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Bessel SPDEs and renormalised local times


In this article, we prove integration by parts formulae (IbPFs) for the laws of Bessel bridges from 0 to 0 over the interval [0, 1] of dimension smaller than 3. As an application, we construct a weak version of a stochastic PDE having the law of a one-dimensional Bessel bridge (i.e. the law of a reflected Brownian bridge) as reversible measure, the dimension 1 being particularly relevant in view of applications to scaling limits of dynamical critical pinning models. We also exploit the IbPFs to conjecture the structure of the stochastic PDEs associated with Bessel bridges of all dimensions smaller than 3.

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  1. 1.

    Ambrosio, L., Savaré, G., Zambotti, L.: Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145(3–4), 517–564 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Amdeberhan, T., Espinosa, O., Gonzalez, I., Harrison, M., Moll, V.H., Straub, A.: Ramanujan’s master theorem. Ramanujan J. 29(1–3), 103–120 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bellingeri, C.: An Itô type formula for the additive stochastic heat equation. arXiv preprint arXiv:1803.01744 (2018)

  4. 4.

    Bruned, Y., Hairer, M., Zambotti, L.: Algebraic renormalisation of regularity structures. Invent. Math. 215(3), 1039–1156 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Caputo, P., Martinelli, F., Toninelli, F.: On the approach to equilibrium for a polymer with adsorption and repulsion. Electron. J. Probab. 13, 213–258 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces, vol. 293. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  7. 7.

    Dalang, R.C., Mueller, C., Zambotti, L.: Hitting properties of parabolic S.P.D.E.’s with reflection. Ann. Probab. 34, 1423–1450 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Deuschel, J.-D., Giacomin, G., Zambotti, L.: Scaling limits of equilibrium wetting models in \((1+1)\)-dimension. Probab. Theory Relat. Fields 132(4), 471–500 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Deuschel, J.-D., Orenshtein, T.: Scaling limit of wetting models in \(1+1\) dimensions pinned to a shrinking strip. Preprint arXiv:1804.02248 (2018)

  10. 10.

    Elad Altman, H.: Bessel SPDEs with general Dirichlet boundary conditions (in preparation)

  11. 11.

    Elad Altman, H.: Bismut–Elworthy–Li Formulae for Bessel Processes. Séminaire de Probabilités XLIX, Lecture Notes in Mathematics, vol. 2215, pp. 183–220. Springer, Cham (2018)

  12. 12.

    Etheridge, A.M., Labbé, C.: Scaling limits of weakly asymmetric interfaces. Commun. Math. Phys. 336(1), 287–336 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fattler, T., Grothaus, M., Voßhall, R.: Construction and analysis of a sticky reflected distorted Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 52(2), 735–762 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, vol. 19. Walter de Gruyter, Berlin (2010)

    Book  Google Scholar 

  15. 15.

    Funaki, T.: Stochastic Interface Models. École d’été de Saint-Flour XXXIII-2003, Lecture Notes in Mathematics, vol. 1869, pp. 103–274. Springer, Berlin (2005)

  16. 16.

    Funaki, T., Ishitani, K.: Integration by parts formulae for Wiener measures on a path space between two curves. Probab. Theory Relat. Fields 137(3–4), 289–321 (2007)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Funaki, T., Olla, S.: Fluctuations for \(\nabla \phi \) interface model on a wall. Stoch. Process. Appl. 94(1), 1–27 (2001)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gelfand, I.M., Shilov, G.E.: Generalized Functions, vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977] (Properties and operations, Translated from the Russian by Eugene Saletan)

  19. 19.

    Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Order. arXiv preprint arXiv:0805.3823 (2008)

  20. 20.

    Grothaus, M., Voßhall, R.: Integration by parts on the law of the modulus of the Brownian bridge. arXiv preprint arXiv:1609.02438 (2016)

  21. 21.

    Grothaus, M., Voßhall, R.: Strong Feller property of sticky reflected distorted Brownian motion. J. Theor. Probab. 31(2), 827–852 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Gubinelli, Massimiliano: Peter Imkeller, and Nicolas Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3, e6, 75 (2015)

  23. 23.

    Hairer, M., Mattingly, J.: The strong Feller property for singular stochastic PDEs. Ann. Inst. Henri Poincaré Probab. Stat. 54(3), 1314–1340 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White Noise, Mathematics and Its Applications, vol. 253. Kluwer Academic Publishers Group, Dordrecht (1993). (An infinite-dimensional calculus)

    MATH  Google Scholar 

  26. 26.

