# Hidden dissipation and convexity for Kimura equations

Jean-Baptiste Casteras, Léonard Monsaingeon

###### Abstract

In this paper we establish a rigorous gradient flow structure for one-
dimensional Kimura equations with respect to some Wasserstein-Shahshahani
optimal transport geometry. This is achieved by first conditioning the
underlying stochastic process to non-fixation in order to get rid of
singularities on the boundaries, and then studying the conditioned $Q$-process
from a more traditional and variational point of view. In doing so we complete
the work initiated in [Chalub et Al., Gradient flow formulations of discrete
and continuous evolutionary models: a unifying perspective. Acta App Math.,
171(1), 1-50], where the gradient flow was identified only formally. The
approach is based on the _Energy Dissipation Inequality_ and _Evolution
Variational Inequality_ notions of metric gradient flows. Building up on some
convexity of the driving entropy functional, we obtain new contraction
estimates and quantitative long-time convergence towards the stationary
distribution.

Keywords: Kimura equation; $Q$-process; gradient flow; optimal transport;
entropy

###### Contents

  1. 1 Introduction
  2. 2 Kimura is NOT a gradient flow
  3. 3 The Kimura process and the $Q$-process
    1. 3.1 Unconditioned dynamics
    2. 3.2 Conditionned dynamics
  4. 4 Metric setting
    1. 4.1 The Wasserstein-Shahshahani distance
    2. 4.2 Absolutely continuous curves
    3. 4.3 Displacement convexity and metric slope
  5. 5 The conditioned gradient flow

##  1 Introduction

The Kimura equation

$\partial_{t}p=\partial^{2}_{xx}(x(1-x)p)+\partial_{x}(x(1-x)p\partial_{x}U),\hskip
28.45274ptt>0,x\in[0,1]$ (1.1)

was derived by Motoo Kimura in the 1950’s [29, 27, 28] as a large population
approximation $N\to\infty$ of discrete Moran-Wright-Fisher models [20, 49,
37]. Roughly speaking, the latter discrete stochastic processes aim at
describing random fluctuations in the genetic expression within a fixed-size
but large population of $N\gg 1$ individuals. Here we consider two alleles
$\mathbb{A},\mathbb{B}$ only, and in (1.1) the variable $x\in[0,1]$ describes
the continuous fraction of the focal allele $\mathbb{A}$ in the population
(and $1-x$ that of $\mathbb{B}$). The function $p(t,x)$ thus gives the
probability of finding a relative composition $x\in[0,1]$ of gene $\mathbb{A}$
within the population at time $t$. The smooth potential $U(x)$ encodes
selective preference (fitness difference) of $\mathbb{A}$ over $\mathbb{B}$ in
the spirit of Darwin’s evolution.

We cannot stress enough the fact that _no boundary conditions are enforced_ in
(1.1). Indeed it is well-known in evolutionary genetics [45] that, once
genetic drift starts, variants tend to either undergo extinction or fixate in
finite time, respectively $x=0$ ($\mathbb{B}$ fixates) and $x=1$ ($\mathbb{A}$
fixates). As a consequence one expects that “solutions” of (1.1), if any,
should develop singular Dirac-deltas at $x=0,1$ in finite time, corresponding
to such extinction/fixation scenarios, and one therefore cannot expect that
biologically relevant solutions remain smooth. This is reminiscent from the
discrete Moran-Wright-Fisher models (with state space $i\in\\{0,\dots,N\\}$
counting the number of type $\mathbb{A}$ individuals), for which $i=0$ and
$i=N$ are _absorbing states_. Note also that the diffusion coefficient
$\theta(x)=x(1-x)$ in (1.1) vanishes on the boundary $x\in\\{0,1\\}$, and,
although formally parabolic in the interior, the problem is degenerate at the
boundary. No matter how one wishes to approach it, the mathematical analysis
(well-posedness, regularity, etc.) of this type of models is bound to fall
outside of the scope of standard parabolic theory, and delicate phenomena must
eventually be dealt with. We refer the reader to [16, 17] and references
therein for the mathematical theory of such degenerate operators.

In the thermodynamic limit $N\to\infty$, and under the so-called _weak
selection_ assumption, it was proved in [7, 9] that solutions of the discrete
Moran-Wright-Fisher process converge towards solutions of (1.1). In order both
to establish this discrete-to-continuous convergence and to account for
finite-time fixation in the limiting model, the right notion of solutions for
(1.1) is that of _measure-valued, very weak solutions_ , see Definition 3.1
later on. These solutions will be of the form

$p_{t}(\mathrm{d}x)=p^{\partial}_{t}(0)\delta_{0}(\mathrm{d}x)+\mathring{p}_{t}(x)\mathrm{d}x+p^{\partial}_{t}(1)\delta_{1}(\mathrm{d}x)$

and, as measures, are really supported on the whole state-space
$\bar{E}=[0,1]$ and not just $E=(0,1)$. The transient part
$\mathring{p}_{t}(x)$ will be a smooth interior solution of (1.1) _without
boundary conditions_ , and the boundary part $p^{\partial}_{t}(x)$ accounts
for fixation at $x=0,1$. Moreover, such solutions automatically satisfy the
two conservation laws

$\frac{d}{dt}\int_{\bar{E}}1\,p_{t}(\mathrm{d}x)=0\qquad\mbox{and}\qquad\frac{d}{dt}\int_{\bar{E}}F(x)\,p_{t}(\mathrm{d}x)=0.$
(1.2)

The first one is nothing but the conservation of total probabilities. In the
second one, $F(x)$ is the so-called _fixation probability_ and encodes the
probability that, starting from an initial composition $x$, allele
$\mathbb{A}$ eventually fixates and $\mathbb{B}$ goes extinct. Both laws are
already present in the discrete models and should therefore naturally be
inherited, in the limit $N\to\infty$, as a necessary and biologically relevant
conservation. Very roughly speaking, these two laws together with the eventual
emergence of fixated Dirac-deltas (both encoded in our choice of a very weak
formulation) act as proxies for the missing boundary conditions and render the
parabolic problem well-posed for measure-valued solutions [7, 13]. It can be
shown [7, 8] that (1.2) is equivalent to the two boundary kinematic relations

$\partial_{t}p^{\partial}_{t}(x)=\mathring{p}_{t}(x),\hskip
28.45274ptt>0,\,x\in\\{0,1\\},$

which will clearly appear in the infinitesimal generators of the underlying
stochastic process in section 3. It turns out that, although degenerate, the
diffusion is still strong enough in the interior so as to force
$\mathring{p}_{t}(x)\geq c_{t}>0$ for any $t>0$, all the way to the boundary
and in particular for $x=0,1$. Hence $p^{\partial}_{t}(x)>0$ for any $t>0$,
and singular Dirac-deltas do develop instantaneously just as suggested from
the discrete models. What is more, fixated states actually attract the whole
dynamics in the long run: the interior part $\|\mathring{p}_{t}\|_{\infty}\to
0$ exponentially fast and

$p_{t}\rightharpoonup
p_{\infty}=a_{\infty}\delta_{0}+b_{\infty}\delta_{1}\hskip 28.45274pt\mbox{as
}t\to\infty,$

where the coefficients $a_{\infty},b_{\infty}\geq 0$ can be computed
explicitly from the two conservation laws (1.2) and $p_{0}$ by solving
$a_{\infty}+b_{\infty}=1,a_{\infty}F(0)+b_{\infty}F(1)=\int_{\bar{E}}F(x)p_{0}(\mathrm{d}x)$.
Similarly, any convex combination $a\delta_{0}+(1-a)\delta_{1}$ yields a
stationary (very weak) solution of (1.1).

On a slightly different note, it has been known for more than twenty years now
[25, 39] that Fokker-Planck equations (as well as many nonlinear drift-
diffusion-aggregation equations) can be interpreted as gradient-flows of
certain natural entropy functionals with respect to Wasserstein optimal
transport geometry. (Here and throughout we use the sign convention opposite
to physics, and entropy _decreases_ in time.) We refer to [47, 42] for a
gentle and applied introduction and to [48, 2] for an extended account of the
theory. This provided deep and geometric insights, in particular convergence
in the long-time regime towards the stationary distribution of the underlying
stochastic process can be reinterpreted as a natural consequence of the
_displacement convexity_ [33] of the driving entropy. This was originally
derived in a purely Euclidean framework (the Wasserstein distance
${\mathcal{W}}$ being built upon the underlying Euclidean cost
$c(x,y)=|x-y|_{\mathbb{R}^{d}}^{2}$), but [31] later dealt with variable
coefficients by viewing $\mathbb{R}^{d}$ as an intrinsic Riemannian manifold
reflecting those coefficients, and consequently adapting the Wasserstein
distance. On the discrete side of things, [32] recently tackled the case of
finite Markov processes by constructing a suitable discrete Wasserstein-like
transportation distance $\mathbb{W}_{N}$, and again identifying the discrete
Markov dynamics as the gradient-flow of the (discrete) entropy in this finite-
dimensional geometry. N. Gigli and J. Maas [22] proved that, as the
discretization $N\to\infty$, the discrete Wasserstein space Gromov-Hausdorff
converges towards the standard continuous Wasserstein space (at least on the
torus). Thus, one may expect that gradient-flows converge accordingly.

This strongly suggests that the convergence of discrete Moran-Wright-Fisher
models towards continuous Kimura equations should admit a deeper
interpretation than just pointwise convergence of solutions, backing up even
more the hope that continuous models can serve as a good approximation of the
discrete ones. In some sense, not only “solutions converge to solutions”, but
in fact “variational gradient structures converge to variational gradient
structures”. A natural notion arising at this point is that of _evolutionary
Gamma-convergence_ of gradient flows [41], which encodes the compatibility
between the discrete and continuous gradient flow structures as $N\to\infty$
(roughly speaking, such a convergence amounts to passing to the limit in the
_Energy Dissipation Inequality_ , as implemented e.g. in [14] for the standard
Euclidean Fokker-Planck equation). This observation was precisely the starting
point of [8], where the discrete and continuous gradient structures as well as
their compatibility were investigated. In order to take into account the
degeneracy at the boundaries, the underlying state space $\bar{E}=[0,1]$ is
not viewed from the usual Euclidean standpoint, but must rather be endowed
with the so-called _Shahshahani distance_ $\mathsf{d}_{\theta}$ [44] much in
the spirit of [31]. The Shahshahani distance is well-known in evolutionary
genetics, particularly in the so-called _replicator dynamics_ [24], and it is
thus no surprise that it arises here. However, the underlying metric space
$(\bar{E},\mathsf{d}_{\theta})$ is not truly speaking a smooth Riemannian
manifold: some singularities arise at the endpoints $x=0,1$, which is of
course reminiscent from the fixation phenomenon, and this poses serious
technical issues. Moreover, and more importantly, the lack of irreducibility
in the discrete models (a crucial assumption in [32]) and the resulting
degeneracy in the continuous diffusions prevent from directly applying the
existing optimal transport theory.

In [8] it was suggested that, in order to make sense of this variational
convergence of gradient flows, one should actually condition both the discrete
and continuous underlying stochastic processes to non-fixation in order to
first obtain proper gradient-flows, and that only those conditioned gradient
flows converge one to another. In fact we shall make a case later on, at least
in the continuous realm, that the unconditioned Kimura diffusion cannot be a
gradient flow in any sense, at least not of the natural entropy functional
with respect to the biologically relevant Wasserstein-Shahshahani distance. It
turns out that this type of conditioning is known in the probabilistic
literature as a _Yaglom-limit_. The resulting _Quasi-Stationary Distribution_
(QSD) and limiting _$Q$ -process_ were studied in a very abstract framework in
[11], see also [34] for a more population-dynamics oriented perspective. We
will heavily rely here on this abstract probabilistic analysis in order to
obtain sufficient analytical information about the PDE, in particular on the
boundary behaviour of solutions. We should also point out that, although the
discrete model $N<\infty$ actually enjoys a rigorous Riemannian interpretation
[32] allowing to truly identify the Moran-Wright-Fisher model as a smooth
gradient flow, the continuous limit is more delicate and the identification of
the (conditioned) Kimura PDE as a Wasserstein-Shahshahani gradient flow was
only formal in [8].

The goal of this work is threefold:

  * •

First, we provide a detailed construction of the conditioned $Q$-process,
which in [8] was merely exploited as a purely analytical change of variables
and not really addressed from a probabilistic standpoint. This is Theorem 2.

  * •

Second, we fill a gap left in [8] by assigning a rigorous gradient flow
structure to the conditioned Kimura equation, in the sense of the Italian
school of abstract metric gradient flows [2] and as stated in Corollary 5.2
and Theorem 6

  * •

Third, we use this gradient-flow structure to derive quantitative contraction
estimates and long-time convergence, both measured in the intrinsic and
biologically relevant Wasserstein-Shahshahani transportation distance. The
corresponding statement is Theorem 6.

In our opinion the main interest of this work is to somehow recast the
problematic unconditioned Kimura equation in a more or less standard optimal
transport (although the Wasserstein-Shahshahani still encompasses the
singularities at the boundary), which then opens the way to well-established
machinery. In this optimal transport setting, one notion usually playing an
important role is that of displacement convexity of the driving entropy
functional, and we will establish this convexity as well. From abstract metric
theory [2], such convexity usually provides Logarithmic Sobolev Inequalities
(LSI in short) and in turn explicit convergence rates towards the stationary
distribution. This convexity approach to contractivity and long-time
convergence is well-known [6] and we do not pretend to be extremely original
here, although it should be pointed out that our logarithmic Sobolev
inequality will include a degenerate weight. One of the biologically
interesting features of our analysis of the conditioned dynamics is that our
construction merely relies on the lowest spectral part of the unconditioned
operator, i.e. the principal eigenvalue $\lambda_{0}>0$ and eigenfunctions. By
contrast, usual spectral theory is expected to predict (optimal) convergence
rates in the conditioned model as well, but this should rely also on the next
spectral level $\lambda_{1}>\lambda_{0}$. In the same spirit, nonlinear
logarithmic Sobolev inequalities are known to be stronger than linear Poincaré
inequalities and associated spectral gaps $\lambda_{0}>0$, but still convey
less information than the next order gap $\lambda_{1}-\lambda_{0}>0$.
Equivalently, our analysis can be considered as a finer nonlinear layer,
constructed above the zero-th order spectral information solely and not
requiring any knowledge of the next modes.

The rest of the paper is organized as follows: In section 2 we briefly explain
why the unconditioned evolution simply cannot be realized as a Wasserstein
gradient flow, thus justifying the need to rather look into the conditioned
process. The probabilistic construction of the $Q$-process is carried out in
section 3, where we also investigate basic properties of the two associated
Fokker-Planck equations (unconditioned and conditioned). In section 4 we study
some features of our Wasserstein-Shahshahani space, compute the derivative
(_metric slope_) of the entropy functional, and crucially establish its
displacement convexity. Our last section 5 finally identifies the conditioned
Kimura PDE as a rigorous metric gradient flow, and as a consequence we deduce
from the abstract theory a contraction estimates, a weighted logarithmic
Sobolev inequality, and the quantitative exponential convergence towards the
unique stationary distribution as $t\to+\infty$.

##  2 Kimura is NOT a gradient flow

Throughout the rest of the paper, and although we shall exclusively work in
dimension one $x\in\bar{E}=[0,1]$, we choose to still write
$\operatorname{div},\nabla,\Delta$ instead of simply
$\partial_{x},\partial^{2}_{xx}$ to ease the comprehension.

Before jumping into the analysis, let us briefly explain why the Kimura
equation cannot be a gradient flow with respect to the Wasserstein distance.
(We defer to Section 4 the precise definition of Wasserstein distances and
prefer to keep the discussion informal at this stage.) Following [31], Fokker-
Planck equations with coefficients

$\partial_{t}\rho=\operatorname{div}(A(x)(\nabla\rho+\rho\nabla V)),\hskip
28.45274ptt>0,\,x\in\mathbb{R}^{d}$ (2.1)

can be interpreted as suitable Wasserstein gradient flows

$``\frac{d\rho}{dt}=-\operatorname{grad}_{{\mathcal{W}}_{g}}\mathcal{S}_{V}(\rho)"$

of the entropy functionals

$\mathcal{S}_{V}(\rho)=\int\rho\log\rho+\int\rho V$

by using some adapted weighted Wasserstein distance ${\mathcal{W}}_{g}$. More
precisely, writing $g_{x}\coloneqq A^{-1}(x)$ (a symmetric, positive-definite
matrix) and viewing the Euclidean space as a Riemannian manifold
$(\mathbb{R}^{d},g)$, the ad-hoc transportation distance is built upon the
corresponding Riemannian distance $c(x,y)=\mathsf{d}_{g}^{2}(x,y)$ instead of
the usual Euclidean cost $c(x,y)=|x-y|^{2}_{\mathbb{R}^{d}}$. Note that (2.1)
admits as a unique stationary measure the usual Gibbs distribution
$\mu=\frac{1}{\mathcal{Z}}e^{-V}$, and the PDE and entropy also read

$\partial_{t}\rho=\operatorname{div}\left(A(x)\rho\nabla\log\left(\frac{\rho}{\mu}\right)\right),\hskip
28.45274pt\mathcal{S}_{V}(\rho)={\mathcal{H}}_{\mu}(\rho)=\int\frac{\rho}{\mu}\log\left(\frac{\rho}{\mu}\right)\mu,$
(2.2)

the relative entropy (Kullback-Leibler divergence) ${\mathcal{H}}_{\mu}(\rho)$
of $\rho$ relatively to the reference measure $\mu$.

Trying to fit this framework, and in view of the $A(x)=x(1-x)$ coefficient in
(1.1), one is immediately tempted to view $E=(0,1)$ as a Riemannian manifold
with tensor $g_{x}=\frac{1}{x(1-x)}$, which is known as the Shahshahani
metrics. It is equally tempting to rewrite (1.1) in a divergence form similar
to (2.2): in doing so, algebra inevitably leads to

$\partial_{t}p=\operatorname{div}\left(x(1-x)p\nabla\log\left(\frac{p}{\frac{e^{-U}}{x(1-x)}}\right)\right),\hskip
28.45274pt\mathcal{S}_{\hat{U}}(p)=\int p\log p+\int
p\underbrace{[U(x)+\log(x(1-x))]}_{\hat{U}(x)}.$

Here a first significant issue immediately arises: the potential
$\hat{U}(x)=U(x)+\log(x(1-x))$ appearing in what _should_ be the right entropy
$\mathcal{S}_{\hat{U}}$ is unbounded from below, and “the unique stationary
measure $\mu$”, if it existed, would necessarily be

$\mu(\mathrm{d}x)=e^{-\hat{U}(x)}\mathrm{d}x=\frac{e^{-U(x)}}{x(1-x)}\mathrm{d}x$

(possibly up to $\frac{1}{\mathcal{Z}}$ scaling factors). But, $U$ being
smooth, this putative probability measure $\mu$ fails to be integrable at the
endpoints $x=0,1$ of the domain. This cannot be fixed by taking principal
parts in any way.

