=\int _{0}^{t}\partial _{s}\Theta (s,x)\,{d}s = - \int _0^t F(s, x) \,{d}s \end{aligned}$$

for all \(x\in {\mathbb {R}}\). After a change of variables in the integral, this is simply (2.21). \(\quad \square \)

Elementary Probabilistic and Analytic Lemmas

In this appendix we prove some elementary technical lemmas that were deferred until this point to avoid disrupting the flow of the paper.

Several symmetry arguments in the paper relied on the following lemma.

Lemma D.1

Let \(X_{1}\) and \(X_{2}\) be random variables such that \(X_{1}\overset{\mathrm {law}}{=}X_{2}\) and \({\mathbb {E}}(X_{2}-X_{1})^{-}>-\infty \). Then \({\mathbb {E}}|X_{2}-X_{1}|<\infty \) and \({\mathbb {E}}(X_{2}-X_{1})=0\).

Proof

Of course the claim is obvious if \({\mathbb {E}}|X_{i}|<\infty \), but for our applications in the paper it will be convenient to not have to assume this. Define \(f_{M}(x)=\max \{\min \{x,M\},-M\}\) and note that \(f_{M}\) is bounded and 1-Lipschitz. The function \(g_{M}(x)=f_{M}(x)-x\) is also 1-Lipschitz. Fix \(R\in (0,\infty ]\) and compute

$$\begin{aligned}&{\mathbb {E}}(|g_{M}(X_{2})-g_{M}(X_{1})| \cdot {\mathbf {1}}\{X_{2}-X_{1}\le R\})\nonumber \\&\quad ={\mathbb {E}}(|g_{M}(X_{2})-g_{M}(X_{1})|{\mathbf {1}}\{\max \{|X_{1}|,|X_{2}|\}\ge M\}{\mathbf {1}}\{X_{2}-X_{1}\le R\})\nonumber \\&\quad \le {\mathbb {E}}(|X_{2}-X_{1}|{\mathbf {1}}\{\max \{|X_{1}|,|X_{2}|\}\ge M\}{\mathbf {1}}\{X_{2}-X_{1}\le R\}). \end{aligned}$$
(D.1)

For each \(0<R<\infty \) fixed, the last expression in (D.1) goes to 0 as \(M\rightarrow \infty \) by the dominated convergence theorem, since \({\mathbb {E}}(|X_{2}-X_{1}|{\mathbf {1}}\{X_{2}-X_{1}\le R\})\) is finite since \((X_{2}-X_{1})^{+}{\mathbf {1}}\{X_{2}-X_{1}\le R\}\) is bounded and \({\mathbb {E}}(X_{2}-X_{1})^{-}\) is finite by assumption. Therefore, we have

$$\begin{aligned} 0&=\lim _{M\rightarrow \infty }{\mathbb {E}}(|g_{M}(X_{2})-g_{M}(X_{1})| \cdot {\mathbf {1}}\{X_{2}-X_{1}\le R\})\nonumber \\&=\lim _{M\rightarrow \infty }{\mathbb {E}}(|f_{M}(X_{2})-f_{M}(X_{1}) -(X_{2}-X_{1})|\cdot {\mathbf {1}}\{X_{2}-X_{1}\le R\})\nonumber \\&\ge \limsup _{M\rightarrow \infty }\big |{\mathbb {E}}((f_{M}(X_{2}) -f_{M}(X_{1})){\mathbf {1}}\{X_{2}-X_{1}\le R\})\nonumber \\&\qquad -{\mathbb {E}}((X_{2}-X_{1}){\mathbf {1}}\{X_{2}-X_{1}\le R\})\big |. \end{aligned}$$
(D.2)

Since \(X_{1}\overset{\mathrm {law}}{=}X_{2}\), we have \({\mathbb {E}}[f_{M}(X_{2})-f_{M}(X_{1})]=0\). Also, \(f_{M}(x)\) is monotone in x, so

$$\begin{aligned}&{\mathbb {E}}[(f_{M}(X_{2})-f_{M}(X_{1})){\mathbf {1}}\{X_{2}-X_{1}\le R\}]\\&\quad =-{\mathbb {E}}[(f_{M}(X_{2})-f_{M}(X_{1})){\mathbf {1}}\{X_{2}-X_{1}>R\}]\le 0 \end{aligned}$$

for all \(M,R\in (0,\infty )\). Using this in (D.2) gives

$$\begin{aligned} {\mathbb {E}}((X_{2}-X_{1}){\mathbf {1}}\{X_{2}-X_{1}\le R\})\le 0 \end{aligned}$$