    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113. Springer, New York (1988)

    Book  Google Scholar 

  27. 27.

    Llavona, J.G.: Approximation of Continuously Differentiable Functions, vol. 130. Elsevier, Amsterdam (1986)

    Book  Google Scholar 

  28. 28.

    Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)

    Book  Google Scholar 

  29. 29.

    Mueller, C., Mytnik, L., Perkins, E.: Nonuniqueness for a parabolic SPDE with \((\frac{3}{4}-\epsilon )\)-Hölder diffusion coefficients. Ann. Probab. 42(5), 2032–2112 (2014)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Mytnik, L., Perkins, E.: Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the White noise case. Probab. Theory Relat. Fields 149(1–2), 1–96 (2011)

    Article  Google Scholar 

  31. 31.

    Nualart, D.: Malliavin Calculus and Its Applications. American Mathematical Society (AMS), Providence, RI (2009)

    Book  Google Scholar 

  32. 32.

    Nualart, D., Pardoux, E.: White noise driven quasilinear SPDEs with reflection. Probab. Theory Relat. Fields 93(1), 77–89 (1992)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Pitman, J., Yor, M.: Sur une décomposition des ponts de Bessel. Functional Analysis in Markov Processes, pp. 276–285. Springer, Berlin (1982)

    MATH  Google Scholar 

  34. 34.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, vol. 293. Springer, Berlin (2013)

    MATH  Google Scholar 

  35. 35.

    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 2, Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000). (Itô calculus, Reprint of the second (1994) edition)

    Book  Google Scholar 

  36. 36.

    Shiga, T., Watanabe, S.: Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheorie verwandte Gebiete 27(1), 37–46 (1973)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Tsatsoulis, P., Weber, H.: Spectral gap for the stochastic quantization equation on the 2-dimensional torus. Ann. Inst. Henri Poincaré Probab. Stat. 54(3), 1204–1249 (2018)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Voßhall, Robert: Sticky reflected diffusion processes in view of stochastic interface models and on general domains. Ph.D. Thesis, Technische Universität Kaiserslautern (2016)

  39. 39.

    Zambotti, L.: A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal. 180(1), 195–209 (2001)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Zambotti, L.: Integration by parts formulae on convex sets of paths and applications to spdes with reflection. Probab. Theory Relat. Fields 123(4), 579–600 (2002)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Zambotti, L.: Integration by parts on \(\delta \)-Bessel bridges, \(\delta > 3\), and related SPDEs. Ann. Probab. 31(1), 323–348 (2003)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Zambotti, L.: Occupation densities for SPDEs with reflection. Ann. Probab. 32(1A), 191–215 (2004)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Zambotti, L.: Integration by parts on the law of the reflecting Brownian motion. J. Funct. Anal. 223(1), 147–178 (2005)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Zambotti, L.: Itô–Tanaka’s formula for stochastic partial differential equations driven by additive space-time White noise. Stoch. Partial Differ. Equ. Appl. VII 245, 337–347 (2006)

    MATH  Google Scholar 

  45. 45.

    Zambotti, L.: Random Obstacle Problems, école d’été de Probabilités de Saint-Flour XLV-2015, vol. 2181. Springer, Berlin (2017)

    Google Scholar 

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The arguments used in Proposition 5.1 below to show quasi-regularity of the form associated with the law of a reflected Brownian bridge were communicated to us by Rongchan Zhu and Xiangchan Zhu, whom we warmly thank. The first author is very grateful to Jean-Dominique Deuschel, Tal Orenshtein and Nicolas Perkowski for their kind invitation to TU Berlin, and for very interesting discussions. We also thank Giuseppe Da Prato for very useful discussion and for his kindness and patience in answering our questions. The authors would finally like to thank the Isaac Newton Institute for Mathematical Sciences for hospitality and support during the programme “Scaling limits, rough paths, quantum field theory” when work on this paper was undertaken: this work was supported by EPSRC Grant Numbers EP/K032208/1 and EP/R014604/1. The second author gratefully acknowledges support by the Institut Universitaire de France and the project of the Agence Nationale de la Recherche ANR-15-CE40-0020-01 grant LSD.