A second issue is that there cannot exist a _unique_ stationary distribution:
as mentioned earlier any convex combination $a\delta_{0}+(1-a)\delta_{1}$
gives a stationary solution of (1.1), and this non-uniqueness stems from the
failure of irreducibility (already in the discrete models). This is in sharp
contrast with the usual picture for standard diffusions and Fokker-Planck
equations, where the stationary distribution is completely determined as the
unique minimizer of the entropy.

What is worse, and as discussed earlier, biologically relevant solutions must
crucially account for fixation and are therefore necessarily measure-valued,
$p=\mathring{p}+p^{\partial}$ (a smooth interior density plus a singular
boundary term). For such measures the very meaning of
$\mathcal{S}_{\hat{U}}(p)$ is unclear due to the presence of atoms at $x=0,1$,
where $\hat{U}=\log 0=-\infty$. A tempting way to easily assign a meaning to
this undetermined integral would be to effectively let it depend on the
interior restriction only, and consider instead

$\mathring{\mathcal{S}}_{\hat{U}}(p)\coloneqq\int\mathring{p}\log\mathring{p}+\int\mathring{p}\,\hat{U},\hskip
28.45274ptp=\mathring{p}+p^{\partial}.$

This is better behaved at least for bounded interior densities $\mathring{p}$,
since $\hat{U}(x)=U(x)+\log(x(1-x))$ is at least $L^{1}$. This is however also
a dead end: Indeed, a minimum requirement for any reasonable notion of
gradient flow is that the driving functional should be non-increasing in time.
In our case it is not difficult to cook-up sequences of atomless initial data
$p^{k}_{0}=\mathring{p}^{k}_{0}$ such that
$\mathring{\mathcal{S}}_{\hat{U}}(p^{k}_{0})\to-\infty$. For example the
restriction
$p^{k}_{0}(\mathrm{d}x)=\frac{1}{\mathcal{Z}^{k}}\chi_{E_{k}}(x)\mu(\mathrm{d}x)$
of $\mu=\frac{e^{-U(x)}}{x(1-x)}\mathrm{d}x$ to a sequence of increasing
intervals $E_{k}=(1/k,1-1/k)\subset E$, suitably renormalized, leads to
$\mathring{\mathcal{S}}_{\hat{U}}(p^{k}_{0})\approx-\log\log k$. In particular
there exists $p_{0}=\mathring{p}_{0}$ with arbitrarily negative initial
entropy $\mathring{\mathcal{S}}_{\hat{U}}(p_{0})=-C<0$. From [9, 7] it is
known that the transient mass decays exponentially fast, and more precisely
$\|\mathring{p}_{t}\|_{L^{\infty}(E)}=\mathcal{O}(e^{-\lambda_{0}t})$ for some
$\lambda_{0}>0$. Now, because $\log(x(1-x))\in L^{1}(0,1)$, it is not
difficult to check that

$\mathring{\mathcal{S}}_{\hat{V}}(p_{t})=\int\mathcal{O}(e^{-\lambda_{0}t})\log\left(\mathcal{O}(e^{-\lambda_{0}t})\right)+\mathcal{O}(e^{-\lambda_{0}t})[U+\log(x(1-x))]\xrightarrow[t\to\infty]{}0.$

When started from $\mathring{\mathcal{S}}_{\hat{U}}(p_{0})=-C<0$, “entropy”
would eventually have to increase at some point in order to reach
$\mathring{\mathcal{S}}_{\hat{U}}(p_{\infty})=0$, and this clearly rules out
the naive gradient flow structure.

Another natural attempt at compensating for singularities was to try and
modify even further the underlying geometry (rather than the entropy
functional), by taking into account the bulk/interface interactions between
$x\in E,x\in\partial E$ directly into a modified Wasserstein-like
transportation distance penalizing the mass flux from the interior to the
boundary. This was initially the motivation for [35] by the second author, but
unfortunately the approach failed to describe the basic Kimura dynamics
discussed herein.

Let us also mention at this stage the work [5], where some Wasserstein
gradient flow structure claimed for (1.1) is leveraged in order to devise
mass-variable numerical schemes accurately capturing the emergence of fixation
$p_{t}\rightharpoonup a\delta_{0}+b\delta_{1}$ as $t\to\infty$ (see also [15]
for a related Lagrangian version and gradient structure). The transportation
distance therein is our exact same Wasserstein-Shahshahani metric (to be
discussed shortly), and the entropy is exactly $\mathcal{S}_{\hat{U}}(p)=\int
p\log p+\int p[U+\log(x(1-x))]$. However, their model is drastically
different, in that the PDE is supplemented with no-flux Neumann boundary
conditions

$[\nabla(x(1-x)p)+x(1-x)p\nabla U]\cdot\nu=0,\hskip 28.45274ptx\in\\{0,1\\}.$

On the mathematical level this dramatically changes the behaviour of
solutions, and atoms do not develop in finite time anymore (although [5,
Theorem 1.1] erroneously appeals to the measure-valued well-posedness from
[7], leading to apparent inconsistencies). As a consequence “their” entropy is
well-defined, since then in our notations $p=\mathring{p}$ and therefore
$\mathcal{S}_{\hat{U}}(p)=\mathring{\mathcal{S}}_{\hat{U}}(p)$ makes sense at
least for atomless and smooth densities $p(x)\mathrm{d}x$. From the modeling
perspective, the absence of atoms makes these solutions biologically
irrelevant: fixations in finite time are prohibited, whereas they do occur for
the discrete Moran-Wright-Fisher processes that they aim at approximating.
However, the dynamics is still attracted towards the same asymptotic behaviour
$p_{t}\rightharpoonup a_{\infty}\delta_{0}+b_{\infty}\delta_{1}$ as
$t\to\infty$, and an intriguing question is to understand how well this
artificial, atomless model approximates the real biological one. Finally, let
us mention for the sake of completeness that a related Wright-Fisher model in
opinion formation is studied in [21], yet with different boundary conditions
and slightly different setup (the drift is repulsive and nonzero at the
boundaries to begin with, and this makes the degeneracy milder in some sense).
In particular long-time convergence is established by entropy methods and
weighted logarithmic Sobolev inequalities, but once again the model does not
account for fixation (either in finite or infinite time).

##  3 The Kimura process and the $Q$-process

We will actually work with the generalization

$\partial_{t}p=\Delta({\theta}(x)p)+\operatorname{div}({\theta}(x)p\nabla
U),\hskip 28.45274ptt>0,\,x\in[0,1]$ (3.1)

of the Kimura equation (1.1), for which boundary conditions again cannot be
stipulated in order to allow for fixation. We write $E=(0,1)$, the state space
will be $\bar{E}=[0,1]$, and in the rest of the paper we enforce without
further mention

######  Assumption A1.

The coefficients ${\theta},U\in C^{\infty}(\bar{E})$, with ${\theta}(x)>0$ in
the interior and

$\mbox{simple zeros}\quad{\theta}(0)={\theta}(1)=0$

######  Assumption A2.

The initial probability distribution $p_{0}\in\mathcal{P}(\bar{E})$ is not
completely fixated to start with, i.e.
$\operatorname{supp}p_{0}\not\subset\\{0,1\\}$.

The strong $C^{\infty}$ regularity is assumed here for all coefficients in
order not to obfuscate the analysis, but can certainly be relaxed. In
Assumption A1 the linear vanishing of ${\theta}$ is really critical: Different
vanishing rates ${\theta}(x)\sim\mathsf{d}_{\mathbb{R}}(x,\partial E)^{\nu}$
for $\nu\neq 1$ would lead to very different behaviours, in the sense that
either the boundaries would end-up at infinite distance of the interior (a
particle starting from $x\in(0,1)$ would not be able to reach the boundary in
finite time anymore), or Brownian particles would be allowed to jump back in
the domain after being fixated for some random but finite time [19] (whereas
linear behaviour here really guarantees eternal fixation). Assumption A2 is
enforced with the sole purpose of ruling out trivial scenarios: For, whenever
$p_{0}=p^{\partial}_{0}$ is initially supported on the absorbing states
$x=0,1$, then $p_{t}\equiv p_{0}$ for later times and no interesting dynamics
can unravel.

For convenience we record here the notion of very weak solutions from [7],
which is the right setting if one wishes to allow for fixation while
maintaining well-posedness. Note that the test-functions below are not
required to vanish on the lateral boundary.

######  Definition 3.1.

Given $p_{0}\in\mathcal{P}(\bar{E})$ we say that a narrowly continuous curve
of measures $t\mapsto p_{t}(\mathrm{d}x)$ is a _very weak solution_ of (3.1)
if

$-\int_{0}^{\infty}\int_{\bar{E}}\partial_{t}\varphi(t,x)\,p_{t}(\mathrm{d}x)\mathrm{d}t-\int_{\bar{E}}\varphi(0,x)\,p_{0}(\mathrm{d}x)\\\
=\int_{0}^{\infty}\int_{\bar{E}}{\theta}(x)[\Delta\varphi(t,x)-\nabla
U(x)\cdot\nabla\varphi(t,x)]\,p_{t}(\mathrm{d}x)\mathrm{d}t$

for every $\varphi\in C^{1,2}_{c}([0,\infty)\times\bar{E})$.

In order to help the reader let us slightly anticipate on our notations:
$(p_{t},({\mathbb{P}}_{x})_{x\in\bar{E}},P_{t},{\mathcal{L}})$ and
$(q_{t},({\mathbb{Q}}_{x})_{x\in E},\tilde{P}_{t},\tilde{\mathcal{L}})$ will
stand for the density, path-measure, semi-group, and generator of the
unconditioned Kimura stochastic process and of the subsequent conditioned
$Q$-process, respectively. The law $p_{t}\in\mathcal{P}(\bar{E})$ of the
unconditioned process will have atoms on the boundaries, and we shall always
decompose accordingly

$p_{t}=\mathring{p}_{t}+p_{t}^{\partial}\quad\in\mathcal{P}(E)+\mathcal{P}(\partial
E)\cong\mathcal{P}(\bar{E}).$

Here we mean of course
$\mathring{p}=p\,\raisebox{-0.5468pt}{\reflectbox{\rotatebox[origin={br}]{-90.0}{$\lnot$}}}\,E$
and
$p^{\partial}=p\,\raisebox{-0.5468pt}{\reflectbox{\rotatebox[origin={br}]{-90.0}{$\lnot$}}}\,\partial
E$ in the sense of restrictions of measures. Typically the interior part
$\mathring{p}_{t}(\mathrm{d}x)=\mathring{p}_{t}(x)\mathrm{d}x$ will have
smooth density w.r.t. the Lebesgue measure, while we write
$p^{\partial}_{t}=p_{t}^{\partial}(0)\delta_{0}+p_{t}^{\partial}(1)\delta_{1}$
with a slight abuse of notations.

Let us also anticipate that we shall have to consider several successive
potentials: starting from the smooth potential $U$ in (3.1), the conditioning
will first induce via some $h$-transform an additional logarithmic drift
$2\frac{\nabla\eta}{\eta}=2\nabla\log\eta$ (here $\eta(x)$ is some
eigenfunction to be defined later on). We will accordingly write
$\tilde{U}=U-2\log\eta$ for the conditioned potential. For the conditioned
process, we will show that there exists a unique stationary measure
$\pi(\mathrm{d}x)=\frac{1}{\mathcal{Z}}\frac{e^{-\tilde{U}(x)}}{{\theta}(x)}\mathrm{d}x=\frac{1}{\mathcal{Z}}e^{-(\tilde{U}+\log{\theta})(x)}\mathrm{d}x$,
and we will write accordingly $V=\tilde{U}+\log{\theta}$. Finally, our
Riemannian viewpoint on $E=(0,1)$ will give rise to the intrinsic Shahshahani
volume form
$\operatorname{Vol}(\mathrm{d}x)=\operatorname{vol}(x)\mathrm{d}x=\frac{1}{\sqrt{{\theta}(x)}}\mathrm{d}x$,
and it will be convenient to take into account this geometric effect by
considering next $W=V+\log\operatorname{vol}=V-\log\sqrt{\theta}$.
Summarizing, the reader may want to keep the following roadmap at hand:

$U\xrightarrow[\mbox{drift
}2\frac{\nabla\eta}{\eta}]{\mbox{conditioning}}\tilde{U}\xrightarrow[\pi=\frac{1}{\mathcal{Z}}e^{-V}\mathrm{d}x]{\mbox{Gibbs}}V\xrightarrow[\operatorname{Vol}(\mathrm{d}x)=\frac{1}{\sqrt{{\theta}(x)}}\mathrm{d}x]{\mbox{geometry
}}W$

A last potential implicitly appearing in section 2 was
$\hat{U}=U+\log{\theta}$, which arised when naively trying to write (3.1) in
divergence form
$\partial_{t}p=\Delta({\theta}p)+\operatorname{div}(p{\theta}\nabla
U)=\operatorname{div}({\theta}\nabla
p)+\operatorname{div}(p{\theta}\nabla\hat{U})$. A crucial feature of these
various potentials will be their boundary behaviours, summarized here for
convenience as well

$\mbox{as }x\to\partial E=\\{0,1\\}:\hskip
28.45274pt\left\\{\begin{array}[]{lcl}U&&\mbox{smooth, bounded}\\\
\hat{U}&&\to-\infty\\\ \tilde{U}&&\to+\infty\\\ V&&\to+\infty\\\
W&&\to+\infty\end{array}\right..$ (3.2)

We wish to emphasize that the probabilistic analysis in this upcoming section
is very strongly inspired from [11], to say the very least, and as such we
tried to stick to the notations thereof as best as possible.

###  3.1 Unconditioned dynamics

From now on we claim full mathematical rigor. Consider the strong
$\bar{E}$-valued Markov process defined by the Stochastic Differential
Equation

$\begin{cases}\mathrm{d}X_{t}=-{\theta}(X_{t})\nabla
U(X_{t})\mathrm{d}t+\sqrt{2{\theta}(X_{t})}\mathrm{d}B_{t}\\\ \mbox{killed on
the boundaries }x\in\partial E=\\{0,1\\}\end{cases}$ (3.3)

(The $\sqrt{2}$ factor is just here to ensure that we end up with the usual
PDE convention for the Laplacian operator $\Delta$ in the sequel, and not the
probabilistic $\frac{1}{2}\Delta$ instead.)

Standard one-dimensional methods [26, Section 5.5] give well-posedness in the
strong sense, and for any initial distribution $p_{0}\in\mathcal{P}(\bar{E})$
the SDE has a pathwise unique global solution. Notably, the linear vanishing
of ${\theta}$ guarantees that $X_{t}$ reaches the boundary in finite time,
almost surely. We also refer to Feller’s work [19] for related one-dimensional
models. By the Itô formula it is easy to check that the infinitesimal
generator ${\mathcal{L}}f(x)\coloneqq\lim\limits_{t\to
0^{+}}\frac{{\mathbb{E}}_{x}f(X_{t})-f(x)}{t}$ is

${\mathcal{L}}f(x)=\begin{cases}{\theta}(x)\Delta f(x)-{\theta}(x)\nabla
U(x)\cdot\nabla f(x)&\mbox{if }x\in E,\\\ 0&\mbox{if }x\in\partial
E,\end{cases}$ (3.4)

at least for $f\in C^{2}(\bar{E})$. In order to compute the adjoint operator
${\mathcal{L}^{*}}$ acting on measures $p\in\mathcal{P}(\bar{E})$, recall that
$p=\mathring{p}+p^{\partial}$ decomposes into an interior part and a boundary
part, and observe that

$\int_{\bar{E}}f(x)({\mathcal{L}^{*}}p)(\mathrm{d}x)=\int_{\bar{E}}({\mathcal{L}}f)(x)p(\mathrm{d}x)=\int_{E}({\mathcal{L}}f)(x)\mathring{p}(\mathrm{d}x)$

because ${\mathcal{L}}f(x)$ vanishes on the boundary. Next, if
$\mathring{p}(\mathrm{d}x)=\mathring{p}(x)\mathrm{d}x$ and $f(x)$ are smooth
enough, integration by parts readily gives

$\int_{E}({\mathcal{L}}f)p=\int_{E}({\mathcal{L}}f)\mathring{p}\,\mathrm{d}x=\int_{E}({\theta}\Delta
f-{\theta}\nabla U\cdot\nabla f)\mathring{p}\,\mathrm{d}x\\\
=\int_{E}[\Delta({\theta}\mathring{p})+\operatorname{div}({\theta}\mathring{p}\nabla
U)]f\,\mathrm{d}x+\int_{\partial E}[{\theta}\mathring{p}\nabla
f-f\nabla({\theta}\mathring{p})-{\theta}\mathring{p}f\nabla
U]\cdot\nu\,\mathrm{d}x.$

(Here and throughout $\nu=\nu_{\mathrm{ext}}$ stands for the outer unit
normal.) Because ${\theta}$ vanishes linearly on $\partial E$ all the boundary
terms where derivatives are not applied to ${\theta}$ cancel, and we are left
with

$\int_{\bar{E}}f({\mathcal{L}^{*}}p)=\int_{E}f\,[\Delta({\theta}\mathring{p})+\operatorname{div}({\theta}\mathring{p}\nabla
U)]\,\mathrm{d}x-\int_{\partial
E}f\mathring{p}\nabla{\theta}\cdot\nu\,\mathrm{d}x$

at least for smooth densities $\mathring{p}$. In more practical terms with
$\partial E=\\{0,1\\}$, this identifies the dual generator as

${\mathcal{L}^{*}}p=\underbrace{\Delta({\theta}\mathring{p})+\operatorname{div}({\theta}\mathring{p}\nabla
U)}_{\mbox{interior}}+\underbrace{\mathring{p}(0)\partial_{x}{\theta}(0)\delta_{0}-\mathring{p}(1)\partial_{x}{\theta}(1)\delta_{1}}_{\mbox{boundary}}.$

Note that

  1. (i)

This only depends on the interior part $\mathring{p}$ of
$p=\mathring{p}+p^{\partial}$

  2. (ii)

The interior and boundary terms consist of, respectively, a usual differential
operator and a singular measure

  3. (iii)

The coefficients
$\mathring{p}(0)\partial_{x}{\theta}(0),-\mathring{p}(1)\partial_{x}{\theta}(1)$
on the boundary are nonnegative (owing to Assumption A1 ${\theta}$ has
positive slope at $x=0$ and negative slope at $x=1$) and depend on the
boundary trace $\mathring{p}(0),\mathring{p}(1)$ of the interior part
$\mathring{p}$ only, but not on the singular part $p^{\partial}$ as one might
have expected.