for all \(R\in (0,\infty )\). Taking \(R\rightarrow +\infty \) and using the monotone convergence theorem and the assumption \({\mathbb {E}}(X_{2}-X_{1})^{-}>-\infty \) yields \({\mathbb {E}}(X_{1}-X_{2})\le 0\), so in particular \({\mathbb {E}}|X_{1}-X_{2}|<\infty \). Thus we can take \(R=+\infty \) in (D.1) and again use the dominated convergence theorem to obtain (D.2) with \(R=+\infty \), namely

$$\begin{aligned} 0\ge \limsup _{M\rightarrow \infty }\left| {\mathbb {E}}[f_{M}(X_{2})-f_{M}(X_{1})] -{\mathbb {E}}(X_{2}-X_{1})\right| =|{\mathbb {E}}(X_{2}-X_{1})|, \end{aligned}$$

and so \({\mathbb {E}}(X_{2}-X_{1})=0\) as claimed. \(\quad \square \)

The following lemma will be used in the proof of Lemma D.3 below.

Lemma D.2

Let \(\eta \in {\mathcal {C}}^{3}(I)\) for some closed interval \(I \subset {\mathbb {R}}\) and define \(A=\Vert \eta \Vert _{{\mathcal {C}}^{3}(I)}\). If \(x_{1}\ne x_{2}\in I\) satisfy \(\eta (x_{1})=\eta (x_{2})\) and \(\eta '(x_{1}),\eta '(x_{2})>\varepsilon \) or \(\eta '(x_{1}),\eta '(x_{2})<-\varepsilon \), then \(|x_{1}-x_{2}|\ge \sqrt{2\varepsilon /A}.\)

Proof

Without loss of generality, we may assume that \(x_2 > x_1\) and \(\eta '(x_{1}),\eta '(x_{2})>\varepsilon \). Let \(\delta =x_{2}-x_{1}\). By Rolle’s theorem, there is a \(y\in (x_{1},x_{2})\) so that \(\eta '(y)=0\). By the mean value theorem, there exist \(z_{1}\in (x_{1},y)\) and \(z_{2}\in (y,x_{2})\) so that

$$\begin{aligned} \eta ''(z_{1})&\le -\delta ^{-1}\varepsilon ,&\eta ''(z_{2})&\ge \delta ^{-1}\varepsilon . \end{aligned}$$

By another application of the mean value theorem, there exists a \(w\in (z_{1},z_{2})\) so that

$$\begin{aligned} \eta '''(w)\ge 2\delta ^{-2}\varepsilon . \end{aligned}$$

This means that \(2\delta ^{-2}\varepsilon \le A\), so \(\delta \ge \sqrt{2\varepsilon /A}\), as claimed. \(\quad \square \)

The following lemma was used in the proof of Lemma 3.5.

Lemma D.3

Suppose that \(\zeta \ge 0\) is in the Schwartz class and \(\eta \) is a smooth function all of whose derivatives have at most polynomial growth at infinity. Define

$$\begin{aligned} g(\lambda )=\sum _{\begin{array}{c} y\in \eta ^{-1}(\lambda )\\ \eta '(y)\ne 0 \end{array} }|\eta '(y)|\zeta (y). \end{aligned}$$

Then g is a continuous function of \(\lambda \).

Proof

Let \(\zeta _{k} \ge 0\) be smooth such that \({\text {supp}}\zeta _{k}\subset [k-1,k+2]\) and \(\sum \limits _{k\in {\mathbb {Z}}}\zeta _{k}=\zeta \). Let

$$\begin{aligned} g_{k}(\lambda )=\sum _{\begin{array}{c} y\in \eta ^{-1}(\lambda )\\ \eta '(y)\ne 0 \end{array} }|\eta '(y)|\zeta _{k}(y). \end{aligned}$$
(D.3)

We first show that \(g_{k}\) is continuous. Let \(A_{k}=\Vert \eta \Vert _{{\mathcal {C}}^{3}([k-1,k+2])}+\Vert \zeta _{k}\Vert _{{\mathcal {C}}^{0}({\mathbb {R}})} +\Vert \eta '\zeta _k\Vert _{{\mathcal {C}}^1({\mathbb {R}})}\) + 1. Define

$$\begin{aligned} S_{k,\ell }(\lambda )=\{y\in (k-1,k+2) \mid \eta (y)=\lambda ,|\eta '(y)|\in [2^{-\ell },2^{-\ell +1})\}. \end{aligned}$$

Lemma D.2 implies

$$\begin{aligned} |S_{k,\ell }(\lambda )|\le 2^3 A_{k}^{1/2}2^{\ell /2}, \end{aligned}$$
(D.4)

so

$$\begin{aligned} \sum _{y\in S_{k,\ell }(\lambda )}|\eta '(y)|\zeta _{k}(y)\le 2^{-\ell + 1} \Vert \zeta _k\Vert _{{\mathcal {C}}^0({\mathbb {R}})} |S_{k, \ell }(\lambda )| \le 2^4 A_{k}^{3/2} 2^{-\ell /2} \end{aligned}$$
(D.5)

for all \(\lambda \).