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Appendix A. Proofs of two technical results

Appendix A. Proofs of two technical results

Proof of Proposition 5.1

Since \(D(\Lambda )\) contains all globally Lipschitz functions on H, for all \(f \in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K)\) we have \(f \circ j \in D(\Lambda )\). A simple calculation shows that for any \(f\in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K)\) of the form (5.4) we have

$$\begin{aligned} \nabla (f\circ j)(z) = \nabla f (j(z)) \, \text {sgn}(z). \end{aligned}$$

Hence, for all \(f,g \in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K)\), we have

$$\begin{aligned} \begin{aligned} {\mathcal {E}}(f,g)&= \frac{1}{2} \int \langle \nabla f(x), \nabla g(x) \rangle \, {\mathrm {d}}\nu (x) = \frac{1}{2} \int \langle \nabla f(j(z)), \nabla g(j(z)) \rangle \, {\mathrm {d}}\mu (z) \\&= \frac{1}{2} \int \langle \nabla (f \circ j)(z), \nabla (g \circ j)(z) \rangle \, {\mathrm {d}}\mu (z) = \Lambda (f \circ j, g \circ j), \end{aligned} \end{aligned}$$

where the third equality follows from (A.1). This shows that the bilinear symmetric form \(({\mathcal {E}},{\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K))\) admits as an extension the image of the Dirichlet form \((\Lambda , D(\Lambda ))\) under the map j. Since \({\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K)\) is dense in \(L^{2}(\nu )\), this extension is a Dirichlet form. In particular, \(({\mathcal {E}},{\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K))\) is closable, its closure \(({\mathcal {E}},D ({\mathcal {E}}))\) is a Dirichlet form, and we have the isometry property (5.6).

There remains to prove that the Dirichlet form \(({\mathcal {E}},D ({\mathcal {E}}))\) is quasi-regular. Since it is the closure of \(({\mathcal {E}},{\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K))\), it suffices to show that the associated capacity is tight. Since K is separable, we can find a countable dense subset \(\{ y_{k}, \, k \in {\mathbb {N}} \} \subset K\) such that \(y_k \ne 0\) for all \(k \in {\mathbb {N}}\).

Let now \(\varphi \in C^{\infty }_{b}({\mathbb {R}})\) be an increasing function such that \(\varphi (t)=t\) for all \(t \in [-1,1]\) and \(\Vert \varphi '\Vert _{\infty } \le 1\). For all \(m \in {\mathbb {N}}\), we define the function \(v_{m} : K \rightarrow {\mathbb {R}}\) by

$$\begin{aligned} v_{m}(z) := \varphi (\Vert z-y_{m}\Vert ), \quad z \in K. \end{aligned}$$

Moreover, we set, for all \(n \in {\mathbb {N}}\)

$$\begin{aligned} w_{n}(z) := \underset{m \le n}{\inf } v_{m}(z), \quad z \in K. \end{aligned}$$

We claim that \(w_{n} \in D({\mathcal {E}})\), \(n \in {\mathbb {N}}\), and that \(w_{n} \underset{n \rightarrow \infty }{\longrightarrow } 0\), \({\mathcal {E}}\) quasi-uniformly in K. Assuming this claim for the moment, for all \(k \ge 1\) we can find a closed subset \(F_{k}\) of K such that \(\text {Cap} (K {\setminus } F_{k}) < 1/k\), and \(w_{n} \underset{n \rightarrow \infty }{\longrightarrow } 0\) uniformly on \(F_{k}\). Hence, for all \(\epsilon >0\), we can find \(n \in {\mathbb {N}}\) such that \(w_{n} < \epsilon \) on \(F_{k}\). Therefore

$$\begin{aligned} F_{k} \subset \underset{m \le n}{\bigcup } B(y_{m}, \epsilon ) \end{aligned}$$

where B(yr) is the open ball in K centered at \(y \in K\) with radius \(r >0\). This shows that \(F_{k}\) is totally bounded. Since it is, moreover, complete as a closed subspace of a complete metric space, it is compact, and the tightness of \(\text {Cap}\) follows.

We now justify our claim. For all \(i \in {\mathbb {N}}\), we set \(l_i := \Vert y_i\Vert ^{-1} \, y_i\). Then for all \(i \ge 1\), \(l_{i} \in K\), \(\Vert l_{i}\Vert = 1\) and, for all \(z \in K\)

$$\begin{aligned} \Vert z\Vert = \underset{i \ge 0}{\sup } \, \langle l_{i}, z \rangle . \end{aligned}$$