As a consequence, and decomposing again $p=\mathring{p}+p^{\partial}$, the
abstract Fokker-Planck equation ${\partial_{t}p_{t}={\mathcal{L}^{*}}p_{t}}$
for the law $p_{t}$ of $X_{t}$ reads in PDE terms as

$\begin{cases}\partial_{t}\mathring{p}_{t}=\Delta({\theta}\mathring{p}_{t})+\operatorname{div}({\theta}\mathring{p}_{t}\nabla
U)\\\
\partial_{t}p^{\partial}_{t}(0)=+\partial_{x}{\theta}(0)\mathring{p}_{t}(0)\\\
\partial_{t}p_{t}^{\partial}(1)=-\partial_{x}{\theta}(1)\mathring{p}_{t}(1).\end{cases}$
(3.5)

This parabolic problem is nonstandard in the sense that no boundary conditions
are imposed for the interior part $\mathring{p}_{t}$. In the boundary flux
conditions, $\mathring{p}_{t}(0),\mathring{p}_{t}(1)$ should be understood as
the interior limit (boundary trace) of $\mathring{p}_{t}(x)$ as
$x\to\\{0,1\\}$. As already noticed in [8, pp. 24] and [7] this, together with
the total probability
$p_{t}(\bar{E})=\mathring{p}_{t}(E)+p^{\partial}_{t}(\partial E)=1$, is
equivalent to the two conservation laws

$\frac{d}{dt}\int_{\bar{E}}1\,p_{t}(\mathrm{d}x)=0\qquad\mbox{and}\qquad\frac{d}{dt}\int_{\bar{E}}F(x)\,p_{t}(\mathrm{d}x)=0,$
(3.6)

where $F(x)$ is the so-called _fixation probability_ defined by

$\begin{cases}{\theta}(\Delta F-\nabla U\cdot\nabla F)=0,\\\
F(0)=0,F(1)=1.\end{cases}$

Biologically speaking, $F(x)$ encodes the probability that, starting from an
initial frequency $x\in[0,1]$ of $\mathbb{A}$ genes in the population, the
process $X_{t}$ gets absorbed at $X_{\tau}=1$ (and thus the $\mathbb{B}$
allele eventually goes extinct in finite time). Note that, here in dimension
1, the pair $(1,F)$ solve the same 2nd order ODE
$G^{\prime\prime}-U^{\prime}G^{\prime}=0$ with different boundary conditions.
They are therefore linearly independent and span the kernel of
${\mathcal{L}}={\theta}[\Delta-\nabla U\cdot\nabla]$. In a more geometric
language, (3.6) thus reads as
$p_{t}\in(\operatorname{Ker}{\mathcal{L}})^{\perp}$ for all $t>0$.

We collect now some useful properties of the Fokker-Planck equation.

######  Lemma 3.2 (adapted from [7]).

Fix an arbitrary
$p_{0}=\mathring{p}_{0}+p^{\partial}_{0}\in\mathcal{P}(\bar{E})$. There exists
a unique global solution $p_{t}=\mathring{p}_{t}+p^{\partial}_{t}\geq 0$ of
(3.5) in the sense of Definition 3.1. This solution satisfies the two
conservation laws (3.6), and $\mathring{p}_{t}$ has density
$\mathring{p}_{t}(x)\in C^{\infty}_{t,x}(\mathbb{R}^{+}\times\bar{E})$. One
has the expansion

$\mathring{p}_{t}(x)=\sum\limits_{j\geq
0}e^{-\lambda_{j}t}\hat{P}_{j}\alpha_{j}(x),\hskip
28.45274pt\hat{P}_{j}=\int_{E}\alpha_{j}(x)\ \mathring{p}_{0}(\mathrm{d}x)$
(3.7)

where $0<\lambda_{0}<\lambda_{1}<\dots$ and $\alpha_{j}(x)$ are a suitable
spectral basis for ${\mathcal{L}^{*}}$. The principal eigenfunction
$\alpha_{0}(x)\geq c_{0}>0$ is bounded away from zero and smooth up to the
boundary.

This follows from a closer inspection of results from [7] (in particular Lemma
2.2, Lemma 2.3, and Theorem 4.3 therein), and essentially consists in
constructing the spectral sequence $(\lambda_{j},\alpha_{j}(x))_{j\geq 0}$ for
a certain Sturm-Liouville problem of _limit-circle-non-oscillatory type_ [50].
The spectral problem is set in a singular weighted space
$L^{2}(\frac{1}{{\theta}(x)}\mathrm{d}x)$ making the corresponding operator
positive-definite (thus resulting in $\alpha_{0}(x)\geq c_{0}>0$), but for the
sake of brevity we skip the details. As far as well-posedness is concerned one
could equally appeal to [16], but since we shall need some spectral
information anyway we opted for the constructive approach here based on the
expansion (3.7).

Note in particular that $\lambda_{1}-\lambda_{0}>0$, and that (3.7) provides
the interior part $\mathring{p}_{t}$ only. The boundary part
$p^{\partial}_{t}$ can be reconstructed a posteriori by integrating the
boundary conditions in (3.5), namely

$p^{\partial}_{t}(x)=p^{\partial}_{0}(x)-\partial_{\nu}{\theta}(x)\int_{0}^{t}\mathring{p}_{s}(x)\,\mathrm{d}s\qquad\mbox{for}\qquad
x\in\partial E=\\{0,1\\}.$ (3.8)

It is also worth stressing that, as is apparent from the explicit
representation (3.7), the evolution of the interior part $\mathring{p}_{t}$ is
completely determined by its initial datum $\mathring{p}_{0}$, and does not
depend on the initial boundary part $p^{\partial}_{0}$.

Let us now setup the probabilistic framework needed to define the $Q$-process
later on. We denote $({\bar{E}},({\mathcal{F}}_{t})_{t\geq 0},(X_{t})_{t\geq
0},(P_{t})_{t\geq 0},({\mathbb{P}}_{x})_{x\in\bar{E}})$ the Markov process
with state space $\bar{E}=E\cup\partial E$, associated to the SDE (3.3)
starting from $X_{0}=x$. This is actually a Feller process. As usual, when
starting from an $X_{0}$ initially distributed according to some
$p_{0}\in\mathcal{P}(\bar{E})$, we write

${\mathbb{P}}_{p_{0}}(\bullet)\coloneqq\int_{\bar{E}}{\mathbb{P}}_{x}(\bullet)p_{0}(\mathrm{d}x)$

and the corresponding expectations are denoted
${\mathbb{E}}_{x},{\mathbb{E}}_{p_{0}}$. Note that this diffusion process
always has continuous paths, hence the absorption time

$\tau\coloneqq\tau_{\partial E}=\inf\left\\{t\geq 0:\quad X_{t}\in\partial
E\right\\}$

is a well-defined stopping time. By definition we have $X_{t}=X_{\tau}$ for
all $t\geq\tau$. It is not difficult to check that $\tau<\infty$,
${\mathbb{P}}_{x}$-almost surely for all $x\in\bar{E}$, and that

${\mathbb{P}}_{x}(t<\tau)>0\qquad\mbox{for all }x\in E\mbox{ and }t\geq 0.$

The following exponential ergodicity of the Kimura process will be crucial for
us, by allowing direct applications of abstract results from [11]

######  Theorem 1.

Let $(\lambda_{0},\alpha_{0}(x))$ be the principal eigenpair as in Lemma 3.2.
The probability measure

$\alpha(\mathrm{d}x)\coloneqq\frac{1}{\int_{E}\alpha_{0}(y)\mathrm{d}y}\alpha_{0}(x)\mathrm{d}x$

is a _quasi-limiting distribution_ , i-e

$\lim\limits_{t\to+\infty}{\mathbb{P}}_{p_{0}}(X_{t}\in
B\,|\,\tau>t)=\alpha(B)$ (3.9)

exponentially fast in time and uniformly w.r.t. both the initial distribution
$p_{0}\in\mathcal{P}(\bar{E})$ and Borel sets $B\in\mathcal{B}(E)$. It is also
the unique _quasi-stationary distribution_ (QSD) [34] of the Markov process
(3.3),

${\mathbb{P}}_{\alpha}(\bullet\,|\,t<\tau)=\alpha(\bullet)\quad\mbox{and}\quad{\mathbb{P}}_{\alpha}(t<\tau)=e^{-\lambda_{0}t}\qquad\mbox{for
all}\qquad t\geq 0.$ (3.10)

It is worth stressing that, as one could have anticipated, the rate
$\lambda_{0}>0$ in (3.10) is exactly the principal eigenvalue. A closer
inspection of the proof below will reveal that the exponential convergence
rate in (3.9) is essentially given by the next order gap
$\lambda_{1}-\lambda_{0}>0$. For the existence and uniqueness of QSD’s for
killed diffusion processes we refer e.g. to [10, 3] and references therein,
see also [23] for more general Feller processes and hypoelliptic diffusions.

###### Proof.

Fix $t>0$ and any Borel set $B\in\mathcal{B}(E)$. Note that, because
$B\cap\partial E=\emptyset$ and by definition of the absorption time $\tau$,
$X_{t}\in B$ implies $t<\tau$ and therefore

${\mathbb{P}}_{p_{0}}(X_{t}\in
B\,|\,t<\tau)=\frac{{\mathbb{P}}_{p_{0}}(\\{X_{t}\in
B\\}\cap\\{t<\tau\\})}{{\mathbb{P}}_{p_{0}}(t<\tau)}=\frac{{\mathbb{P}}_{p_{0}}(X_{t}\in
B)}{{\mathbb{P}}_{p_{0}}(X_{t}\not\in\partial
E)}=\frac{{\mathbb{P}}_{p_{0}}(X_{t}\in B)}{{\mathbb{P}}_{p_{0}}(X_{t}\in
E)}.$

Now, since the law $p_{t}\in\mathcal{P}(\bar{E})$ of $X_{t}$ decomposes as
$p_{t}=\mathring{p}_{t}+p^{\partial}_{t}$ and $B\subset E$, we conclude that

${\mathbb{P}}_{p_{0}}(X_{t}\in
B\,|\,t<\tau)=\frac{{\mathbb{P}}_{p_{0}}(X_{t}\in
B)}{{\mathbb{P}}_{p_{0}}(X_{t}\in
E)}=\frac{\int_{B}p_{t}(\mathrm{d}x)}{\int_{E}p_{t}(\mathrm{d}x)}=\frac{\int_{B}\mathring{p}_{t}(\mathrm{d}x)}{\int_{E}\mathring{p}_{t}(\mathrm{d}x)}.$
(3.11)

In order to take the limit in this last quotient, we leverage the expansion
(3.7) of $\mathring{p}_{t}$ in order to control

$\left|\mathring{p}_{t}(x)-e^{-\lambda_{0}t}\hat{P}_{0}\alpha_{0}(x)\right|\leq\sum\limits_{j\geq
1}e^{-\lambda_{j}t}\left|\hat{P}_{j}\right|\,\|\alpha_{j}\|_{\infty}$

uniformly in $x$. Notice that
$|\hat{P}_{j}|=\left|\int_{E}\alpha_{j}(x)\mathring{p}_{0}(\mathrm{d}x)\right|\leq\|\alpha_{j}\|_{\infty}\mathring{p}_{0}(E)\leq\|\alpha_{j}\|_{\infty}$.
From [7] it is known that $\|\alpha_{j}\|_{\infty}\lesssim j^{3/4}$ and
$\lambda_{j}\gtrsim j^{2}$ (only depending on the operator
${\mathcal{L}^{*}}$). This controls

$\left|\mathring{p}_{t}(x)-e^{-\lambda_{0}t}\hat{P}_{0}\alpha_{0}(x)\right|\leq
e^{-\lambda_{1}(t-1)}\sum_{j}e^{-Cj^{2}}j^{3/2}\leq Ce^{-\lambda_{1}t}$

for all $t\geq 1$. Since all the constants involved at this stage depend only
on ${\mathcal{L}^{*}}$ through its spectral sequence
$(\lambda_{j},\alpha_{j})$, one has in (3.11)

${\mathbb{P}}_{p_{0}}(X_{t}\in
B\,|\,t<\tau)=\frac{e^{-\lambda_{0}t}\hat{P}_{0}\int_{B}\alpha_{0}(x)\mathrm{d}x+\mathcal{O}(e^{-\lambda_{1}t})}{e^{-\lambda_{0}t}\hat{P}_{0}\int_{E}\alpha_{0}(x)\mathrm{d}x+\mathcal{O}(e^{-\lambda_{1}t})}\\\
=\frac{\int_{B}\alpha_{0}(x)\mathrm{d}x}{\int_{E}\alpha_{0}(x)\mathrm{d}x}+\mathcal{O}\left(|\hat{P}_{0}|e^{-(\lambda_{1}-\lambda_{0})t}\right)=\alpha(B)+\mathcal{O}\left(e^{-(\lambda_{1}-\lambda_{0})t}\right),$

where all the $\mathcal{O}$ terms are uniform in $p_{0},B$. Here we also used
again that $|\hat{P}_{0}|\leq\|\alpha_{0}\|_{\infty}\leq C$ uniformly in
$p_{0}$ as required.

The fact that this quasi-limiting distribution $\alpha$ is indeed the unique
QSD and satisfies (3.10) readily follows from [11, Theorem 2.1] and the proof
is complete. ∎

An immediate consequence is the existence of a principal eigenpair:

######  Proposition 3.3.

The QSD $\alpha$ satisfies

${\mathcal{L}^{*}}\alpha=-\lambda_{0}\alpha.$

The limit

$\eta(x)=\lim\limits_{t\to+\infty}\frac{{\mathbb{P}}_{x}(t<\tau)}{{\mathbb{P}}_{\alpha}(t<\tau)}\geq
0$ (3.12)

is finite, converges uniformly in $x\in\bar{E}$, belongs to the domain
$\mathcal{D}({\mathcal{L}})$, and provides a simple eigenfunction

${\mathcal{L}}\eta=-\lambda_{0}\eta.$

Moreover $\eta\in C^{\infty}(\bar{E})$ is positive in $E$ and vanishes
linearly on the boundary.

###### Proof.

First of all, our Theorem 1 allows to apply any result from [11, section 3] by
validating condition (iv) in Theorem 2.1 therein. In order to get
${\mathcal{L}^{*}}\alpha=-\lambda_{0}\alpha$, recall that by definition
$\alpha$ is simply a renormalization of $\alpha_{0}$ from Lemma 3.2, and that
the latter was precisely the principal eigenfunction
${\mathcal{L}^{*}}\alpha_{0}=-\lambda_{0}\alpha_{0}$ from the Sturm-Liouville
spectral analysis in [7]. This would also have followed from the well-known
fact [34] that a QSD always satisfies
$P_{t}^{*}\alpha=e^{-\lambda_{0}t}\alpha$, which entails the claim by simple
differentiation in time at $t=0^{+}$.

That $\eta>0$ in (3.12) belongs to the domain, is an eigenfunction
${\mathcal{L}}\eta=-\lambda_{0}\eta$, and vanishes at the boundary follows
from [11, Prop. 2.3]. The multiplicity one is exactly the content of [11, Cor.
2.4].

Observe next that $\eta\in\mathcal{D}({\mathcal{L}})$ is at least continuous
up to the boundary. Usual local elliptic regularity for the elliptic equation
${\mathcal{L}^{*}}\eta=-\lambda_{0}\eta$ guarantees $C^{\infty}$ regularity in
the interior. As for the boundary regularity we only discuss here $x=0^{+}$,
the right boundary $x=1^{-}$ can be handled in the exact same way. A close
inspection of the asymptotic behaviour for the ODE
${\mathcal{L}}\eta+\lambda_{0}\eta=0$ shows that solutions all behave, for
$x\to 0^{+}$, as linear combinations $\eta(x)=C_{1}xF(x)+C_{2}G(x)$ of a
suitable Kummer confluent hypergeometric function $F$ and a Meijer
$G$-function $G$. The former $xF(x)$ is smooth up to the boundary and vanishes
linearly, while the latter has non-trivial limit $G(0)\neq 0$. But then
$\eta(0)=0$ implies $C_{2}=0$. This simultaneously entails the full boundary
regularity and the linear behaviour, and the proof is complete. ∎

###  3.2 Conditionned dynamics

Roughly speaking, the $Q$-process is obtained by conditioning the previous
Kimura process to non-absorption $\tau=+\infty$. However, since $\tau<\infty$
almost surely, this conditioning is too singular and one needs to work out a
little more.

######  Theorem 2.

Let $p_{0}\in\mathcal{P}(\bar{E})$ be any initial distribution, and define

$q_{0}(\mathrm{d}x)\coloneqq\frac{\eta(x)p_{0}(\mathrm{d}x)}{\int_{E}\eta(y)p_{0}(\mathrm{d}y)}\in\mathcal{P}(E).$
(3.13)

Then

${\mathbb{Q}}_{q_{0}}(A_{s})\coloneqq\lim\limits_{t\to\infty}{\mathbb{P}}_{p_{0}}(A_{s}|t<\tau),\hskip
28.45274pt\forall A_{s}\in{\mathcal{F}}_{s}$ (3.14)

is a well-defined probability measure, with moreover

$\left.\frac{\mathrm{d}{\mathbb{Q}}_{q_{0}}}{\mathrm{d}{\mathbb{P}}_{p_{0}}}\right|_{{\mathcal{F}}_{s}}=\frac{e^{\lambda_{0}s}\eta(X_{s})}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}\eqqcolon
M_{s}$ (3.15)

The conditioned semi-group is

$\tilde{P}_{t}f(x)=\frac{e^{\lambda_{0}t}}{\eta(x)}P_{t}(\eta f)(x).$ (3.16)

Denoting ${\mathbb{Q}}_{x}\coloneqq{\mathbb{Q}}_{\delta_{x}}$ the
corresponding measure obtained by taking $p_{0}=q_{0}=\delta_{x}$, the
$Q$-process $({\bar{E}},({\mathcal{F}}_{t})_{t\geq 0},(X_{t})_{t\geq
0},(\tilde{P}_{t})_{t\geq 0},({\mathbb{Q}}_{x})_{x\in E})$ is an $E$-valued
strong Markov process with generator

$\tilde{\mathcal{L}}f=\lambda_{0}f+\frac{{\mathcal{L}}(\eta f)}{\eta}.$ (3.17)

Finally, for any $s\geq 0$, the law $q_{s}\in\mathcal{P}(E)$ of $X_{s}$ under
${\mathbb{Q}}_{q_{0}}$ is given by

$q_{s}(\mathrm{d}x)=\frac{\eta(x)p_{s}(\mathrm{d}x)}{\int_{E}\eta(y)p_{s}(\mathrm{d}y)},$
(3.18)

where $p_{s}\in\mathcal{P}(\bar{E})$ denotes the law of $X_{s}$ under the
unconditioned measure ${\mathbb{P}}_{p_{0}}$.

Note that, since $\eta$ vanishes on the boundary, we have
$\eta(x)p_{s}(\mathrm{d}x)=\eta(x)\mathring{p}_{s}(\mathrm{d}x)$ and in
particular $q_{s}$ does not charge the boundary $\partial E$. This justifies
writing $q_{s}\in\mathcal{P}(E)$ instead of $\mathcal{P}(\bar{E})$, and is
also of course consistent with the fact that our conditioning to
“$\tau=+\infty$” precisely prevents $X_{t}$ from hitting the boundary (when
viewed from the ${\mathbb{Q}}_{q_{0}}$ perspective, not from
${\mathbb{P}}_{p_{0}}$). It is also worth pointing out that, owing to (3.18),
the initial distribution of $X_{0}$ under the conditioned measure is really
$q_{0}$ as defined in (3.13). This is of course consistent with our notation
${\mathbb{Q}}_{q_{0}}$ in (3.14) as well as the customary relation
${\mathbb{Q}}_{q_{0}}=\int{\mathbb{Q}}_{x}\,q_{0}(\mathrm{d}x)$.