Now fix \(\varepsilon >0\) and choose \(\ell \) so large that \(2^9 A_k^{3/2} 2^{-\ell /2} < \varepsilon .\) Define

$$\begin{aligned} T_{k, \ell }^{+}(\lambda )&=\{y\in (k - 1, k + 2) \mid \eta (y)=\lambda , \eta '(y)\ge 2^{-\ell }\}, \\ T_{k, \ell }^{-}(\lambda )&=\{y\in (k - 1, k + 2) \mid \eta (y)=\lambda , \eta '(y)\le -2^{-\ell }\}. \end{aligned}$$

Suppose that \(\lambda _{1}<\lambda _{2}\) satisfy \(\lambda _{2}-\lambda _{1} < 2^{-2\ell - 2}A_k^{-1}\). Take \(x\in T_{k, \ell }^{+}(\lambda _{1})\). On the interval \({[x - 2^{-\ell -1} A_k^{-1}, x + 2^{-\ell -1} A_k^{-1}]}\), we must have

$$\begin{aligned} \eta ' \ge \eta '(x) - \Vert \eta ''\Vert _{{\mathcal {C}}^0([k-1, k+2])} 2^{-\ell -1} A_k^{-1} \ge 2^{-\ell - 1}. \end{aligned}$$

Thus

$$\begin{aligned} \eta (x + 2^{-\ell -1} A_k^{-1}) \ge \lambda _{1} + 2^{-\ell - 1} \cdot 2^{-\ell -1} A_k^{-1} > \lambda _2. \end{aligned}$$

Since \(\eta \) is continuous, there exists \(y \in T_{k, \ell +1}^{+}(\lambda _{2})\cap (x,x+ 2^{-\ell -1} A_k^{-1}]\). We say that y is “paired to x.” Notice that \(y - x < 2^{-\ell -1} A_k^{-1} \ge 2^{-\ell - 1}\) while \(\eta \le \lambda _1\) on \([x - 2^{-\ell -1} A_k^{-1}, x]\), so y is not paired to any other \(x \in T_{k, \ell }^{+}(\lambda _{1})\). Thus to each \(x \in T_{k, \ell }^{+}(\lambda _{1})\) we have paired a unique \(y \in T_{k, \ell +1}^{+}(\lambda _{2})\). Also,

$$\begin{aligned} |\eta '(x)|\zeta _{k}(x)-|\eta '(y)|\zeta _{k}(y) \le \Vert \eta ' \zeta _k\Vert _{{\mathcal {C}}^1({\mathbb {R}})} 2^{-\ell -1} A_k^{-1} \le 2^{-\ell - 1}. \end{aligned}$$

Now consider the difference

$$\begin{aligned} \sum _{x\in T_{k, \ell }^{+}(\lambda _{1})}|\eta '(x)|\zeta _{k}(x)-\sum _{y\in T_{k, \ell }^{+}(\lambda _{2})}|\eta '(y)|\zeta _{k}(y). \end{aligned}$$
(D.6)

Each \(x \in T_{k, \ell - 1}^+(\lambda _1)\) is paired to a unique \(y \in T_{k, \ell }^{+}(\lambda _{2})\), and the corresponding terms’ difference is at most \(2^{-\ell - 1}\). On the other hand, if \(x \in T_{k, \ell }^+(\lambda _1) \setminus T_{k, \ell - 1}^+(\lambda _1)\), we have

$$\begin{aligned} |\eta '(x)|\zeta _{k}(x) \le 2^{-\ell + 1} A_k. \end{aligned}$$

Decomposing (D.6) into these two cases, we obtain

$$\begin{aligned} \sum _{x\in T_{k, \ell }^{+}(\lambda _{1})}|\eta '(x)|\zeta _{k}(x)-\sum _{y\in T_{k, \ell }^{+}(\lambda _{2})}|\eta '(y)|\zeta _{k}(y)&\le \sum _{x \in T_{k, \ell }^+(\lambda _1)} (2^{-\ell - 1} + 2^{\ell + 1}A_k) \\ {}&\le 2^2 A_k 2^{-\ell } |T_{k, \ell }^+(\lambda _1)|. \end{aligned}$$

