A market correction for an industry which has had long-run negative externalities Take a market where there have been long-run negative externalities. That is to say, the negative externalities have been in place long enough to have played a part in the investment decisions for all extant fixed capital.

Capital costs are a significant part of total costs.

The proposal now is to correct this market inefficiency, efficiently and quickly. Pricing in the negative externality should eventually, in the long term, correct this efficiently. However, as all the extant capital is a sunk cost, and can continue to operate at low operating cost, it can hinder and crowd out new entrants.

So the ongoing costs by the incumbents are low: their fixed capital was built and initially operated in a world of negative externalities, and those capital costs are both (a) sunk costs and (b) fully written off by now. Merely pricing in the negative externalities from their ongoing activities won't change the effective subsidy they've received that allowed them to build up their incumbency in the first place; so they'll have no incentive to leave the market.

What else can be done, in addition to simply pricing in the negative externality, to ensure prompt replacement of fixed capital which only exists because it was built in a time when it did not need to bear its full cost? Differential taxation for incumbents versus new entrants, or something else? How can this be done with economic efficiency?

The market in this case is fossil fuels for transport, heat and electricity generation; so the long-term is of the order of decades, whereas the need is for faster capital replacement. Additionally, several large suppliers exert oligopolistic influence, and there is a global cartel operating (OPEC), and has been for over 50 years.
An aspect of the matter could be described as follows:
We want


  prompt replacement of (existing) fixed capital


because, I guess, it creates currently "unacceptable" levels of negative externalities, and we know better than to think that through the pricing of the externalities we will be able to reverse the damages, and all swell.  

From this point of view, we don't care about distribution between suppliers: it should be equivalent to us whether the existing companies change their capital base, or whether there appear new entrants with new cleaner technologies which will somehow drive out of business (or marginalize) the incumbents.  

Let's describe the situation with very simple relations:

Prior to any pricing of externalities, the typical incumbent (symbolized by $h$ where needed) has a after-income-tax profit function at current time $t$

$$(1-\tau)\pi_{t} = (1-\tau)\big[p_{t}q_{t} -c(q_t)\big] \tag{1}$$

The OP writes that installed capital is "fully written off by now". This means that there is no income tax offset from accounting depreciation, Of course there is a capital base here that operates, needs maintenance, some marginal replacements etc. These costs are captured in the operational cost function (i.e. we assume 100% depreciation rate for them).

Assume now that we declare that anyone who continues to use the existing technology will pay an enviromental tax that is linear in quantity, $T_t = \xi\cdot q_t$ (naturally, this tax does not decrease taxable profits). Alternatively, he will be exempt from the tax if he completely renews his capital base, with new technology. 

The incumbent now has to compare two discounted cash-flow scenaria:

$$\text {OLD} : NPV_{Oh} = \sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}-\xi q_t\big]$$

and

$$\text {NEW} : NPV_{Nh} = -C_R-I_N+\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}+\tau\cdot\frac 1T(C_R+I_N)\big] \tag{2}$$

where $C_R$ measure the (capitalized) side-costs of renewal (disposal, disruption of operations etc), while $I_N$ is the needed new investment (including perhaps adjustment costs). $r$ is some average measure of opportunity cost (and not interest rate). We have assumed that the operational costs of the new technology are the same. $\tau\cdot\delta(C_R+I_N)$ is the tax benefit from accounting depreciation of the new investments. $\delta$ is the depreciation rate. We use the Fixed percentage method of Accounting depreciation here.

What the situation is for a prospective new entrant? It is easily seen that if he attempts to install Old Technology, he will make less profits than the incumbent (due to the necessary investments). So $NPV_{Oe} &lt; NPV_{Oh}$. If he opts for the new technology for production capacity equal to that of the incumbent described above, we have

$$\text {NEW} : NPV_{Ne} = -I_N+\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}+\tau\cdot\frac 1TI_N\big] \tag{3}$$

It is easy again to determine that, here, $NPV_{Ne} &gt; NPV_{Nh}$ i.e. the entrant is in a better position from the point of view of investment evaluation, because he does not have to bear the side-costs of renewal The tax benefits of the incumbent for depreciation of $C_R$ do not offset in NPV terms $C_R$.

Why do we care whether $NPV_{Ne} &gt; NPV_{Nh}$? Because we want, conditional on both using the new technology, the new entrant to have an advantage over the incumbent in terms of financial strength, in order to have the resources to sustain a possible war from the part of the latter.  

If we want that everybody has an incentive to use the new technology, then we must have

$$NPV_{Oe} &lt; NPV_{Oh}&lt; NPV_{Nh} &lt; NPV_{Ne}$$
The first and third inequalities already we already have. For the middle one, which will connect the chain, we need to set the tax rate accordingly:

$$NPV_{Oh}- NPV_{Nh} &lt;0 \Rightarrow -\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\xi q_t +C_R+I_N-\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\tau\cdot\frac 1T(C_R+I_N)\big]$$

Using an average period quantity produced $\bar q$, and denoting

$$R\equiv  \frac {(1+r)^{T+1}-1}{r(1+r)}$$

this becomes

$$NPV_{Oh}- NPV_{Nh} &lt;0 \Rightarrow - R \cdot\xi\cdot \bar q +C_R+I_N-R\cdot\tau\cdot\frac 1T(C_R+I_N) &lt; 0 $$

$$\Rightarrow \left(1-R\cdot\tau\cdot\frac 1T\right)(C_R+I_N) &lt; R \cdot\xi\cdot \bar q $$

$$\Rightarrow \xi &gt; \left(R^{-1}-\frac {\tau}T\right)\frac {C_R+I_N}{\bar q} \tag{4}$$

Inequality $(4)$ determines the minimum level of the environmental tax rate from the point of view of financial incentives in order for the incumbents to switch technologies.  

So, does it matter whether this minimum level is consistent with the tax rate that would reflect the proper pricing of negative externalities, since by setting it above it, we expect that the old technology will be wiped out? It doesn't seem so. This is important: it says that we can see the environmental tax rate as a threat that it won't be necessary to realize.
From a "political point of view", is this minimum tax rate feasible in view of possible reaction, lobbying etc? This is a different chapter.

Note that it does not affect our goal what will happen in an oligopolistic framework, where possibly, the market demand cannot sustain both the incumbent and the new entrant (i.e. double output): whatever happens, the one, the other, or both, the appropriately set tax rate ensures that we will see only new technology in operation.