###### Proof.

The proof (and the result) is a minor extension of [11, Theorem 3.1], where
the same statement was proved only for $p_{0}=\delta_{x}$. For the reader’s
convenience we only reproduce the main steps, slightly adapted in order to
cover the case of arbitrary initial distributions $p_{0}$.

Write $\Gamma_{t}\coloneqq{\boldsymbol{1}}_{t<\tau}$ and let

${\mathbb{Q}}^{t}_{q_{0}}\coloneqq\frac{\Gamma_{t}}{{\mathbb{E}}_{p_{0}}(\Gamma_{t})}{\mathbb{P}}_{p_{0}}.$

We need to show that ${\mathbb{Q}}^{t}_{p_{0}}$ has a proper limit, whose
density with respect to ${\mathbb{P}}_{p_{0}}$ restricted to
${\mathcal{F}}_{s}$ is precisely $M_{s}$. By the Markov property, and because
by definition
${\mathbb{P}}_{p_{0}}=\int{\mathbb{P}}_{y}\,{p_{0}}(\mathrm{d}y)$, we have
first for any $t\geq s$

$\frac{{\mathbb{E}}_{{p_{0}}}(\Gamma_{t}|{\mathcal{F}}_{s})}{{\mathbb{E}}_{p_{0}}(\Gamma_{t})}=\frac{{\boldsymbol{1}}_{s<\tau}{\mathbb{P}}_{X_{s}}(t-s<\tau)}{{\mathbb{P}}_{p_{0}}(t<\tau)}=\frac{{\boldsymbol{1}}_{s<\tau}{\mathbb{P}}_{X_{s}}(t-s<\tau)}{\int_{E}{\mathbb{P}}_{y}(t<\tau){p_{0}}(\mathrm{d}y)}.$

By Proposition 3.3 we have

${\mathbb{P}}_{x}(t<\tau)\underset{t\to\infty}{\sim}e^{-\lambda_{0}t}\eta(x)$

uniformly in $x\in\bar{E}$. As a consequence
${\mathbb{P}}_{X_{s}}(t-s<\tau)\sim e^{-\lambda_{0}(t-s)}\eta(X_{s})$ almost
surely in the numerator, while by dominated convergence
$\int_{E}{\mathbb{P}}_{y}(t<\tau){p_{0}}(\mathrm{d}y)\sim\int_{E}e^{-\lambda_{0}t}\eta(y){p_{0}}(\mathrm{d}y)$
in the denominator. As a consequence

$\frac{{\mathbb{E}}_{{p_{0}}}(\Gamma_{t}|{\mathcal{F}}_{s})}{{\mathbb{E}}_{p_{0}}(\Gamma_{t})}\xrightarrow[t\to\infty]{}M_{s}\eqqcolon\frac{{\boldsymbol{1}}_{s<\tau}e^{\lambda_{0}s}\eta(X_{s})}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}$

almost surely. By dominated convergence we also get that

${\mathbb{E}}_{p_{0}}(M_{s})=\lim\limits_{t\to\infty}{\mathbb{E}}_{p_{0}}\left(\frac{{\mathbb{E}}_{{p_{0}}}(\Gamma_{t}|{\mathcal{F}}_{s})}{{\mathbb{E}}_{p_{0}}(\Gamma_{t})}\right)=1.$

By [40, Theorem 2.1] this implies that $M_{s}$ is a martingale under
${\mathbb{P}}_{p_{0}}$ and that

${\mathbb{Q}}^{t}_{q_{0}}(A_{s})\xrightarrow[t\to\infty]{}{\mathbb{E}}_{p_{0}}(M_{s}{\boldsymbol{1}}_{A_{s}})$

for any $A_{s}\in{\mathcal{F}}_{s}$. This implies that
${\mathbb{Q}}_{q_{0}}\coloneqq\lim{\mathbb{Q}}^{t}_{p_{0}}$ is well-defined,
and moreover

${\mathbb{Q}}_{q_{0}}(A_{s})={\mathbb{E}}_{p_{0}}(M_{s}{\boldsymbol{1}}_{A_{s}})=\int_{A_{s}}M_{s}(\omega)\,{\mathbb{P}}_{p_{0}}(\mathrm{d}\omega)\quad\Rightarrow\quad\left.\frac{\mathrm{d}{\mathbb{Q}}_{q_{0}}}{\mathrm{d}{\mathbb{P}}_{p_{0}}}\right|_{{\mathcal{F}}_{s}}=M_{s}.$

Notice moreover that ${\boldsymbol{1}}_{s<\tau}={\boldsymbol{1}}_{X_{s}\in
E}$. Recalling that $\eta$ vanishes on $\partial E$, we thus have that
${\boldsymbol{1}}_{s<\tau}\eta(X_{s})=\eta(X_{s})$ and therefore

$M_{s}=\frac{{\boldsymbol{1}}_{s<\tau}e^{\lambda_{0}s}\eta(X_{s})}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}=\frac{e^{\lambda_{0}s}\eta(X_{s})}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}$

as claimed. The explicit characterizations (3.16) and (3.17) of
$\tilde{P}_{t}$ and $\tilde{\mathcal{L}}$ immediately follow from from this
expression for $\frac{\mathrm{d}{\mathbb{Q}}}{\mathrm{d}{\mathbb{P}}}$, see
also [11, Theorems 3.1 and 3.2]. The fact that the conditioned process is
strong Markov (in fact, Feller continuous) is proved exactly as in [11].

In order to compute now the law $q_{s}$ of $X_{s}$ under the conditioned
measure ${\mathbb{Q}}_{q_{0}}$, take $A_{s}=\\{X_{s}\in B\\}$ for any Borel
set $B\in\mathcal{B}(E)$. Because $A_{s}\in{\mathcal{F}}_{s}$ we have from the
previous computation

$q_{s}(B)={\mathbb{Q}}_{q_{0}}(X_{s}\in
B)=\int_{A_{s}}M_{s}(\omega)\,{\mathbb{P}}_{p_{0}}(\mathrm{d}\omega).$

Substituting for $M_{s}$ from the previous equality readily gives

$q_{s}(B)=\frac{e^{\lambda_{0}s}}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}\int_{X_{s}\in
B}\eta(X_{s}(\omega)){\mathbb{P}}_{p_{0}}(\mathrm{d}\omega)=\frac{e^{\lambda_{0}s}}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}\int_{B}\eta(x)p_{s}(\mathrm{d}x),$

which means exactly

$q_{s}(\mathrm{d}x)=\frac{e^{\lambda_{0}s}}{\int_{E}\eta(y){p_{0}}(\mathrm{d}y)}\eta(x)p_{s}(\mathrm{d}x).$

Our claim (3.18) finally follows from the fact that $\eta$ is an
eigenfunction, resulting in

$\frac{d}{ds}\int_{\bar{E}}\eta(x)p_{s}(\mathrm{d}x)=\int_{\bar{E}}\eta(x){\mathcal{L}^{*}}p_{s}(\mathrm{d}x)=\int_{\bar{E}}{\mathcal{L}}\eta(x)p_{s}(\mathrm{d}x)=-\lambda_{0}\int_{\bar{E}}\eta(x)p_{s}(\mathrm{d}x)$

and therefore
$\int_{E}\eta(y){p_{0}}(\mathrm{d}y)=e^{\lambda_{0}s}\int_{E}\eta(y)p_{s}(\mathrm{d}y)$
in the previous denominator. ∎

In more tractable SDE terms, the conditioned process is characterized by

######  Corollary 3.4.

The probability measure

$\pi(\mathrm{d}x)\coloneqq\frac{1}{\int_{E}\eta(y)\alpha(\mathrm{d}y)}\eta(x)\alpha(\mathrm{d}x)\qquad\in\mathcal{P}(E)$
(3.19)

is the unique stationary distribution of the conditioned $Q$-process. The
conditioned generator is given by

$\tilde{\mathcal{L}}f(x)={\theta}\left[\Delta f(x)+\left(-\nabla
U+2\frac{\nabla\eta}{\eta}\right)\cdot\nabla f(x)\right],\hskip 28.45274ptx\in
E,$ (3.20)

and the conditioned dynamics is given by

$\mathrm{d}X_{t}=\sqrt{2{\theta}(X_{t})}\mathrm{d}\tilde{B}_{t}+{\theta}(X_{t})\left(-\nabla
U(X_{t})+2\frac{\nabla\eta(X_{t})}{\eta(X_{t})}\right)\mathrm{d}t$ (3.21)

under ${\mathbb{Q}}_{q_{0}}$ for any $q_{0}\in\mathcal{P}(E)$. (Here
$\tilde{B}_{t}$ is a standard Brownian motion.)

The main point here is that, compared to (3.3), the conditioning induces an
additional logarithmic drift $2\nabla\log\eta=2\frac{\nabla\eta}{\eta}$
rendering the boundaries strongly repulsive. Indeed, since $\eta$ vanishes
linearly at both endpoints we have that $\frac{\nabla\eta}{\eta}$ blows-up and
eventually overtakes both the $-\nabla U$ drift and the Brownian fluctuation.
In particular particles never reach the boundary, as they should since the
$Q$-process was precisely conditioned to never being absorbed, and therefore
only takes values in $E$ (and not in $\bar{E}$). More rigorously: from (3.15)
it is not difficult to show that $\tau=+\infty$, ${\mathbb{Q}}$-almost surely,
and therefore no particular boundary conditions are needed in (3.21) (although
we will see later on that at the PDE level the corresponding Fokker-Planck
equation should be interpreted with Neumann boundary conditions – see Remark
3.7, so the SDE can also be written more rigorously with reflection if really
needed). This also explains why, contrarily to (3.4), the new generator
$\tilde{\mathcal{L}}$ only needs to be defined for $x\in E$ in (3.20), and not
$x\in\bar{E}$.

###### Proof.

The fact that $\pi$ is stationary follows from [11, Theorem 3.1].
Alternatively this can be checked by hand, simply choosing $p_{0}=\alpha$ in
(3.18). Indeed in this case $q_{0}=\pi$, and since $\alpha$ is an
eigenfunction we have $\mathring{p}_{t}=e^{-\lambda_{0}t}\alpha$ whence $\eta
p_{t}=\eta\mathring{p}_{t}=e^{-\lambda_{0}t}\eta\alpha$ and $q_{t}=\frac{\eta
e^{-\lambda_{0}t}\alpha}{\int_{E}\eta
e^{-\lambda_{0}t}\alpha}=\frac{1}{\int_{E}\eta\alpha}\eta\alpha=\pi$. From
(3.17) we compute

$\tilde{\mathcal{L}}f=\lambda_{0}f+\frac{1}{\eta}{\mathcal{L}}(\eta
f)=\lambda_{0}f+\frac{1}{\eta}{\theta}\left[\Delta(\eta f)-\nabla
U\cdot\nabla(\eta f)\right]\\\
=\lambda_{0}f+\frac{1}{\eta}{\theta}\left[\Big{(}\eta\Delta
f+2\nabla\eta\cdot\nabla f+f\Delta\eta\Big{)}-\nabla U\cdot\Big{(}\eta\nabla
f+f\nabla\eta\Big{)}\right]\\\
=\lambda_{0}f+\frac{f}{\eta}\underbrace{{\theta}\left[\Delta\eta-\nabla
U\cdot\nabla\eta\right]}_{{\mathcal{L}}\eta=-\lambda_{0}\eta}+{\theta}\Big{[}\Delta
f+\left(-\nabla U+2\frac{\nabla\eta}{\eta}\right)\cdot\nabla f\Big{]}.$

This entails (3.20) and the SDE (3.21) immediately follows. ∎

A more variational and PDE-oriented characterization of the conditioning is as
follows:

######  Corollary 3.5.

Let

$\tilde{U}\coloneqq U-2\log\eta\qquad\mbox{and}\qquad
V\coloneqq\tilde{U}+\log{\theta}$ (3.22)

Then $\frac{e^{-\tilde{U}}}{{\theta}}=e^{-V}\in L^{1}(\mathrm{d}x)$, the
stationary measure $\pi=\eta\alpha$ can be written as the Gibbs distribution

$\pi(\mathrm{d}x)=\frac{1}{\mathcal{Z}}\frac{e^{-\tilde{U}(x)}}{{\theta}(x)}\mathrm{d}x=\frac{1}{\mathcal{Z}}e^{V(x)}\mathrm{d}x,$
(3.23)

and the adjoint generator reads

$\widetilde{\mathcal{L}}^{*}q=\operatorname{div}({\theta}\nabla
q)+\operatorname{div}({\theta}q\nabla V).$ (3.24)

The previous repulsive effect of the conditioning should also be observed at
this energetic level: Since $\pi(\mathrm{d}x)=\eta(x)\alpha(\mathrm{d}x)$
vanishes linearly at the boundaries the new effective potential

$V=\tilde{U}+\log{\theta}=cst-\log\pi=\to+\infty$

blows-up logarithmically, while previously $\hat{U}=U+\log{\theta}\to-\infty$
was blowing _down_ as anticipated in (3.2). Before conditioning, the
boundaries were energetically attracting the dynamics towards
$\hat{U}(\partial E)=-\infty$, while after conditioning, the $-2\log\eta$ term
overtakes $\log{\theta}$ and the boundaries become repulsive
$\tilde{U}(\partial E)=+\infty$ (recall once again that both $\eta$ and
${\theta}$ vanish linearly, while $U$ is at least bounded). The law
$q_{s}(\mathrm{d}x)=Ce^{-\lambda_{0}s}\eta(x)p_{s}(\mathrm{d}x)$ accordingly
does not charge $\partial E$.

###### Proof.

Note first that $\eta,{\theta}$ are smooth up to the boundary, where both
vanish linearly. As a consequence
$\frac{e^{-\tilde{U}}}{{\theta}}=\frac{e^{-U+\log\eta^{2}}}{{\theta}}=\frac{\eta^{2}}{{\theta}}e^{-U}$
is actually an honest continuous function, and as such belongs to
$L^{1}(\mathrm{d}x)$.

From (3.20) the generator also reads

$\tilde{\mathcal{L}}f=\theta\left[\Delta f+\left(-\nabla
U+2\frac{\nabla\eta}{\eta}\right)\cdot\nabla f\right]=\theta\Big{[}\Delta
f+\nabla(-U+2\log\eta)\cdot\nabla f\Big{]},$

whence the adjoint

$\tilde{\mathcal{L}^{*}}q=\Delta({\theta}q)+\operatorname{div}\Big{(}q{\theta}\nabla(\underbrace{U-2\log\eta}_{=\tilde{U}})\Big{)}=\operatorname{div}({\theta}\nabla
q)+\operatorname{div}\Big{(}q{\theta}\nabla(\underbrace{\tilde{U}+\log{\theta}}_{=V})\Big{)}$

as claimed in (3.24). In order to derive the Gibbs relation (3.23), we use the
standard identity $\nabla q=q\nabla\log q$ to rewrite
$\tilde{\mathcal{L}^{*}}$ as

$\tilde{\mathcal{L}^{*}}q=\operatorname{div}({\theta}\nabla
q)+\operatorname{div}(q{\theta}\nabla
V)=\operatorname{div}\left({\theta}q\nabla\log\left(\frac{q}{e^{-V}}\right)\right).$

It is now immediate to check that $Q=e^{-V}$ gives a stationary solution
$\tilde{\mathcal{L}^{*}}Q=0$, and the result follows by uniqueness of the
stationary distribution $\pi$. ∎

We finish this section with a PDE characterization of solutions to the
abstract conditioned Fokker-Planck equation

$\partial_{t}q_{t}=\tilde{\mathcal{L}^{*}}q_{t}.$ (3.25)

######  Proposition 3.6.

Let $V$ be as in (3.22). The density $q_{t}(x)\in
C^{\infty}(\mathbb{R}^{+}\times\bar{E})$ of the $Q$-process in (3.18) solves

$\begin{cases}\partial_{t}q_{t}=\operatorname{div}({\theta}\nabla
q_{t})+\operatorname{div}(q_{t}{\theta}\nabla V)\hskip 28.45274pt&t>0,x\in
E\\\ q_{t}=0&x\in\partial E\end{cases}$ (3.26)

in the classical sense. Moreover $q_{t}\to q_{0}$ narrowly as $t\to 0^{+}$,
and for all $t_{0}>0$ there exist $0<c_{0}<C_{0}$ depending on $t_{0}$ such
that

$c_{0}\pi(x)\leq q_{t}(x)\leq C_{0}\pi(x),\hskip 28.45274pt\forall\,t\geq
t_{0},x\in\bar{E}.$ (3.27)

Note at this stage that we do not claim uniqueness of solutions to (3.26).
This will be proved later in section 5 in a very general sense, namely in the
sense of $\mathsf{EVI}$ metric gradient flows. The lower and upper bounds on
$q_{t}(x)$ will be technically useful later on.

######  Remark 3.7.

The Dirichlet boundary condition can (and should) actually be interpreted as a
natural Neumann boundary condition $J_{t}\cdot\nu|_{\partial_{E}}=0$, where

$J_{t}\coloneqq{\theta}\nabla q_{t}+q_{t}{\theta}\nabla V$

is the effective flux $J_{t}$ in (3.26). This is consistent with our
interpretation of the conditioned PDE as a Wasserstein gradient flow, which
usually comes with no-flux boundary conditions so as to comply with the built-
in mass conservation. To see this, let us take for granted that $q_{t}(x)$ is
smooth and vanishes linearly for $x\to\partial E$ as claimed. Whence the first
term ${\theta}\nabla q_{t}$ vanishes linearly, while ${\theta}q_{t}$ vanishes
quadratically in the second factor. Moreover, since $U$ is smooth and
$\frac{\eta^{2}}{{\theta}}$ vanishes linearly, we see that $\nabla V=\nabla
U-\nabla\log\left(\frac{\eta^{2}}{{\theta}}\right)$ behaves as
$-\frac{1}{{\theta}(x)}$ as $x\to\partial E$. Therefore we get that
$J_{t}\cdot\nu=0$ is actually equivalent to $q_{t}\theta\frac{1}{\theta}\to 0$
on the boundaries, i-e

$J_{t}\cdot\nu=0\quad\Leftrightarrow\quad q_{t}=0,\hskip
28.45274ptx\in\partial E.$

###### Proof.

We first observe that, owing to our standing assumption A2 that
$\mathring{p}_{0}\not\equiv 0$, and due to $P_{t}\eta=e^{-\lambda_{0}t}\eta$
from the previous section, there holds

$\int_{E}\eta(x)\,p_{t}(\mathrm{d}x)=\int_{\bar{E}}P_{t}\eta(x)\,p_{0}(\mathrm{d}x)=e^{-\lambda_{0}t}\int_{E}\eta(x)\,p_{0}(\mathrm{d}x)>0$

in the denominator of (3.18), with continuity as $t\to 0$. The
$C^{\infty}(\mathbb{R}^{+}\times\bar{E})$ smoothness and narrow continuity at
$t=0^{+}$ of $q_{t}$ thus follow from those of $\eta(x)$ and $p_{t}$ (note
that narrow continuity of $t\mapsto p_{t}$ is part of the requirement for very
weak solutions in Definition 3.1, hence
$\eta(x)p_{t}(\mathrm{d}x)\rightharpoonup\eta(x)p_{0}(\mathrm{d}x)$ in the
numerator of (3.18)). The PDE in (3.26) is given by our explicit computation
(3.24) for $\tilde{\mathcal{L}^{*}}$. As for the Dirichlet condition, the
continuity of $x\mapsto\mathring{p}_{t}(x)$ for fixed $t>0$ and the fact that
$\eta(x)$ vanishes continuously on the boundary guarantees that $q_{t}(x)$
vanishes as well.