Let \(m \in {\mathbb {N}}\) be fixed. For all \(i \ge 0\), let \(u_{i}(z) := \underset{j \le i}{\sup } \, \, \varphi ( \, \langle l_{j}, z- y_{m} \rangle \, )\), \(z \in K\). We have \(u_{i} \in D({\mathcal {E}})\), and, for \(\nu \) - a.e. \(z \in K\)

$$\begin{aligned} \sum _{k=1}^{\infty } \frac{\partial u_{i}}{\partial e_{k}} (z) ^{2} \le \underset{j \le i}{\sup } \left( \sum _{k=1}^{\infty } \varphi '(\langle l_{j}, z - y_{m} \rangle )^{2} \, \langle l_{j}, e_{k} \rangle ^{2} \right) \le 1, \end{aligned}$$

whence \({\mathcal {E}}(u_{i}, u_{i})\le 1\). By the definition of \(v_{m}\), as \(i \rightarrow \infty \), \(u_{i} \uparrow v_{m}\) on K, hence in \(L^{2}(K, \nu )\). By [28, I.2.12], we deduce that \(v_{m} \in D({\mathcal {E}})\), and that \( {\mathcal {E}}(v_{m}, v_{m}) \le 1. \) Therefore, for all \(n \in {\mathbb {N}}\), \(w_{n} \in D({\mathcal {E}})\), and \( {\mathcal {E}}(w_{n}, w_{n}) \le 1. \) But, since \(\{ y_{k}, \, k \in {\mathbb {N}} \}\) is dense in K, as \(n \rightarrow \infty \), \(w_{n} \downarrow 0\) on K. Hence \(w_{n} \underset{n \rightarrow \infty }{\longrightarrow } 0\) in \(L^{2}(K, \nu )\). This and the previous bound imply, by [28, I.2.12], that the Cesàro means of some subsequence of \((w_{n})_{n \ge 0}\) converge to 0 in \(D({\mathcal {E}})\). By [28, III.3.5], some subsequence thereof converges \({\mathcal {E}}\) quasi-uniformly to 0. But, since \((w_{n})_{n \ge 0}\) is non-increasing, we deduce that it converges \({\mathcal {E}}\)-quasi-uniformly to 0. The claimed quasi-regularity follows. There finally remains to check that \(({\mathcal {E}}, D({\mathcal {E}}))\) is local in the sense of Definition [28, V.1.1]. Let \(u,v \in D({\mathcal {E}})\) satisfying \(\text {supp}(u) \cap \text {supp}(v) = \emptyset \). Then, \(u \circ j\) and \(v \circ j\) are two elements of \(D(\Lambda )=W^{1,2}(\mu )\) with disjoint supports, and, recalling (5.6), we have

$$\begin{aligned} {\mathcal {E}}(u,v) = \Lambda (u \circ j,v \circ j) = \frac{1}{2} \int _{H} \nabla (u \circ j) \cdot \nabla (v \circ j) \, {\mathrm {d}}\mu =0. \end{aligned}$$

The claim follows. \(\square \)

Proof of Lemma 5.3

Recall that \(D({\mathcal {E}})\) is the closure under the bilinear form \({\mathcal {E}}_{1}\) of the space \({\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(K)\) of functionals of the form \(F = \Phi \bigr |_{K}\), where \(\Phi \in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(H)\). Therefore, to prove the claim, it suffices to show that for any functional \(\Phi \in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(H)\) and all \(\epsilon >0\), there exists \(\Psi \in {\mathscr {S}}\) such that \({\mathcal {E}}_1(\Phi -\Psi ,\Phi -\Psi ) < \epsilon \).

Let \(\Phi \in {\mathcal {F}} {\mathcal {C}}^{\infty }_{b}(H)\). We set for all \(\epsilon > 0\)

$$\begin{aligned} \Phi _{\epsilon }(\zeta ) := \Phi (\sqrt{\zeta ^{2} + \epsilon }), \quad \zeta \in H. \end{aligned}$$

A simple calculation shows that \(\Phi _{\epsilon } \underset{\epsilon \rightarrow 0}{\longrightarrow } \Phi \) and \(\nabla \Phi _{\epsilon } \underset{\epsilon \rightarrow 0}{\longrightarrow } \nabla \Phi \) pointwise, with uniform bounds \(\Vert \Phi _{\epsilon }\Vert _{\infty } \le \Vert \Phi \Vert _{\infty }\) and \( \Vert \nabla \Phi _{\epsilon } \Vert _{\infty } \le \Vert \nabla \Phi \Vert _{\infty }\). Hence, by dominated convergence, \({\mathcal {E}}_1 (\Phi _{\epsilon } - \Phi , \Phi _{\epsilon } - \Phi ) \underset{\epsilon \rightarrow 0}{\longrightarrow } 0\). Then, introducing for all \(d \ge 1\)\((\zeta ^{d}_{i})_{1 \le i \le d}\) the orthonormal family in \(L^{2}(0,1)\) given by

