Bessel SPDEs and renormalised local times

Abstract

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.

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

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

(A.1)

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(y, r) 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. $\square$