Let us now turn to the bounds (3.27). A naive argument would consist in trying
to apply a comparison principle for (3.26), using as sub and supersolutions
$c_{0}\pi(x)$ and $C_{0}\pi(x)$ for small and large constants $c_{0},C_{0}>0$,
respectively (observe of course that
$(\partial_{t}-\tilde{\mathcal{L}^{*}})\pi=0$). However this requires

  1. (i)

that $\partial_{t}-\tilde{\mathcal{L}^{*}}$ satisfy the weak maximum principle

  2. (ii)

initially ordering $c_{0}\pi(x)\leq q_{t_{0}}(x)\leq C_{0}\pi(x)$ at time
$t_{0}>0$ for some well-chosen $c_{0},C_{0}>0$

Unfortunately, due to the new logarithmic drift $2\frac{\nabla\eta}{\eta}$ the
operator $\tilde{\mathcal{L}^{*}}$ has unbounded coefficients (in particular
the zero-th order term), and is not even uniformly elliptic due to the lateral
vanishing of ${\theta}$. Hence (i) is questionable to start with. As for (ii),
note that we actually allow $p_{0}\in\mathcal{P}(\bar{E})$ and
$q_{0}\in\mathcal{P}(E)$ to be singular in the interior: one expects that
usual smoothing properties and the strong maximum principle would rather lead
to $q_{t}(x)\geq c_{0}>0$ isntantaneously in time. However this is
incompatible with our statement, and clearly there is also more than meets the
eye in (ii) due to the ${\theta}$ degeneracy. Instead we go back to the
unconditioned level in order to retrieve information on $p$, which will be
then translated to $q$.

More precisely, recall from (3.5) that the evolution for $\mathring{p}$ can be
considered as a nonstandard, degenerate parabolic problem without boundary
conditions

$\partial_{t}\mathring{p}={\mathcal{L}^{*}}\mathring{p}=\Delta({\theta}\mathring{p})+\operatorname{div}({\theta}\mathring{p}\nabla
U),$

but now with smooth coefficients. Results from [18] guarantee that, for such
degenerate operators, boundary conditions are only needed on the possibly
nondegenerate portion of the boundary (where the diffusion does not vanish) in
order to enable the (weak and strong) maximum principle. Here in our one-
dimensional framework with $\theta(0)=\theta(1)=0$ this means that the
comparison principle holds regardless of any ordering on the boundary, and
only the initial ordering should be taken into account when applying the
comparison principle (see also [16, 17] for related topics). Since
$\mathring{p}_{0}\geq 0$ is nontrivial, a first application of the strong
maximum principle guarantees that $\mathring{p}_{t_{0}}(x)\geq c(t_{0})>0$ for
$t_{0}>0$ and any $x\in\bar{E}$. Recalling that the principal eigenfunction
$\alpha_{0}(x)$ is smooth and bounded away from zero, we can therefore
sandwich $c\alpha_{0}(x)\leq\mathring{p}_{t_{0}}(x)\leq C\alpha_{0}(x)$ upon
choosing $c,C$ are small and large enough. Observing that
$P(t,x)=e^{-\lambda_{0}t}\alpha_{0}(x)$ is a solution of
$(\partial_{t}-{\mathcal{L}^{*}})P=0$, the weak maximum principle therefore
guarantees

$ce^{-\lambda_{0}(t-t_{0})}\alpha_{0}(x)\leq\mathring{p}_{t}(x)\leq
Ce^{-\lambda_{0}(t-t_{0})}\alpha_{0}(x),\hskip 28.45274ptt\geq
t_{0}>0,\,x\in\bar{E}.$

Multiplying by $\eta(x)$ and dividing by
$\int_{E}\eta(x)p_{t}(\mathrm{d}x)=e^{-\lambda_{0}t}\int_{E}\eta(x)p_{0}(\mathrm{d}x)>0$
finally gives the desired result (recall that
$\pi=\eta\alpha=C\eta\alpha_{0}$). ∎

From a purely analytical perspective, once $q_{t}$ has been determined by
solving the conditioned Fokker-Planck equation (3.26), one can reconstruct the
full unconditioned distribution $p_{t}$ by first undoing the change of
variables (3.18) as

$q_{t}(\mathrm{d}x)=\frac{\eta(x)p_{t}(\mathrm{d}x)}{\int_{E}\eta(y)p_{s}(\mathrm{d}y)}=\frac{\eta(x)\mathring{p}_{t}(\mathrm{d}x)}{e^{-\lambda_{0}t}\int_{E}\eta(y)\mathring{p}_{0}(\mathrm{d}y)}\quad\Rightarrow\quad\mathring{p}_{t}(\mathrm{d}x)=e^{-\lambda_{0}t}\int_{E}\eta(y)\mathring{p}_{0}(\mathrm{d}y)\frac{q_{t}(\mathrm{d}x)}{\eta(x)},$

then leveraging the boundary kinematics (3.8) to compute $p^{\partial}_{t}$,
and finally retrieve $p_{t}=p^{\partial}_{t}+\mathring{p}_{t}$. This may
appear slightly intriguing at first sight: Indeed, the $Q$-process was
obtained by conditioning to non-absorption, a ${\mathbb{P}}$-negligible set of
paths. This construction therefore comes at the price of a total loss of
probabilistic information, and it seems very surprising that one can
reconstruct any statistics of the unconditioned dynamics from this negligible
knowledge. This actually goes even further: Not only the time marginal $p_{t}$
can be reconstructed from that of $q_{t}$, but in view of (3.15) even the
path-measure ${\mathbb{P}}$ can be inferred from ${\mathbb{Q}}$, so in some
sense it seems that the fullest possible unconditioned information can be
retrieved. As pointed out to us by N. Champagnat, this picture is however not
completely correct: The above reconstruction actually requires the explicit
knowledge of the $\eta(x)$ eigenfunction, which comes from the unconditioned
realm exclusively and cannot be determined from the conditioned generator
$\tilde{\mathcal{L}}$ in any way. In some sense, the eigenfunction $\eta$
really encodes much of the unconditioned information, at least sufficiently to
bridge the two worlds, and there is no paradox.

##  4 Metric setting

Let us first motivate informally the metric theory of gradient flows [2].
Given a smooth function $F:\mathbb{R}^{n}\to\mathbb{R}$, the Cauchy-Schwarz
inequality gives

$\frac{d}{dt}F(x_{t})=\nabla
F(x_{t})\cdot\dot{x}_{t}\geq-\frac{1}{2}|\dot{x}_{t}|^{2}-\frac{1}{2}|\nabla
F(x_{t})|^{2}$

for _any_ arbitrary $C^{1}$ curve $(x_{t})_{t\geq 0}$. Assume now that a
particular curve $x_{t}$ satisfies the reverse inequality

$\frac{d}{dt}F(x_{t})\leq-\frac{1}{2}|\dot{x}_{t}|^{2}-\frac{1}{2}|\nabla
F(x_{t})|^{2}.$ (4.1)

Then equality $\nabla
F(x_{t})\cdot\dot{x}_{t}=-\frac{1}{2}|\dot{x}_{t}|^{2}-\frac{1}{2}|\nabla
F(x_{t})|^{2}$ holds in the Cauchy-Schwarz inequality, which implies

$\dot{x}_{t}=-\nabla F(x_{t}).$

In other words, the _Energy Dissipation Inequality_ (4.1) ($\mathsf{EDI}$ in
short) is equivalent to the gradient flow.

Another useful characterization of the gradient-flow relies instead on
convexity: Assume now that the driving functional $F$ is in addition
$\lambda$-convex, in the sense that the Hessian
$D^{2}F(x)\geq\lambda\operatorname{Id}$ for some $\lambda\in\mathbb{R}$. By
convexity the gradient $\nabla F(x)$ is completely characterized by

$\left\\{F(y)\geq F(x)+\frac{\lambda}{2}|y-x|^{2}+p\cdot(y-x)\mbox{ for all
}y\right\\}\qquad\mbox{if and only if}\qquad\\{p=\nabla F(x)\\}.$

If now $x_{t}$ is an arbitrary $C^{1}$ curve we have
$\frac{1}{2}\frac{d}{dt}|y-x_{t}|^{2}=\dot{x}_{t}\cdot(y-x_{t})$, and imposing
the _Evolution Variational Inequality_ ($\mathsf{EVI}_{\lambda}$ in short)

$\frac{1}{2}\frac{d}{dt}|y-x_{t}|^{2}+\frac{\lambda}{2}|y-x_{t}|^{2}+F(x_{t})\leq
F(y),\hskip 28.45274pt\mbox{for all }y\in R^{n}\mbox{ and }t\geq 0$ (4.2)

is therefore also equivalent to $\dot{x}_{t}=-\nabla F(x_{t})$. Although
convexity is not really essential for existence purposes, it is more suited
for quantitative contraction and stability: For, whenever $F$ is
$\lambda$-convex and $x^{1}_{t},x^{2}_{t}$ are two solutions, computing

$\frac{1}{2}\frac{d}{dt}|x^{1}_{t}-x^{2}_{t}|^{2}=(x^{1}_{t}-x^{2}_{t})\cdot(\dot{x}^{1}_{t}-\dot{x}^{2}_{t})=-(x^{1}_{t}-x^{2}_{t})\cdot(\nabla
F(x^{1}_{t})-\nabla F(x^{2}_{t}))\leq-\lambda|x^{1}_{t}-x_{t}^{2}|^{2}$

gives exponential contraction

$|x^{1}_{t}-x^{2}_{t}|\leq e^{-\lambda t}|x^{1}_{0}-x^{2}_{0}|.$

The fundamental observation is that both (4.1) and (4.2) have a clear
counterpart in a purely metric setting, and can serve as a starting point for
a theory of gradient flows in metric spaces. This is developed in [2] (see
also [43] for a gentle introduction), and we collect below (without proofs)
the bare minimum material to serve our purpose later on.

Let $(\mathsf{X},\mathsf{d})$ be a geodesic, complete metric space and
$\mathsf{E}:\mathsf{X}\to(-\infty,+\infty]$ a function with proper domain
$\operatorname{Dom}(\mathsf{E})=\\{x\in\mathsf{X}:\,\mathsf{E}(x)<+\infty\\}$.
Mostly following [2, 38], we have

  * •

An _absolutely continuous_ curve $\gamma:[0,T]\to\mathsf{X}$ belongs to
$AC^{p}(0,T)$, $p\in[1,+\infty]$, if there exists $m\in L^{p}(0,T)$ such that

$\mathsf{d}(\gamma_{t_{0}},\gamma_{t_{1}})\leq\int_{t_{0}}^{t_{1}}m(t)\,\mathrm{d}t,\hskip
28.45274pt\forall\,t_{0},t_{1}\in[0,T].$

In this case $\gamma$ is absolutely continuous, the _metric speed_

$|\gamma^{\prime}|(t)\coloneqq\lim\limits_{h\to
0}\frac{\mathsf{d}(\gamma_{t+h},\gamma_{t})}{h}$ (4.3)

exists for a.e. $t\in(0,T)$, belongs to $L^{p}(0,T)$, and is the smallest
function $m$ satisfying the above inequality. For $p=1$ we simply denote
$AC=AC^{1}$, and the local-in time counterparts are denoted
$AC^{p}_{\mathrm{loc}}$ with obvious definitions.

  * •

We say that $g:\mathsf{X}\to[0,+\infty]$ is a _strong upper gradient_ for
$\mathsf{E}$ if, for every absolutely continuous curve $\gamma\in AC(0,T)$,
the function $g\circ\gamma$ is Borel and

$|\mathsf{E}(\gamma_{t_{0}})-\mathsf{E}(\gamma_{t_{1}})|\leq\int_{t_{0}}^{t_{1}}g(\gamma_{t})|\gamma^{\prime}|(t)\,\mathrm{d}t,\hskip
28.45274pt\forall\,0<t_{0}\leq t_{1}<T.$ (4.4)

In particular if $g\circ\gamma|\gamma^{\prime}|\in L^{1}(0,T)$ then
$\mathsf{E}\circ\gamma$ is absolutely continuous and

$\left|\frac{d}{dt}\mathsf{E}\circ\gamma(t)\right|\leq
g(\gamma_{t})|\gamma^{\prime}|(t),\hskip 28.45274pt\mbox{for a.e. }t\in(0,T).$

  * •

The _local slope_ is

$|\partial\mathsf{E}|(x)\coloneqq\limsup\limits_{y\to
x}\frac{(\mathsf{E}(x)-\mathsf{E}(y))^{+}}{\mathsf{d}(x,y)}$ (4.5)

  * •

The function $\mathsf{E}$ is _$\lambda$ -geodesically convex_ (or
$\lambda$-convex, in short) if, for any $x,y\in\mathsf{X}$, there exists a
geodesic $(\gamma_{t})_{t\in[0,1]}$ joining $\gamma_{0}=x,\gamma_{1}=y$ along
which

$\mathsf{E}(\gamma_{t})\leq(1-t)\mathsf{E}(x)+t\mathsf{E}(y)-\lambda\frac{t(1-t)}{2}\mathsf{d}^{2}(x,y),\qquad\forall
t\in[0,1].$

  * •

Whenever $\mathsf{E}$ is $\lambda$-convex the local slope admits the global
representation

$|\partial\mathsf{E}|(x)=\sup\limits_{y\neq
x}\frac{(\mathsf{E}(x)-\mathsf{E}(y)+\lambda\mathsf{d}^{2}(x,y))^{+}}{\mathsf{d}(x,y)},$

has dense domain, is automatically lower semi-continuous, and is a strong
upper gradient for $\mathsf{E}$. This relies on the fact that the global slope
as above is always l.s.c. and a strong upper gradient, and $\lambda$-convexity
guarantees that _local=global_. We omit the subtle details and refer instead
to [2, 38].

  * •

A locally absolutely continuous map $\gamma\in
AC^{2}_{\mathrm{loc}}([0,\infty))$ satisfies the _Energy-Dissipation
Inequality_ $\mathsf{EDI}_{t_{0}}$ started from $t_{0}\geq 0$ if
$\gamma_{t_{0}}\in\operatorname{Dom}(\mathsf{E})$ and

$\int_{t_{0}}^{t}\left(\frac{1}{2}|\gamma^{\prime}|^{2}(s)+\frac{1}{2}|\partial\mathsf{E}|^{2}(\gamma_{s})\right)\mathrm{d}s+\mathsf{E}(\gamma_{t})\leq\mathsf{E}(\gamma_{0}),\qquad\forall\,t>t_{0}.$
($EDI_{t_{0}}$)

Whenever equality is achieved we say that $\gamma$ satisfies the _Energy
Dissipation Equality_ $\mathsf{EDE}$.

  * •

Fix $\lambda\in\mathbb{R}$. A solution of the _Evolution Variational
Inequality_ $\mathsf{EVI}_{\lambda}$ is a continuous curve
$\gamma:(0,\infty)\to\mathsf{X}$ such that $\mathsf{E}\circ\gamma\in
L^{1}_{\mathrm{loc}}$ and, for every $y\in\operatorname{Dom}(\mathsf{E})$,

$\frac{1}{2}\frac{d}{dt}\mathsf{d}^{2}(\gamma_{t},y)+\frac{\lambda}{2}\mathsf{d}^{2}(\gamma_{t},y)+\mathsf{E}(\gamma_{t})\leq\mathsf{E}(y)$
($\mathsf{EVI}_{\lambda}$)

for a.e. $t>0$.

An _$\mathsf{EVI}_{\lambda}$ -gradient flow_ of $\mathsf{E}$ on a (possibly
strict) subdomain $D\subseteq\overline{\operatorname{Dom}(\mathsf{E})}$ is a
family of continuous maps $(\mathsf{S}_{t})_{t\geq 0}:D\to D$ such that, for
every $\gamma_{0}\in D$, the strong semi-group property holds

$\mathsf{S}_{t+h}\gamma_{0}=\mathsf{S}_{h}\mathsf{S}_{t}\gamma_{0}\mbox{ for
}t,h\geq 0\qquad\mbox{with}\qquad\lim\limits_{t\to
0^{+}}\mathsf{S}_{t}\gamma_{0}=\mathsf{S}_{0}\gamma_{0}=\gamma_{0},$

and

$t\mapsto\gamma_{t}=\mathsf{S}_{t}\gamma_{0}\qquad\mbox{is an
}\mathsf{EVI}_{\lambda}\mbox{ solution}.$

  * •

Is is known [2] that $\lambda$-convexity (almost) guarantees the well-
posedness of an $\mathsf{EVI}_{\lambda}$ flow, but the converse is also true:
If $\mathsf{E}$ generates an $\mathsf{EVI}_{\lambda}$-flow then necessarily
$\mathsf{E}$ is $\lambda$-convex along _all_ geodesics, see [12, 36].

Fundamental properties of $\mathsf{EVI}_{\lambda}$ solutions are as follows

######  Theorem 3 (Properties of $\mathsf{EVI}_{\lambda}$-solutions, [38,
Theorem 3.5,]).

1

$\lambda$-contraction and uniqueness

$\mathsf{d}(\gamma^{1}_{t},\gamma^{2}_{t})\leq e^{-\lambda
t}\mathsf{d}(\gamma^{1}_{0},\gamma^{2}_{0}),\hskip 28.45274pt\forall\,t\geq 0$

for any two solutions $\gamma^{1},\gamma^{2}$.

Regularizing effect: $\gamma$ is locally Lipschitz,
$\gamma_{t}\in\operatorname{Dom}(|\partial\mathsf{E}|)\subset\operatorname{Dom}(\mathsf{E})$
for all $t>0$, and

$t\in[0,\infty)\mapsto\mathsf{E}(\gamma_{t})\mbox{ is nonincreasing}$

Functional inequalities and asymptotic behaviour: if $\lambda>0$ and
$\mathsf{E}$ has complete sublevels, then it admits a unique minimizer
$\gamma_{\infty}\in\mathsf{X}$ satisfying

$\frac{\lambda}{2}\mathsf{d}^{2}(x,\gamma_{\infty})\leq\mathsf{E}(x)-\mathsf{E}(\gamma_{\infty})\leq\frac{1}{2\lambda}|\partial\mathsf{E}|^{2}(x),$

for all $x\in\mathsf{X}$ and

$\mathsf{E}(\gamma_{t})-\mathsf{E}(\gamma_{\infty})\leq
e^{-2\lambda(t-t_{0})}\left(\mathsf{E}(\gamma_{t_{0}})-\mathsf{E}(\gamma_{t_{\infty}})\right)$
$\mathsf{d}(\gamma_{t},\gamma_{\infty})\leq
e^{-\lambda(t-t_{0})}\mathsf{d}(\gamma_{t_{0}},\gamma_{\infty})$

for all $t\geq t_{0}\geq 0$.