$$\begin{aligned} \zeta ^{d}_{i} := \sqrt{d} \ {\mathbf {1}}_{[\frac{i-1}{d}, \frac{i}{d}[}, \quad i = 1, \ldots , d, \end{aligned}$$

and setting

$$\begin{aligned} \Phi ^{d}_{\epsilon }(\zeta ) := \Phi _{\epsilon } \left( \left( \sum _{i=1}^{d} \langle \zeta _{d,i}, \zeta ^{2} \rangle \right) ^{\frac{1}{2}} \right) = \Phi \left( \left( \sum _{i=1}^{d} \langle \zeta _{d,i}, \zeta ^{2} \rangle + \epsilon \right) ^{\frac{1}{2}} \right) , \quad \zeta \in H, \end{aligned}$$

again we obtain the convergence \({\mathcal {E}}_1(\Phi ^{d}_{\epsilon } - \Phi _{\epsilon }, \Phi ^{d}_{\epsilon } - \Phi _{\epsilon }) \underset{d \rightarrow \infty }{\longrightarrow } 0\).

There remains to show that any fixed functional of the form

$$\begin{aligned} \Phi (\zeta ) = f\left( \langle \zeta _{1}, \zeta ^{2} \rangle , \ldots , \langle \zeta _{d}, \zeta ^{2} \rangle \right) , \quad \zeta \in H \end{aligned}$$

with \(d \ge 1\), \(f \in C^{1}_{b}({\mathbb {R}}_{+}^{d})\), and \((\zeta _{i})_{i=1, \ldots , d}\) a family of elements of K, can be approximated by elements of \({\mathscr {S}}\). Again by dominated convergence, we can suppose that f has compact support in \({\mathbb {R}}_{+}^{d}\). We define \(g\in C^{1}_{b}([0,1]^{d})\),

$$\begin{aligned} g(y) := f(-\ln (y_{1}), \ldots , -\ln (y_{d})), \quad y \in \,]0,1]^{d}, \end{aligned}$$

and \(g(y):=0\) if \(y_i=0\) for any \(i=1,\ldots ,d\). By a differentiable version of the Weierstrass Approximation Theorem (see Theorem 1.1.2 in [27]), there exists a sequence \((p_{k})_{k \ge 1}\) of polynomial functions converging to g for the \(C^{1}\) topology on \([0,1]^{d}\). Defining for all \(k \ge 1\) the function \(f_{k}: {\mathbb {R}}_{+}^{d} \rightarrow {\mathbb {R}}\) by

$$\begin{aligned} f_{k}(x) = p_{k}(e^{-x_{1}}, \ldots , e^{-x_{d}}), \quad x \in {\mathbb {R}}_{+}^{d}, \end{aligned}$$

we define \(\Phi _{k} \in {\mathscr {S}}\) by

$$\begin{aligned} \Phi _{k} (\zeta ) = f_{k} \left( \langle \zeta _{1}, \zeta ^{2} \rangle , \ldots , \langle \zeta _{d}, \zeta ^{2} \rangle \right) , \quad \zeta \in H. \end{aligned}$$

Since \(p_{k} \underset{k \rightarrow \infty }{\longrightarrow } g\) for the \(C^{1}\) topology on \([0,1]^{d}\), \(f_{k} \underset{k \rightarrow \infty }{\longrightarrow } f\) uniformly on \({\mathbb {R}}_{+}^{d}\) together with its first order derivatives. Hence, it follows that \(\Phi _{k} \underset{k \rightarrow \infty }{\longrightarrow } \Phi \) pointwise on K together with its gradient. It also follows that there is some \(C>0\) such that for all \(k \ge 1\)

$$\begin{aligned} \forall \zeta \in K, \quad |\Phi _{k}(\zeta )|^{2} + \Vert \nabla \Phi _{k}(\zeta )\Vert ^{2} \le C(1+ \Vert \zeta \Vert ^{2}). \end{aligned}$$

Since the quantity in the right-hand side is \(\nu \) integrable in \(\zeta \), it follows by dominated convergence that \({\mathcal {E}}_1(\Phi _{k}-\Phi , \Phi _{k}-\Phi ) \underset{k \rightarrow \infty }{\longrightarrow } 0\). This yields the claim. \(\square \)

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Elad Altman, H., Zambotti, L. Bessel SPDEs and renormalised local times. Probab. Theory Relat. Fields 176, 757–807 (2020).

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Mathematics Subject Classification

  • 60H15
  • 60J55