Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}.$$
**Question.** Is there a name for such ideals? Have they been studied? When is this satisfied? Do you know examples other than those mentioned below?

**Background**. This paper provides good background *related* to this question. In particular, Brodmann proved that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n$ always exists and in fact the sequence becomes constant for $n\gg0$. Brodmann also proved that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n\leq\dim R-\ell(\mathfrak{a})$, where $\ell(\mathfrak{a})$ is the *analytic spread* of $\mathfrak{a}$, i.e., $\ell(\mathfrak{a})=\dim\mathcal{R}(\mathfrak{a})/\mathfrak{m}\mathcal{R}(\mathfrak{a})$, where $$\mathcal{R}(\mathfrak{a})=\bigoplus_n\mathfrak{a}^nt^n$$ is the Rees ring of $\mathfrak{a}$. Eisenbud and Huneke showed that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n=\dim R-\ell(\mathfrak{a})$, if the associated graded ring $\mathrm{gr}_{\mathfrak{a}}(R)$ is Cohen-Macaulay, which is the case, for instance, when $R$ and $\mathcal{R}(\mathfrak{a})$ are Cohen-Macaulay. Theorem 1.2 in *loc. cit.* summarizes and re-proves these results of Brodmann and Eisenbud-Huneke. Another related concept is the concept of *equimultiple ideal*. An ideal $\mathfrak{a}$ of $R$ is called equimultiple, if $\mathrm{height}\:\mathfrak{a}=\ell(\mathfrak{a})$.

**Examples.** 1) The maximal ideal $\mathfrak{m}$ has this property. 2) Any principal ideal has this property. 3) If $R$ and $R/\mathfrak{a}$ and the Rees ring $\mathcal{R}(\mathfrak{a})$ are Cohen-Macaulay and $\mathfrak{a}$ is an equimultiple ideal, then $\mathfrak{a}$ satisfies the property.

* Proof.* If $R$ and $\mathcal{R}(\mathfrak{a})$ (and therefore $\mathrm{gr}_{\mathfrak{a}}(R)$) are Cohen-Macaulay and $\mathfrak{a}$ is an equimultiple ideal then $$\dim R/\mathfrak{a}=\dim R-\mathrm{height}\:\mathfrak{a}=\dim R-\ell(\mathfrak{a})=\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n.$$
If $R/\mathfrak{a}$ is also Cohen-Macaulay, then we get an example of what I want.

**P.S.** The problem with this last example is that it seems too technical. It involves too many conditions and I don't know which ideals are equimultiple.