Finally, a useful connection between notions of solutions of metric gradient
flows will be for us

######  Theorem 4 ($\mathsf{EDI}\Rightarrow\mathsf{EVI}$).

Assume that an $\mathsf{EVI}_{\lambda}$ gradient flow $\mathsf{S}_{t}$ of
$\mathsf{E}$ exists on a (possibly strict) subdomain
$D\subseteq\overline{\operatorname{Dom}(\mathsf{E})}$. Let
$\gamma:[0,\infty)\to\mathsf{E}$ be an $\mathsf{EDI}_{t_{0}}$ solution for all
$t_{0}>0$, whose image is contained in $D$. If $\gamma$ is continuous at
$t=0^{+}$ then $\gamma_{t}=\mathsf{S}_{t}\gamma_{0}$.

Since it is well-known that
$\mathsf{EVI}_{\lambda}\Rightarrow\mathsf{EDE}\Rightarrow\mathsf{EDI}$, this
means that all three notions are in fact equivalent, provided that the
solution under consideration remains in the domain of the flow $\mathsf{S}$.

###### Proof.

This exact statement can be found in [38, theorem 4.2], assuming that
$\mathsf{EDI}_{0}$ holds up to $t=0$ and not up to any $t_{0}>0$ (note however
that the continuity of $\gamma$ at $t=0^{+}$ is contained in the definition of
$\mathsf{EDI}_{0}$). Observe that ($\mathsf{EVI}_{\lambda}$) is in fact local
in time: applying [38, theorem 4.2] thus shows that $\gamma$ is an
$\mathsf{EVI}_{\lambda}$ solution in the interval $[t_{0},+\infty)$ for any
$t_{0}>0$, thus in $(0,+\infty)$. The continuity $\gamma_{t}\to\gamma_{0}$
imposed at $t\to 0^{+}$ allows to conclude. ∎

###  4.1 The Wasserstein-Shahshahani distance

Let us recall that our whole analysis relies upon choosing an ad-hoc geometry,
that is optimal transport over the Shahshahani space. More precisely, recall
that we choose to view $E=(0,1)$ as a Riemannian manifold with metrics induced
by ${\theta}$

$\langle\xi_{1},\xi_{2}\rangle_{\theta}\coloneqq\frac{\langle\xi_{1},\xi_{2}\rangle}{{\theta}(x)},\qquad\forall\,\xi_{1},\xi_{2}\in
T_{x},\,x\in E.$

We denote by

$|\xi|^{2}_{{\theta}}\coloneqq\frac{|\xi|^{2}}{{\theta}(x)}$

the corresponding norm on $T_{x}$. For the specific choice
${\theta}(x)=x(1-x)$, which is our guiding thread in this work, this is known
as the Shahshahani metrics and was introduced in [44] to study a Replicator
Equation from evolutionary genetics as a gradient system. Given the biological
background of our Kimura model, it is no surprise that this Shahshahani
structure should appear again here and we stick to the terminology. Although
singular at $x=0,1$, this induces a proper distance on $\bar{E}$ [8, Lemma 10]

$\mathsf{d}_{\theta}(x,y)\coloneqq\inf\limits_{\begin{subarray}{c}u_{0}=x\\\
u_{1}=y\end{subarray}}\int_{0}^{1}|\dot{u}_{t}|^{2}_{{\theta}}\,\mathrm{d}t=\int_{x}^{y}\frac{1}{\sqrt{{\theta}(z)}}\,\mathrm{d}z.$
(4.6)

Note that the linear behaviour of ${\theta}$ at the boundaries makes this
quantity finite even for $x=0$ or $y=1$, and that

$\mathsf{d}_{\theta}(x,\partial E)\sim
C\sqrt{\mathsf{d}_{\mathbb{R}}(x,\partial E)}\qquad\mbox{as }x\to\partial E.$
(4.7)

The space $(\bar{E},\mathsf{d}_{\theta})$ has finite diameter
$D=\int_{0}^{1}\frac{\mathrm{d}z}{\sqrt{{\theta}(z)}}$, and it is not
difficult to check that it is complete. The corresponding _Wasserstein-
Shahshahani distance_ over $\mathcal{P}(\bar{E})$ is then simply defined as
the usual quadratic Wasserstein distance [48] over the Polish space
$(\bar{E},\mathsf{d}_{\theta})$,

${\mathcal{W}}_{\theta}^{2}(q,r)\coloneqq\min\limits_{\gamma\in\Gamma(q,r)}\iint_{\bar{E}^{2}}\mathsf{d}_{\theta}^{2}(x,y)\mathrm{d}\gamma(x,y)$

for any probability measures $q,r\in\mathcal{P}(\bar{E})$. Here the
minimization runs over admissible plans

$\gamma\in\Gamma(q,r)=\left\\{\pi(\mathrm{d}x,\mathrm{d}y)\in\mathcal{P}(\bar{E}\times\bar{E})\mbox{
with marginals }\pi_{x}=q,\pi_{y}=r\right\\}.$

General Optimal Transport theory [48, 47] gives that
$(\mathcal{P}(\bar{E}),{\mathcal{W}}_{\theta})$ turns into a Polish space of
its own. Note also that, since $\mathsf{d}_{\theta}$ is locally equivalent to
the Euclidean metric and $1/2$-Hölder at the boundaries, we have that
$|x_{n}-x|_{\mathbb{R}}\to 0$ if and only if $\mathsf{d}_{\theta}(x_{n},x)\to
0$ and therefore the set of real-valued $\mathsf{d}_{\theta}$-continuous
functions over $\bar{E}$ coincides with the usual set $C(\bar{E})$ of
continuous functions up to the boundary. In particular
${\mathcal{W}}_{\theta}$ is well-known to metrize the narrow convergence of
measures [48], i.e.

$\int_{\bar{E}}\varphi(x)q_{n}(\mathrm{d}x)\to\int_{\bar{E}}\varphi(x)q(\mathrm{d}x)$
for all $\varphi\in C(\bar{E})$ if and only if
${\mathcal{W}}_{\theta}(q_{n},q)\to 0$.

In order to emphasize the dependence on the Riemannian structure and
differentiate from the Euclidean framework we use ${\theta}$-subscripts in the
sequel for any quantity computed with respect or related to this geometry. For
example we write $\mathcal{P}_{{\theta}}=\mathcal{P}(\bar{E})$ equipped with
${\mathcal{W}}_{\theta}$, the metric speed of a curve of measures $q_{t}$
computed w.r.t ${\mathcal{W}}_{\theta}$ is denoted
$|q^{\prime}|_{{\mathcal{W}}_{\theta}}(t)$, we write $AC_{\theta}$ for
${\theta}$-absolutely continuous curves of measures, and so on (with sometimes
a slight abuse of notations). By contrast and whenever needed, we use the
$\mathbb{R}$ subscript to denote any quantity measured using the Euclidean
norm $|.|_{\mathbb{R}}$.

###  4.2 Absolutely continuous curves

A fundamental result in Optimal Transport is the Benamou-Brenier formula [4],
yielding a dynamical interpretation of the static Wasserstein distance. This
allows the representation of $AC^{2}$ curves in the Wasserstein space as
solutions of continuity equations

$\partial_{t}q_{t}+\operatorname{div}(q_{t}v_{t})=0,$ (CE)

which in turn reveals crucial in identifying solutions of abstract metric
gradient-flows in the Wasserstein space as actual PDEs. However, since our
pseudo-Riemannian manifold is singular on the boundaries, we cannot simply
apply the existing theory. We opted instead for a self-contained presentation,
mostly relying on [30] and trying to keep the technical details to a bare
minimum (many of the arguments below are mere adaptations of known results,
fitted to our purposes).

######  Definition 4.1.

We say that a curve of measures $(q_{t})_{t\in I}$ on $\mathcal{P}(\bar{E})$
and a (Borel) velocity field $v_{t}(x)$ solve the continuity equation (CE) in
$(t,x)\in I\times\bar{E}$ if

$\frac{d}{dt}\int_{\bar{E}}\varphi(x)\,q_{t}(\mathrm{d}x)=\int_{\bar{E}}\nabla\varphi(x)\cdot
v_{t}(x)\,q_{t}(\mathrm{d}x)$

holds in the sense of distributions in $I$ for all $\varphi\in
C^{1}(\bar{E})$.

Note that this encodes a no-flux boundary condition $qv\cdot\nu=0$ on
$\partial E$ in a weak sense. We denote the weighted $L^{2}$ norm (kinetic
energy) as

$\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\coloneqq\int_{\bar{E}}|v_{t}(x)|_{\theta}^{2}\,q_{t}(\mathrm{d}x)=\int_{\bar{E}}\frac{|v_{t}(x)|^{2}}{{\theta}(x)}\,q_{t}(\mathrm{d}x)$

whenever this quantity is finite. Note that the very meaning of this integral
is far from being clear: Since $\frac{1}{{\theta}(x)}\approx\frac{1}{x(1-x)}$
is not continuous up to the boundary, it may a priori not be legitimate to
integrate w.r.t to an arbitrary measure $q\in\mathcal{P}(\bar{E})$, in
particular if $q$ has atoms on the boundary. This can be made rigorous upon
understanding the quotient $\frac{|\xi|^{2}}{{\theta}(x)}$ in a particular
sense, which will be done very soon in Lemma 4.11 – see in particular (4.10).
Leaving this issue aside, the Benamou-Brenier formula is expected to read at
least formally

${\mathcal{W}}^{2}_{\theta}(q_{0},q_{1})=\min\Bigg{\\{}\int_{0}^{1}\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\mathrm{d}t\mbox{
s.t. }(q,v)\mbox{ solve \eqref{eq:CE} with
}q|_{t=0}=q_{0},q|_{t=1}=q_{1}\Bigg{\\}},$

and suggests that $AC^{2}(I;\mathcal{P}_{\theta})$ curves should be
represented by suitable velocity fields solving (CE) with finite weighted
kinetic action. This is indeed the case:

######  Theorem 5.

For any curve $q\in AC^{2}(I;\mathcal{P}_{\theta})$ there exists a minimal
Borel vector-field $\tilde{v}_{t}(x):I\times\bar{E}\to\mathbb{R}$ such that
$(q,\tilde{v})$ solves the continuity equation and

$\|\tilde{v}_{t}\|_{L^{2}_{\theta}(q_{t})}\leq|q^{\prime}|_{{\mathcal{W}}_{\theta}}(t)\hskip
28.45274pt\mbox{for a.e. }t\in I.$ (4.8)

Conversely, any solution $(q,v)$ of the continuity equation with finite
kinetic energy
${\int_{I}\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\mathrm{d}t<\infty}$ gives an
absolutely continuous curve $q\in AC^{2}(I;\mathcal{P}_{\theta})$ with

$|q^{\prime}|_{{\mathcal{W}}_{\theta}}(t)\leq\|v_{t}\|_{L^{2}_{\theta}(q_{t})}.\hskip
28.45274pt\mbox{for a.e. }t\in I.$ (4.9)

In particular, for any given $q\in AC^{2}(I;\mathcal{P}_{\theta})$ the minimal
vector-field $\tilde{v}$ as above is unique, and such that
$|q^{\prime}|_{{\mathcal{W}}_{\theta}}(t)=\|\tilde{v}_{t}\|_{L^{2}_{\theta}(q_{t})}$
for a.e. $t$.

The proof below will rely on the following elementary lemma:

######  Lemma 4.2.

Define the Lagrangian

$L(x,\xi)=\begin{cases}\frac{|\xi|_{\mathbb{R}}^{2}}{{\theta}(x)}&\mbox{if
}x\in E,\\\ 0&\mbox{if }x\in\partial E\mbox{ and }\xi=0,\\\ +\infty&\mbox{if
}x\in\partial E\mbox{ but }\xi\neq 0.\end{cases}$ (4.10)

For any curve $u\in AC^{2}(I;\bar{E}_{\theta})$ and a.e. $t\in I$, the
classical derivative

$\dot{u}_{t}=\lim\limits_{h\to 0}\frac{u_{t+h}-u_{t}}{h}$

exists and

$|u^{\prime}|_{\theta}^{2}(t)=L(u_{t},\dot{u}_{t}).$ (4.11)

Let us emphasize once again that, in the left-hand side of (4.11),
$|u^{\prime}|_{\theta}$ denotes the metric speed computed w.r.t. the
Shahshahani distance $\mathsf{d}_{\theta}$.

###### Proof.

Observe first that $u\in AC^{2}(I;\bar{E}_{\theta})$ is
$\mathsf{d}_{\theta}$-continuous in time, hence also continuous in the
classical sense (the $\mathsf{d}_{\theta}$ topology being equivalent to the
Euclidean topology). As a consequence $u_{t_{0}}$ can be unambiguously
evaluated at any $t_{0}\in I$, and we treat separately $u_{t_{0}}\in\partial
E$ and $u_{t_{0}}\in E$. The case of an interior point $u_{t_{0}}\in E$ falls
within standard, smooth Riemannian geometry and the proof is immediate, see
e.g. [31, Section 2.1]. For a boundary point, assume e.g. $u_{t_{0}}=0$ (the
case $u_{t_{0}}=1$ is similar), and assume without loss of generality that
$t_{0}$ is a “metric differentiability point” for $u$ (i-e the metric
derivative $|u^{\prime}|_{\theta}(t_{0})$ exists, which is indeed guaranteed
for a.e. $t$ by standard properties of the metric speed). Then by (4.7) we
have that

$\frac{\sqrt{u_{t_{0}+h}}-\sqrt{u_{t_{0}}}}{h}=\frac{\sqrt{u_{t_{0}+h}}}{h}\sim
C\frac{\mathsf{d}_{\theta}(u_{t_{0}+h},u_{t_{0}})}{h}\to
C|u^{\prime}|_{\theta}(t_{0})<+\infty,$

i-e $t\mapsto\sqrt{u_{t}}$ is differentiable at $t_{0}$. By standard chain-
rule we see that the usual derivative

$\dot{u}_{t_{0}}=2\sqrt{u_{t_{0}}}\left.\frac{d}{dt}\right|_{t_{0}}\sqrt{u_{t}}=0$

exists and vanishes, hence by definition $L(u_{t_{0}},\dot{u}_{t_{0}})=0$. The
subtle issue here is that, although $L(u_{t_{0}},\dot{u}_{t_{0}})=0$ at that
particular time $t=t_{0}$, it might happen that
$|u^{\prime}|_{\theta}(t_{0})\neq 0$ there, thus violating our claim that
$|u^{\prime}|_{\theta}^{2}(t)=L(u_{t},\dot{u}_{t})$. Note however that we only
claim equality for almost every $t$: Hence, it suffices to prove that the “bad
set” $N\subset I$ of such times $t_{0}\in I$ where $u_{t_{0}}=0$ (or
$u_{t_{0}}=1$) with $|u^{\prime}|_{\theta}(t_{0})\neq 0$ is negligible. By
definition, having non-zero metric speed implies that the curve is really
moving, and more precisely
$\mathsf{d}_{\theta}(u_{t},u_{t_{0}})\sim|u^{\prime}|_{\theta}(t_{0})|t-t_{0}|\neq
0$ for $t\neq t_{0}$ close enough to $t_{0}$. As a result $u_{t}\neq 0$ for
such times, which means that $N$ is actually a discrete set and therefore
Lebesgue-negligible as claimed. ∎

###### Proof of Theorem 5.

With Lemma 4.11 now at hand, the proof is almost identical to [30, Theorem
2.4] so we briefly give the main lines and only highlight the slight
differences. The main idea is that, in an abstract metric framework,
absolutely continuous curves in an overlying Wasserstein space can always be
recovered as suitable superposition of absolutely continuous curves in the
associated underlying space [30]. Whenever that underlying metric space is in
fact a Riemannian manifold with induced Riemannian distance, one can go one
step further and relate the metric speed of an underlying curve to the
Riemannian norm of its usual velocity (its time derivative, i-e a tangent
vector). This allows reconstructing the velocity field $\tilde{v}_{t}(x)$ in
Theorem 5 essentially by superposing the velocities of all Lagrangian
particles simultaneously sitting at position $x$ at time $t$. The only subtle
point here, compared to [30], is that our underlying metric space
$(\bar{E},\mathsf{d}_{\theta})$ is not properly speaking a smooth Riemannian
manifold and therefore the singular boundary points must be dealt with
separately, which is done thanks to Lemma 4.11.

More precisely, let $\Gamma=C(I;\bar{E}_{{\theta}})$ be the space of
continuous curves in $\bar{E}$ with induced $\mathsf{d}_{\theta}-\sup$
distance. Owing to [30, Theorem 5], valid in general metric spaces, and given
a curve $q\in AC^{2}(I;\mathcal{P}_{\theta})$ as in our statement, there is a
path-measure $\eta\in\mathcal{P}(\Gamma)$ such

  1. (i)

$\eta$ is concentrated on $AC^{2}(I;\bar{E}_{\theta})$

  2. (ii)

$q_{t}={e_{t}}_{\\#}\eta$ for all $t\in I$, where $e_{t}(u)=u_{t}$ is the
time-$t$ evaluation map

  3. (iii)

the metric speed is represented by superposition

$|q^{\prime}|^{2}_{{\mathcal{W}}_{{\theta}}}(t)=\int_{\Gamma}|u^{\prime}|^{2}_{{\theta}}(t)\,\eta(\mathrm{d}u)$

By disintegration, there exists a (unique) Borel family of probability
measures $\eta_{t,x}\in\mathcal{P}(\Gamma)$ and concentrated on
$\Gamma_{t,x}=\\{u\in\Gamma:\quad u_{t}=x\\}$ such that

$\int_{I\times\Gamma}\varphi(t,u)\,\mathrm{d}t\,\eta(\mathrm{d}u)=\int_{I\times\bar{E}}\left(\int_{\Gamma}\varphi(t,u)\,\eta_{t,x}(\mathrm{d}u)\right)q_{t}(\mathrm{d}x)\mathrm{d}t$

for all $\varphi\in C(I\times\Gamma)$. The velocity field $\tilde{v}$ is then
defined as the superposition of all Lagrangian speeds

$\tilde{v}_{t}(x)\coloneqq\int_{\Gamma}\dot{u}_{t}\,\eta_{t,x}(\mathrm{d}u).$
(4.12)

A keypoint is that, exactly as in [31], this is unambiguously defined for
$\mathrm{d}q_{t}\otimes\mathrm{d}t$-almost every $x,t$ because the paths $u$
charged by $\eta$ (or $\eta_{t,x}$) are $AC^{2}$, here specifically
$AC^{2}(I;\bar{E}_{\theta})$. Hence the classical derivative $\dot{u}_{t}$ is
well-defined for a.e. $t\in I$ according to Lemma 4.11. By Jensen’s inequality
and our representation of the metric speed
$|u^{\prime}|_{\theta}^{2}(t)=\frac{|\dot{u}_{t}|_{\mathbb{R}}^{2}}{{\theta}(u_{t})}$
from Lemma 4.11 we get

