1 Erratum to: Probab. Theory Relat. Fields (2013) 155:71–126 DOI 10.1007/s00440-011-0392-1

Unfortunately, the proof of Proposition 4.8 in the original article is not correct. In fact, equation (4.60), where \(R^\theta \) is rewritten as a stochastic integral, has no meaning because the integrand is not adapted. Hence in (4.61) we are not allowed to apply the Burkholder–Davies–Gundy inequality.

The argument was corrected in [1, Lemma 3.6 and Corollary 3.7] in a more complicated situation. In order to transfer these statement to the situation in the original article some small changes are needed. First of all, the regularity of the linearised process \(X\) should be measured in a Hölder norm with slightly bigger index \(\alpha _\star \) than the solution \(u\). This can be done without further problems. With this change, the definition of the stopping time \(\tau _K^X\) in (4.53) should be replaced by the following.

For \(K >0\) and for an \(\alpha _\star \in (\alpha ,1/2)\) we introduce the stopping time

$$\begin{aligned} \tau _{K}^{X} = \inf \left\{ t \in [0,T] :\sup _{\begin{array}{c} x_1 \ne x_2 \\ 0 \le s_1< s_2 \le t \end{array}} \frac{\big |X(s_1,x_1) - X(s_2,x_2) \big | }{|s_1-s_2|^{\alpha _\star /2} + |x_1-x_2|^{\alpha _\star } } > K \right\} . \end{aligned}$$
(1)

With this changed definition, Lemma 3.6 of [1] implies, in the notation of the original article, the following result.

Lemma 1

Suppose that \(0<\alpha <\alpha _\star <\frac{1}{2} \) and let \(\tau \) be a stopping time that almost surely satisfies

$$\begin{aligned} 0 \le \tau \le \tau _{K}^{X} \wedge T. \end{aligned}$$

For every \(0 \le t \le T\) we set

$$\begin{aligned} \tilde{\theta }(t)&:= \theta (t \wedge \tau ),\\ \tilde{\Psi }^\theta (t)&:= \int \limits _0^{t\wedge \tau } S(t-r) \, \tilde{\theta }(r) \, dW(r),\\ \tilde{X}(t)&:= \int \limits _{0}^{t \wedge \tau } S(t-r) \, dW(r),\\ \tilde{R}^\theta (t;x,y)&:= \delta \tilde{\Psi }^\theta (t;x,y) - \tilde{\theta }(t ,x) \, \delta \tilde{X} (t;x,y). \end{aligned}$$

Then, for any \(p\) large enough and for any \(\gamma > 0\) such that

$$\begin{aligned} \gamma < \alpha _\star +\alpha - \frac{1}{p} - \sqrt{\frac{1}{2p} (1 + \alpha - \alpha _\star ) } , \end{aligned}$$

the following bound holds true:

$$\begin{aligned} \sup _{0 < t \le T} \, {\mathbb E}\big | \tilde{R}^\theta (t) |_{\Omega \mathcal {C}^\gamma }^p \, \lesssim \, |\!|\!| \theta |\!|\!|_{p,\alpha }^{p} . \end{aligned}$$
(2)

The statement given here is actually slightly stronger than the bound stated in [1] because the norm appearing on the left hand side of (2) is bounded uniformly in \(t\) instead of allowing a blow up near 0. In [1] we had to introduce this blowup due to a slightly modified definition of the Gaussian process \(X\): the process used in [1] does not start at 0, but with stationary initial condition, which was convenient for other reasons. When going through the proof given in [1], one realises that when considering the process with zero initial condition, one can apply bound (3.74) for all times \(t\) and there is no need to use (3.75) for small times.

Based on this version of Lemma 1, it is then straightforward to use the a priori information on the time regularity of \(R^\theta \), combined with the fact that the “tilde” processes coincide with the “non-tilde” processes before time \(\tau \), to obtain the bound

$$\begin{aligned} {\mathbb E}\left[ \big \Vert R^\theta \big \Vert _{C^{\kappa } \left( [0,\tau ];\Omega C^{2\alpha } \right) }^p \right] \le C(K,T) |\!|\!| \theta |\!|\!|_{p,\alpha }^{p}, \end{aligned}$$

for sufficiently small values of \(\kappa \) and sufficiently large values of \(p\), as required.