Now (D.4) implies

$$\begin{aligned} |T_{k, \ell }^+(\lambda _1)| \le \sum _{\ell ' \le \ell } |S_{k, \ell '}(\lambda _1)| \le 2^3 A_k^{1/2} 2^{\ell /2} \sum _{m \ge 0} 2^{-m/2} \le 2^5 A_k^{1/2} 2^{\ell /2}. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{x\in T_{k, \ell }^{+}(\lambda _{1})}|\eta '(x)|\zeta _{k}(x)-\sum _{y\in T_{k, \ell }^{+}(\lambda _{2})}|\eta '(y)|\zeta _{k}(y)&\le 2^2 A_k 2^{-\ell } |T_{k, \ell }^+(\lambda _1)| \\&\le 2^7 A_k^{3/2} 2^{-\ell /2}. \end{aligned}$$

Symmetric arguments show that in fact

$$\begin{aligned} \Bigg |\sum _{x\in T_{k, \ell }^{\pm }(\lambda _{1})}|\eta '(x)|\zeta _{k}(x)-\sum _{y\in T_{k, \ell }^{\pm }(\lambda _{2})}|\eta '(y)|\zeta _{k}(y)\Bigg |\le 2^7 A_{k}^{3/2} 2^{-\ell /2}. \end{aligned}$$
(D.7)

Now consider \(g_k(\lambda _1) - g_k(\lambda _2)\). We divide (D.3) into terms in \(T_{k, \ell }^\pm (\lambda _i)\) or \(S_{k, \ell '}(\lambda _i)\) for \(\ell ' > \ell \) and \(i \in \{1, 2\}\). We pair terms in \(T_{k, \ell }^\pm (\lambda _1)\) with those in \(T_{k, \ell }^\pm (\lambda _2)\) to take advantage of the cancellation in (D.7). We treat the terms in \(S_{k, \ell '}(\lambda _i)\) as error and use (D.5). Then

$$\begin{aligned} |g_k(\lambda _1) - g_k(\lambda _2)| \le 2^8 A_k^{3/2} 2^{-\ell /2} + 2^5 A_k^{3/2} \sum _{\ell ' > \ell } 2^{-\ell '/2} \le 2^9 A_k^{3/2} 2^{-\ell /2} < \varepsilon . \end{aligned}$$

It follows that \(g_k\) is uniformly continuous.

We now wish to sum over k to conclude the same for g. To do so, we bound \(g_k\). Let \({\ell _k = -\log _2 A_k + 1}\). Then \(S_{k, \ell }(\lambda ) = \emptyset \) for all \(\lambda \in {\mathbb {R}}\) and \(\ell < \ell _k\). Hence (D.4) implies

$$\begin{aligned} g_k(\lambda )&= \sum _{\ell \ge \ell _k} \sum _{y \in S_{k, \ell }(\lambda )} |\eta '(y)| \zeta _k(y) \\&\le 2\Vert \zeta _k\Vert _{{\mathcal {C}}^0({\mathbb {R}})} \sum _{\ell \ge \ell _k} 2^{-\ell } |S_{k, \ell }(\lambda )| \\&\le 2^4 \Vert \zeta _k\Vert _{{\mathcal {C}}^0({\mathbb {R}})} A_k^{1/2} 2^{-\ell _k/2} \sum _{m \ge 0} 2^{-m/2}. \end{aligned}$$

Using the definition of \(\ell _k\), this yields

$$\begin{aligned} g_k(\lambda ) \le 2^5 \Vert \zeta _k\Vert _{{\mathcal {C}}^0({\mathbb {R}})} A_k. \end{aligned}$$

By hypothesis, there exists \(C \ge 1\) independent of k such that \(A_k \le C \langle k \rangle ^C\) for all \(k \in {\mathbb {Z}}\). But \(\zeta _k \le \zeta \) decays super-polynomially as \(|k| \rightarrow \infty \). It follows that \(\Vert g_k\Vert _{{\mathcal {C}}^0({\mathbb {R}})}\) itself decays super-polynomially in k. Therefore \(g=\sum \limits _{k\in {\mathbb {Z}}}g_{k}\) is an absolutely and uniformly convergent sum of uniformly continuous functions and hence is (uniformly) continuous.

\(\square \)

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Dunlap, A., Graham, C. & Ryzhik, L. Stationary Solutions to the Stochastic Burgers Equation on the Line. Commun. Math. Phys. 382, 875–949 (2021). https://doi.org/10.1007/s00220-021-04025-x

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