$\int_{I}\|\tilde{v}_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\mathrm{d}t=\int_{I\times\bar{E}}\frac{|\tilde{v}_{t}(x)|^{2}}{{\theta}(x)}q_{t}(\mathrm{d}x)\mathrm{d}t\\\
=\int_{I\times\bar{E}}\frac{\left|\int_{\Gamma}\dot{u}_{t}\,\eta_{t,x}(\mathrm{d}u)\right|^{2}}{{\theta}(x)}q_{t}(\mathrm{d}x)\mathrm{d}t\leq\int_{I\times\bar{E}}\left(\int_{\Gamma}\frac{|\dot{u}_{t}|^{2}}{{\theta}(x)}\eta_{t,x}(\mathrm{d}u)\right)q_{t}(\mathrm{d}x)\mathrm{d}t\\\
=\int_{I\times\Gamma}\frac{|\dot{u}_{t}|^{2}}{{\theta}(u_{t})}\,\mathrm{d}t\,\eta(\mathrm{d}u)\overset{\eqref{eq:udot_Luxi}}{=}\int_{I\times\Gamma}|u^{\prime}|^{2}_{{\theta}}(t)\mathrm{d}t\,\eta(\mathrm{d}u)=\int_{I}|q^{\prime}|^{2}_{{\mathcal{W}}_{\theta}}(t)\,\mathrm{d}t.$

One can actually carry over the exact same computation in any time interval
$[t_{0},t_{1}]\subset I$, which allows to conclude that
$\int_{t_{0}}^{t_{1}}\|\tilde{v}_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\mathrm{d}t\leq\int_{t_{0}}^{t_{1}}|q^{\prime}|^{2}_{{\mathcal{W}}_{\theta}}(t)\mathrm{d}t$
and thus

$\|\tilde{v}_{t}\|_{L^{2}_{\theta}(\mathrm{d}q_{t})}\leq|q^{\prime}|(t)\qquad\mbox{for
a.e. }t\in I.$

In order to prove that $(q,\tilde{v})$ solve the continuity equation as
claimed, observe first that, from (4.6) and the mean value Theorem, we have
$\mathsf{d}(x,y)=\int_{x}^{y}\frac{1}{\sqrt{{\theta}(z)}}\mathrm{d}z=\frac{|x-y|}{\sqrt{{\theta}(z)}}$
for some $z=z_{x,y}\in(x,y)$ and as a consequence

$|x-y|\leq\|\sqrt{\theta}\|_{\infty}\mathsf{d}(x,y)$

for any $x,y\in\bar{E}$. In particular any $\mathsf{d}_{\mathbb{R}}$-Lipschitz
map $\varphi:\bar{E}\to\mathbb{R}$ is $\mathsf{d}_{\theta}$-Lipschitz as well.
Take now any $\varphi\in C^{1}(\bar{E})$ and write, for any optimal plan
$\gamma_{t_{0},t_{1}}$ from $q_{t_{0}}$ to $q_{t_{1}}$,

$\left|\int_{\bar{E}}\varphi\,q_{t_{1}}-\int_{\bar{E}}\varphi\,q_{t_{0}}\right|=\left|\int_{{\bar{E}}\times{\bar{E}}}\left(\varphi(x)-\varphi(y)\right)\gamma_{t_{0},t_{1}}(\mathrm{d}x,\mathrm{d}y)\right|\leq\int_{{\bar{E}}\times{\bar{E}}}|\varphi(x)-\varphi(y)|\gamma_{t_{0},t_{1}}(\mathrm{d}x,\mathrm{d}y)\\\
\leq
L\int_{{\bar{E}}\times{\bar{E}}}\mathsf{d}_{\theta}(x,y)\gamma_{t_{0},t_{1}}(\mathrm{d}x,\mathrm{d}y)\leq
L\left(\int_{{\bar{E}}\times{\bar{E}}}\mathsf{d}_{\theta}^{2}(x,y)\gamma_{t_{0},t_{1}}(\mathrm{d}x,\mathrm{d}y)\right)^{\frac{1}{2}}=L\,{\mathcal{W}}_{{\theta}}(q_{t_{0}},q_{t_{1}}),$

where of course $L=\operatorname{Lip}_{\mathsf{d}_{\theta}}(\varphi)$. Since
$q\in AC^{2}([0,1];\mathcal{P}_{\theta})$ this shows that
$t\mapsto\int_{\bar{E}}\varphi(x)q_{t}(\mathrm{d}x)$ is $AC^{2}$ (in the
classical sense), hence we can legitimately differentiate under the integral
sign as

$\frac{d}{dt}\int_{\bar{E}}\varphi(x)\,q_{t}(\mathrm{d}x)=\frac{d}{dt}\int_{\Gamma}\varphi(u_{t})\,\eta(\mathrm{d}u)=\int_{\Gamma}\left[\frac{d}{dt}\varphi(u_{t})\right]\,\eta(\mathrm{d}u)\\\
=\int_{\Gamma}\nabla\varphi(u_{t})\cdot\dot{u}_{t}\,\eta(\mathrm{d}u)=\int_{0}^{1}\int_{\bar{E}}\nabla\varphi(x)\cdot\tilde{v}_{t}(x)\,q_{t}(\mathrm{d}x)\mathrm{d}t.$

Here the last equality follows from the very definition (4.12) of $\tilde{v}$
as $\eta$-superposition of Lagrangian velocities $\dot{u}$.

Let us now turn to the converse implication, and assume that $(q,v)$ solve the
continuity equation in $I\times{\bar{E}}$ with finite $L^{2}_{\theta}$ kinetic
energy. Without loss of generality we assume that $I=(0,T)$. The key step is
to apply a probabilistic representation for solutions of the continuity
equation, allowing somehow to retrieve a notion of Lagrangian trajectories
although the driving velocity field $v_{t}(x)$ could be very rough, see [2,
Section 8.2] and [1]. Observe that
$\frac{|v_{t}(x)|^{2}}{\|{\theta}\|_{\infty}}\leq\frac{|v_{t}(x)|^{2}}{{\theta}(x)}$,
hence $(q,v)$ solve the continuity equation with finite Euclidean kinetic
energy
$\int_{I}\int_{\bar{E}}|v_{t}(x)|_{\mathbb{R}}^{2}\,q_{t}(\mathrm{d}x)\,\mathrm{d}t<\infty$.
This allows a direct application of [2, Theorem 8.2.1.i] and we conclude that
there exists a probability measure $\eta\in P({\bar{E}}\times\Gamma)$ (here
$\Gamma=C(I;{\bar{E}})$), supported on pairs $(x,u)$ such that $u\in
AC^{2}(I;\bar{E}_{\mathbb{R}})$ with $u_{0}=x$ and solving
$\dot{u}_{t}=v_{t}(u_{t})$ in the integral sense. Moreover for all $t\in I$
the measure $q_{t}$ is the pushforward of $\eta$ by the evaluation map
$e_{t}$, i-e

$\int_{\bar{E}}\varphi(x)q_{t}(\mathrm{d}x)=\int_{{\bar{E}}\times\Gamma}\varphi(u_{t})\,\eta(\mathrm{d}x,\mathrm{d}u)$

for all $\varphi\in C({\bar{E}})$. The second marginal
$\zeta\in\mathcal{P}(\Gamma)$ of $\eta$ is thus supported on
$AC^{2}_{\mathbb{R}}$ solutions of $\dot{u}_{t}=v_{t}(u_{t})$ and such that

$\int_{\bar{E}}\varphi(x)q_{t}(\mathrm{d}x)=\int_{\Gamma}\varphi(u_{t})\,\zeta(\mathrm{d}u).$
(4.13)

Now for any $t_{0}<t_{1}$ the measure
$\gamma_{t_{0},t_{1}}\coloneqq(e_{t_{0}},e_{t_{1}})_{\\#}\zeta\in\mathcal{P}({\bar{E}}\times{\bar{E}})$
is an admissible coupling between $q_{t_{0}},q_{t_{1}}$ and therefore

${\mathcal{W}}_{\theta}^{2}(q_{t_{0}},q_{t_{1}})\leq\int_{{\bar{E}}\times{\bar{E}}}\mathsf{d}_{\theta}^{2}(x,y)\mathrm{d}\gamma_{t_{0},t_{1}}(x,y)=\int_{\Gamma}\mathsf{d}_{\theta}^{2}(u_{t_{0}},u_{t_{1}})\,\zeta(\mathrm{d}u).$

Since $\dot{u}_{t}=v_{t}(u_{t})$ for $\zeta$-a.e. $u\in\Gamma$, we have, by
definition (4.6) of $\mathsf{d}_{\theta}$, interpreting the quotient
$\frac{|\xi|^{2}}{{\theta}(x)}$ as before, and scaling time appropriately,

$\int_{\Gamma}\mathsf{d}_{\theta}^{2}(u_{t_{0}},u(t_{1}))\,\zeta(\mathrm{d}u)\leq\int_{\Gamma}\left((t_{1}-t_{0})\int_{t_{0}}^{t_{1}}\frac{|\dot{u}_{t}|^{2}}{{\theta}(u_{t})}\,\mathrm{d}t\right)\zeta(\mathrm{d}u)\\\
=(t_{1}-t_{0})\int_{\Gamma}\int_{t_{0}}^{t_{1}}\frac{|v_{t}(u_{t})|^{2}}{{\theta}(u_{t})}\,\mathrm{d}t\,\zeta(\mathrm{d}u)=(t_{1}-t_{0})\int_{t_{0}}^{t_{1}}\int_{\Gamma}\frac{|v_{t}(u_{t})|^{2}}{{\theta}(u_{t})}\,\zeta(\mathrm{d}u)\,\mathrm{d}t\\\
=(t_{1}-t_{0})\int_{t_{0}}^{t_{1}}\int_{{\bar{E}}}\frac{|v_{t}(x)|^{2}}{{\theta}(x)}\,q_{t}(\mathrm{d}x)\,\mathrm{d}t.$

Here we made a crucial use of (4.13) for the last equality. Since
$t_{0}<t_{1}$ were arbitrary this shows that $q\in
AC^{2}(I;\mathcal{P}_{\theta})$ and entails (4.9).

Finally, uniqueness of the minimal velocity field $\tilde{v}$ is an easy
consequence of the linearity of the continuity equation constraint and of the
strong convexity of the $L^{2}$ norm. ∎

###  4.3 Displacement convexity and metric slope

Let us recall from Section 3 the expressions

$\pi(\mathrm{d}x)=\frac{1}{\int_{\bar{E}}\eta(x)\alpha(\mathrm{d}x)}\eta(x)\alpha(\mathrm{d}x)=\frac{1}{\mathcal{Z}}e^{-(\tilde{U}+\log{\theta})(x)}\mathrm{d}x=\frac{1}{\mathcal{Z}}e^{-V(x)}\mathrm{d}x$

for our stationary measure of the $Q$-process. For reasons that shall become
clear later on, let us denote

$W\coloneqq V-\log\sqrt{\theta}.$ (4.14)

Typically writing

$\sigma(x)=\frac{\mathrm{d}q}{\mathrm{d}\pi}(x)$

for the Radon-Nykodim derivative w.r.t. the reference measure, the relative
entropy ${\mathcal{H}_{\pi}}:\mathcal{P}(\bar{E})\to(-\infty,\infty]$ is
classically defined as

${\mathcal{H}_{\pi}}(q)\coloneqq\begin{cases}\int_{\bar{E}}\sigma(x)\log\sigma(x)\,\pi(\mathrm{d}x)&\mbox{if
}q(\mathrm{d}x)=\sigma(x)\pi(\mathrm{d}x),\\\ +\infty&\mbox{else}.\end{cases}$
(4.15)

This is well-known to be lower semi-continuous w.r.t. narrow convergence of
measures. Moreover it is not difficult to check that ${\mathcal{H}}_{\pi}$ has
dense domain
$\overline{\operatorname{Dom}({\mathcal{H}}_{\pi})}=\mathcal{P}_{\theta}(\bar{E})$.
Note that, since $\pi(\mathrm{d}x)=C\eta(x)\alpha_{0}(x)\mathrm{d}x$ is
absolutely continuous, any $q\ll\pi$ is also absolutely continuous w.r.t. to
the Lebesgue measure. Thus writing $q(\mathrm{d}x)=q(x)\mathrm{d}x$, our
entropy (4.15) can also be written as

${\mathcal{H}}_{\pi}(q)=\int_{\bar{E}}q(x)\log
q(x)\,\mathrm{d}x+\int_{\bar{E}}q(x)V(x)\,\mathrm{d}x.$

(Here we assumed w.l.o.g. that the normalizing factor $\mathcal{Z}=1$ and
$V=-\log\pi$.)

######  Proposition 4.3.

Let $W$ be as in (4.14). Then

$\lambda\coloneqq\min\limits_{x\in(0,1)}\left\\{\sqrt{{\theta}}\frac{d}{dx}\left(\sqrt{{\theta}}\frac{d}{dx}W\right)\right\\}.$
(4.16)

is a finite real number and the entropy $q\mapsto{\mathcal{H}}_{\pi}(q)$ is
$\lambda$-displacement convex along any ${\mathcal{W}}_{\theta}$-geodesic
$(q_{t})_{t\in[0,1]}$ between $q_{0},q_{1}\in\mathcal{P}_{\theta}(E)$,

${\mathcal{H}}_{\pi}(q_{t})\leq(1-t){\mathcal{H}}_{\pi}(q_{0})+t{\mathcal{H}}_{\pi}(q_{1})-\lambda\frac{t(1-t)}{2}{\mathcal{W}}_{\theta}^{2}(q_{0},q_{1}),\qquad\forall\,t\in[0,1].$
(4.17)

###### Direct but formal proof.

It is by now well understood [46, Theorem 1.3] that, on a smooth, complete
Riemannian manifold $(M,g)$, and given a smooth potential $W:M\to\mathbb{R}$,
the functional

$\mathcal{S}_{W}(q)=\int_{M}\rho(x)\log\rho(x)\,\operatorname{Vol}(\mathrm{d}x)+\int_{M}W(x)\,q(\mathrm{d}x),\qquad\mbox{with}\qquad
q(\mathrm{d}x)=\rho(x)\operatorname{Vol}(\mathrm{d}x)$ (4.18)

is $\lambda$-geodesically convex in the Wasserstein space $\mathcal{P}_{g}(M)$
(built upon the Riemannian distance induced by $g$) if and only if

$\operatorname{Ric}_{x}+\operatorname{Hess}_{x}W\geq\lambda,\hskip
28.45274pt\forall\,x\in M.$ (4.19)

Note now that the standard formula
$\operatorname{Vol}(\mathrm{d}x)=\sqrt{\operatorname{Det}g_{x}}\mathrm{d}x$
gives here
$\operatorname{Vol}(\mathrm{d}x)=\frac{1}{\sqrt{{\theta}(x)}}\mathrm{d}x$.
Changing reference measure
$q(\mathrm{d}x)=\sigma(x)\pi(\mathrm{d}x)=\rho(x)\operatorname{Vol}(\mathrm{d}x)$,
straightforward algebra shows that our entropy reads in the more intrinsically
Riemannian form

${\mathcal{H}}_{\pi}(q)=\mathcal{S}_{W}(q)\qquad\mbox{with}\qquad W\mbox{ as
in }\eqref{eq:def_V_W}.$

In our flat one-dimensional geometry we have of course
$\operatorname{Ric}_{x}\equiv 0$ hence it suffices to check
$\operatorname{Hess}W\geq\lambda$. In intrinsic Riemannian coordinates $y$ we
have
$\partial_{y}=\frac{1}{\sqrt{g}_{x}}\partial_{x}=\sqrt{{\theta}(x)}\partial_{x}$,
hence (4.16) is exactly $\lambda\coloneqq\min\operatorname{Hess}W$ expressed
in Euclidean coordinates. ∎

This proof is of course not acceptable because $(\bar{E},\mathsf{d}_{\theta})$
is not a smooth, complete Riemannian manifold. We shall bypasses this issue
thanks to a purely one-dimensional isometry argument.

###### Proof.

To begin with, let us briefly justify that (4.16) is indeed finite. Rewinding
the two successive steps (3.22)(4.14) relating the potentials
$U,\tilde{U},V,W$ gives $W=U+\log\left(\frac{\sqrt{\theta}}{\eta^{2}}\right)$.
Since $U$ is smooth up to the boundary and $\eta,{\theta}>0$ in the interior
this is a smooth function in the interior. On the left boundary $x\to 0$ (the
case $x\to 1$ is identical) we recall that $\eta,{\theta}$ vanish linearly.
Hence as $x\to 0$ we have $W\sim\log\left(x^{-3/2}\right),{\theta}(x)\sim Cx$
and therefore

$x\to
0:\qquad\sqrt{{\theta}}\frac{d}{dx}\left(\sqrt{{\theta}}\frac{d}{dx}W\right)\sim
C\sqrt{x}\frac{d}{dx}\left(\sqrt{x}\frac{d}{dx}\log\left(x^{-3/2}\right)\right)=\frac{C^{\prime}}{x}\to+\infty.$

(this formal differentiation can be made completely rigorous by expanding out
and keeping track of the leading terms). Thus this function blows-up to
$+\infty$ as $x\to 0^{+}$, and similarly for $x\to 1^{-}$. As a consequence
$\inf=\min$ is attained for a finite value at an interior point in (4.16).

In order to establish the more difficult displacement convexity, let
$D=\operatorname{diam}(\bar{E})=\mathsf{d}_{\theta}(0,1)$ and $F=(0,D)$. In
dimension $d=1$ the arclength map

$\begin{array}[]{crcl}i:&\bar{E}&\rightarrow&\bar{F}\\\
&x&\mapsto&\mathsf{d}_{\theta}(0,x)\end{array}$

gives an isometry between the metric spaces $(\bar{E},\mathsf{d}_{\theta})$
and $(\bar{F},\mathsf{d}_{\mathbb{R}})$, the latter being equipped with the
Euclidean distance. This induces in turn an isometry

$\begin{array}[]{crcl}I:&\mathcal{P}_{\theta}(\bar{E})&\rightarrow&\mathcal{P}_{\mathbb{R}}(\bar{F})\\\
&q&\mapsto&i_{\\#}q\end{array}$

between the Wasserstein spaces
$(\mathcal{P}(\bar{E}),{\mathcal{W}}_{{\theta}})$ and
$(\mathcal{P}(\bar{F}),{\mathcal{W}}_{\mathbb{R}})$. Moreover, by standard
properties of the pushforward operation, it is easy to check that, if

$\bar{q}\coloneqq I(q)\coloneqq
i_{\\#}q\qquad\mbox{and}\qquad\bar{\pi}\coloneqq I(\pi)\coloneqq i_{\\#}\pi,$

then the Radon-Nykodim derivative
$\bar{\sigma}(y)\coloneqq\frac{\mathrm{d}\bar{q}}{\mathrm{d}\bar{\pi}}(y)=\frac{\mathrm{d}q}{\mathrm{d}\pi}(i^{-1}(y))=\sigma(i^{-1}(y))$.
This gives

$\bar{{\mathcal{H}}}_{\bar{\pi}}(\bar{q})\coloneqq\int_{\bar{F}}\bar{\sigma}(y)\log\bar{\sigma}(y)\bar{\pi}(\mathrm{d}y)=\int_{\bar{E}}\sigma(x)\log\sigma(x)\pi(\mathrm{d}x)={\mathcal{H}}_{\pi}(q).$

Since $y=i(x)$ is a bijection so is $\bar{q}=I(q)$, and because the isometry
$I$ maps ${\mathcal{W}}_{\theta}$-geodesics to
${\mathcal{W}}_{\mathbb{R}}$-geodesics, clearly (4.17) will be established if
we can show that

$\bar{{\mathcal{H}}}_{\bar{\pi}}(\bar{q}_{t})\leq(1-t)\bar{{\mathcal{H}}}_{\bar{\pi}}(\bar{q}_{0})+t\bar{{\mathcal{H}}}_{\bar{\pi}}(\bar{q}_{1})-\lambda\frac{t(1-t)}{2}{\mathcal{W}}_{\mathbb{R}}^{2}(\bar{q}_{0},\bar{q}_{1}),\qquad
t\in[0,1]$

for all ${\mathcal{W}}_{\mathbb{R}}$-geodesic $(\bar{q}_{t})_{t\in[0,1]}$
joining arbitrary measures
$\bar{q}_{0},\bar{q}_{1}\in\mathcal{P}_{\mathbb{R}}(\bar{F})$. The main
advantage is that this is now written with respect to the Wasserstein distance
based upon $\bar{F}=[0,D]$ endowed with the smooth, complete, Euclidean
metric. This allows the completely rigorous application of [46, Theorem 1.3],
in the simplest of all settings since there is no Ricci curvature in dimension
$d=1$. More precisely, if $\bar{W}(y)\coloneqq cst-\log\bar{\pi}(y)$ is such
that

$\bar{\pi}(\mathrm{d}y)=\frac{1}{\mathcal{Z}}\exp(-\bar{W}(y))\mathrm{d}y\qquad\mbox{with}\qquad\bar{\mathcal{H}}_{\bar{\pi}}(\bar{q})=\int_{\bar{F}}\bar{q}(y)\log\bar{q}(y)\,\mathrm{d}y+\int_{\bar{F}}\bar{W}(y)\bar{q}(y)\,\mathrm{d}y-cst,$

the modulus of displacement-convexity $\lambda$ of
$\bar{{\mathcal{H}}}_{\bar{\pi}}$ (hence of ${\mathcal{H}}_{\pi}$) is given by
any modulus of (linear) convexity $\frac{d^{2}}{dy^{2}}\bar{W}(y)\geq\lambda$.
The point here is that $\mathrm{d}y$ is now really the volume form on
$\bar{F}$, and all the pre-isometry issues have now been encoded in the
explicit potential $\bar{W}$. By definition of the pushforward
$\bar{\pi}=i{}_{\\#}\pi$ it is easy to see that
$\bar{\pi}(y)=\frac{\pi}{|\partial_{x}i|}\circ
i^{-1}(y)=[\sqrt{{\theta}}\pi]\circ i^{-1}(y)$. In particular

$\bar{W}(y)=cst-\log\bar{\pi}(y)=cst-\log(\pi\sqrt{{\theta}})(i^{-1}(y))=cst-\log(\pi(i^{-1}(y)))-\frac{1}{2}\log{\theta}(i^{-1}(y)).$

With this new explicit formula at hand and $y=i(x)$,
$\frac{d}{dy}=\sqrt{{\theta}(x)}\frac{d}{dx}$, straightforward calculus
finally confirms that $\frac{d^{2}}{dy^{2}}\bar{W}(y)\geq\lambda$ is exactly
equivalent to (4.16). ∎

When attempting to naively investigate displacement convexity for the
unconditioned model, one would immediately face the potential
$\hat{U}(x)=U(x)+\log{\theta}(x)$ as in section 2. The effect of the
Shahshahani geometry would induce the very same
$-\log\operatorname{vol}(x)=-\frac{1}{2}\log{\theta}(x)$ compression (that
appeared in the previous proof by isometry), but any putative modulus of
displacement convexity would still be given in the end by the modulus of
Riemannian convexity of
$U+\log{\theta}-\frac{1}{2}\log{\theta}=U+\frac{1}{2}\log{\theta}$. The
$+\log{\theta}$ term blowing down to $-\infty$ on the boundary actually makes
this singularly concave as a function of $x$ (even in Riemannian coordinates),
and as a result the naive entropy from section 2 would not be
$\lambda$-displacement convex for any $\lambda\in\mathbb{R}$. This is yet
another hint that conditioning is really needed.

With convexity now being settled, let us turn to an equally fundamental
question, the practical computation of the metric slope w.r.t. the
${\mathcal{W}}_{\theta}$ metric.

######  Proposition 4.4.

Consider the relative Fisher information

$\mathcal{I}_{\pi}(q)\coloneqq\begin{cases}\int_{E}{\theta}(x)\left|\frac{\nabla\sigma}{\sigma}(x)\right|^{2}q(\mathrm{d}x)&\mbox{if
}q(\mathrm{d}x)=\sigma(x)\pi(\mathrm{d}x)\in\operatorname{Dom}(\mathcal{I}_{\pi}),\\\
+\infty&\mbox{otherwise}\end{cases}$ (4.20)

with domain

$\operatorname{Dom}(\mathcal{I}_{\pi})\coloneqq\left\\{q=\sigma\cdot\pi\in
D(\mathcal{H}_{\pi})\quad\mbox{s.t.}\quad\sigma\in
W^{1,1}_{\mathrm{\mathrm{loc}}}(E),\quad\frac{\nabla\sigma}{\sigma}\in
L^{2}({\theta}q)\right\\}.$ (4.21)

Then the local slope

$|\partial{\mathcal{H}}_{\pi}|_{{\mathcal{W}}_{\theta}}(q)=\sqrt{\mathcal{I}_{\pi}(q)}$
(4.22)

is narrowly lower semi-continuous and provides a strong upper gradient for
$\mathcal{H}_{\pi}$.

###### Proof.

As before we argue via isometry, and for any
$q\in\mathcal{P}_{\theta}(\bar{E})$ we write
$\bar{q}=i_{\\#}q\in\mathcal{P}_{\mathbb{R}}(\bar{F})$. From (4.5) clearly
isometry preserves the metric slope

$|\partial{\mathcal{H}}_{\pi}|_{{\mathcal{W}}_{\theta}}(q)=|\partial\bar{\mathcal{H}}_{\bar{\pi}}|_{{\mathcal{W}}_{\mathbb{R}}}(\bar{q})$

and it suffices to compute
$|\partial\bar{\mathcal{H}}_{\bar{\pi}}|_{{\mathcal{W}}_{\mathbb{R}}}(\bar{q})$.
In the standard, Euclidean $(\mathcal{P}(\bar{F}),{\mathcal{W}}_{\mathbb{R}})$
framework we can rigorously apply [2, Theorem 10.4.9] to conclude that

$|\partial\bar{\mathcal{H}}_{\bar{\pi}}|_{{\mathcal{W}}_{\mathbb{R}}}^{2}(\bar{q})=\int_{{F}}\left|\frac{\nabla_{y}\bar{\sigma}}{\bar{\sigma}}(y)\right|^{2}\,\bar{q}(\mathrm{d}y),\hskip
28.45274pt\bar{q}(\mathrm{d}y)=\bar{\sigma}(y)\cdot\bar{\pi}(\mathrm{d}y)$

with corresponding domain. Here of course we implicitly mean that
$\bar{q}\ll\bar{\pi}$ with density
$\bar{\sigma}(y)=\frac{\mathrm{d}\bar{q}}{\mathrm{d}\bar{\pi}}(y)$ belonging
to $W^{1,1}_{\mathrm{loc}}({F})$ and making the right-hand side integral
finite. Since $\bar{q}=i_{\\#}q$ and $\bar{\pi}=i_{\\#}\pi$ the corresponding
densities are again related through

$\bar{\sigma}(y)=\sigma(i^{-1}(y)).$

In particular, since $i$ and $i^{-1}$ are locally smooth, $\bar{\sigma}\in
W^{1,1}_{\mathrm{loc}}({F})$ if and only if $\sigma\in
W^{1,1}_{\mathrm{loc}}(E)$. Moreover, owing to
$i(x)=\mathsf{d}_{\theta}(0,x)=\int_{0}^{x}\frac{1}{\sqrt{{\theta}(z)}}\mathrm{d}z$
hence $\frac{di}{dx}=\frac{1}{\sqrt{{\theta}(x)}}$, we get

$\nabla_{y}\bar{\sigma}(y)=\frac{di^{-1}}{dy}(y)\nabla_{x}\sigma(i^{-1}(y))=\frac{1}{\frac{di}{dx}(i^{-1}(y))}\nabla_{x}\sigma(i^{-1}(y))=\sqrt{{\theta}(i^{-1}(y))}\nabla_{x}\sigma(i^{-1}(y)).$

By definition of the pushforward $\bar{q}=i_{\\#}q$ this gives

$|\partial{\mathcal{H}}_{\pi}|_{{\mathcal{W}}_{\theta}}^{2}(q)=|\partial\bar{\mathcal{H}}_{\bar{\pi}}|_{{\mathcal{W}}_{\mathbb{R}}}^{2}(\bar{q})=\int_{{F}}\left|\frac{\sqrt{{\theta}}\nabla_{x}\sigma}{\sigma}(i^{-1}(y))\right|^{2}\,\bar{q}(\mathrm{d}y)=\int_{E}{\theta}(x)\left|\frac{\nabla_{x}\sigma}{\sigma}(x)\right|^{2}\,q(\mathrm{d}x)$

as in our statement.

In order to conclude the proof, we finally recall from Proposition 4.3 that
${\mathcal{H}}_{\pi}$ is $\lambda$-geodesically convex: [2, Corollary 2.4.10]
guarantees that the local slope
$|\partial{\mathcal{H}}_{\pi}|_{{\mathcal{W}}_{\theta}}$ coincides with the
_global slope_ , and as such it is automatically lower-semicontinuous and a
_strong_ -upper gradient. ∎

##  5 The conditioned gradient flow

The goal of this section is to identify the (smooth) solutions $q_{t}(x)$ of
the conditioned Fokker-Planck equation (3.26), constructed in Proposition 3.6
by purely PDE techniques, as $\mathsf{EVI}$ solutions of the abstract gradient
flow

$``\frac{d}{dt}q_{t}=-\operatorname{grad}_{{\mathcal{W}}_{\theta}}{\mathcal{H}}_{\pi}(q_{t})".$

Let us first recall that the potential $V$ from (4.14) was defined so that

$\pi(\mathrm{d}x)=\frac{1}{\mathcal{Z}}e^{-V(x)}\mathrm{d}x\qquad\mbox{and}\qquad\partial_{t}q_{t}=\operatorname{div}\left(q_{t}{\theta}\nabla\log\left(\frac{q_{t}}{\pi}\right)\right).$

Writing as before

$\sigma_{t}(x)\coloneqq\frac{q_{t}(x)}{\pi(x)}=\frac{dq_{t}}{d\pi}(x)$

for the density w.r.t. our stationary measure, we can recast the PDE as the
continuity equation

$\partial_{t}q_{t}+\operatorname{div}(q_{t}v_{t})=0\qquad\mbox{with}\qquad
v_{t}(x)=-{\theta}(x)\nabla\log\sigma_{t}(x).$

This resembles both the Fisher information (4.20) and the metric speed from
Theorem 5. Indeed we have:

######  Proposition 5.1.

Let $q_{t}(x)$ be the smooth solution from Proposition 3.6 and
$\sigma_{t}(x)=\frac{q_{t}(x)}{\pi(x)}$. Then for all $T>t_{0}>0$ there exists
$C=C(t_{0},T)>0$ such that

$|\nabla\log\sigma_{t}(x)|=\left|\frac{\nabla\sigma_{t}(x)}{\sigma_{t}(x)}\right|\leq\frac{C}{\pi(x)},\hskip
28.45274pt\forall x\in E$ (5.1)

and

$|q^{\prime}|^{2}_{{\mathcal{W}}_{\theta}}(t)\leq\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}=\int_{E}\frac{|v_{t}|^{2}}{{\theta}}q_{t}=\int_{E}{\theta}\left|\frac{\nabla\sigma_{t}}{\sigma_{t}}\right|^{2}q_{t}=|\partial{\mathcal{H}}_{\pi}|^{2}(q_{t})\leq
C$ (5.2)

for all $t\in[t_{0},T]$.

###### Proof.

Fix any $0<t_{0}<T$. We recall from Proposition 3.6 that $q_{t}(x),\pi(x)>0$
are smooth up to the boundary, vanish linearly, and that
$0<c_{0}\leq\frac{q_{t}(x)}{\pi(x)}=\sigma_{t}(x)\leq C_{0}$ for $t\geq
t_{0}$. In particular $\sigma\in C([t_{0},+\infty)\times\bar{E})\cap
C^{\infty}([t_{0},\infty)\times E)$, and the chain rule
$\nabla\log\sigma_{t}=\frac{\nabla\sigma_{t}}{\sigma_{t}}$ is completely
rigorous at least in the interior. Recall moreover that $\pi\in
C^{\infty}(\bar{E})$ and that $q_{t}\in
C^{\infty}(\mathbb{R}^{+}\times\bar{E})$. As a consequence $|\nabla
q_{t}(x)|+|\nabla\pi(x)|\leq C$ for any $(t,x)\in[t_{0},T]\times\bar{E}$.
Writing $\nabla\sigma_{t}=\nabla\left(\frac{q_{t}}{\pi}\right)=\frac{\pi\nabla
q_{t}-q_{t}\nabla\pi}{\pi^{2}}$ and recalling that $q_{t}\asymp\pi$ are
comparable up to the boundary, we see that
$|\nabla\sigma_{t}(X)|\leq\frac{C}{\pi(x)}$ and our bound (5.1) for
$\nabla\log\sigma_{t}=\frac{\nabla\sigma_{t}}{\sigma_{t}}$ follows from the
lower bound $\sigma_{t}(x)\geq c_{0}>0$.

Let us now focus on (5.2) and relate $\nabla\log\sigma$ to the metric speed as
in Proposition 4.4. Since $\sigma_{t}(x)$ is smooth and positive in the
interior we have of course $\frac{\nabla\sigma_{t}}{\sigma_{t}}\in
W^{1,1}_{\mathrm{loc}}(E)$. In order to check the upper bound in (5.2) we
simply use the previous estimate on $\nabla\log\sigma$ to get

$|\partial{\mathcal{H}}_{\pi}|^{2}(q_{t})=\int_{E}{\theta}\left|\frac{\nabla\sigma_{t}}{\sigma_{t}}\right|^{2}q_{t}=\int_{E}{\theta}\left|\nabla\log\sigma_{t}\right|^{2}q_{t}\leq\int_{E}{\theta}\frac{C}{\pi^{2}}q_{t}\leq
C,$

where the last inequality follows again from (3.27) ($q_{t},\pi,{\theta}$
being all comparable one to another). This also entails

$\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}=\int_{E}\frac{|v_{t}|^{2}}{{\theta}}q_{t}=\int_{E}{\theta}\left|\frac{\nabla\sigma_{t}}{\sigma_{t}}\right|^{2}q_{t}\leq
C$

locally uniformly in time $t\in[t_{0},T]$. Our estimate (5.1) also controls

$|q_{t}(x)v_{t}(x)|=\left|q_{t}(x){\theta}(x)\nabla\log\sigma_{t}(x)\right|=\mathcal{O}\left(\pi^{2}(x)\frac{1}{\pi(x)}\right)\xrightarrow[x\to\partial
E]{}0$ (5.3)

locally uniformly in time. As a consequence the flux
$q_{t}v_{t}\cdot\nu|_{\partial E}=0$ vanishes in the strong (continuous)
sense, and a fortiori $(q,v)$ solve the continuity equation in the sense of
Definition 4.1. Our characterization of absolutely continuous curves from
Theorem 5 finally guarantees that $q\in
AC^{2}([t_{0},T];\mathcal{P}_{{\mathcal{W}}_{\theta}})$ with
$|q^{\prime}|^{2}_{{\mathcal{W}}_{t}}(t)\leq\|v_{t}\|^{2}_{L^{2}_{\theta}(q_{t})}\leq
C$ and the proof is complete. ∎

######  Corollary 5.2.

The solution $q$ from Proposition 3.6 is an $\mathsf{EDI}_{t_{0}}$ solution
for any $t_{0}>0$.

###### Proof.

We will actually establish a stronger differential version of ($EDI_{t_{0}}$).
For this we simply use the PDE to compute, for $t\geq t_{0}>0$,

$\frac{d}{dt}{\mathcal{H}}_{\pi}(q_{t})=\frac{d}{dt}\int_{E}\sigma_{t}\log\sigma_{t}\,\pi=\int_{E}(1+\log\sigma_{t})\partial_{t}\sigma_{t}\,\pi=\int_{E}(1+\log\sigma_{t})\partial_{t}q_{t}\\\
=\int_{E}(1+\log\sigma_{t})\operatorname{div}\left({\theta}q_{t}\nabla\log\sigma_{t}\right)=-\int_{E}{\theta}|\nabla\log\sigma_{t}|^{2}\,q_{t}.$

The first differentiation w.r.t. time under the integral sign is completely
rigorous since $q,\sigma$ are smooth for $t>0$, and $0<c\leq\sigma_{t}(x)\leq
C$ from (3.27). Our last integration by parts in space is also legitimate,
since as already observed from (5.3) $q_{t}{\theta}\nabla\log\sigma_{t}=0$ in
the boundary term

$\int_{\partial
E}(1+\log\sigma_{t}){\theta}q_{t}\nabla\log\sigma_{t}\cdot\nu=0$

($\log\sigma$ being bounded uniformly in $[t_{0},T]\times\bar{E}$). Finally
leveraging (5.2) from the previous Proposition, this yields

$\frac{d}{dt}{\mathcal{H}}_{\pi}(q_{t})=-\frac{1}{2}\int_{E}{\theta}|\nabla\log\sigma_{t}|^{2}q_{t}-\frac{1}{2}\int_{E}{\theta}|\nabla\log\sigma_{t}|^{2}q_{t}\\\
=-\frac{1}{2}\underbrace{\int_{E}\frac{1}{{\theta}}|v_{t}|^{2}q_{t}}_{\geq|q^{\prime}|^{2}_{{\mathcal{W}}_{\theta}}(t)}-\frac{1}{2}\underbrace{\int_{E}{\theta}\left|\frac{\nabla\sigma_{t}}{\sigma_{t}}\right|^{2}q_{t}}_{=|\partial{\mathcal{H}}_{\pi}|^{2}(q_{t})}\leq-\frac{1}{2}|q^{\prime}|^{2}_{{\mathcal{W}}_{\theta}}(t)-\frac{1}{2}|\partial{\mathcal{H}}_{\pi}|^{2}(q_{t}).$

and ($EDI_{t_{0}}$) immediately follows. ∎

We are now in position of proving our main result.

######  Theorem 6.

Let $\lambda\in\mathbb{R}$ be as in (4.16). Then

  1. 1.

${\mathcal{H}}_{\pi}$ generates an $\mathsf{EVI}_{\lambda}$-gradient flow
$(\mathsf{S}_{t})_{t\geq 0}$ on
$\mathcal{P}_{\theta}(\bar{E})=\overline{\operatorname{Dom}({\mathcal{H}}_{\pi})}$

  2. 2.

For any $q_{0}\in\mathcal{P}(\bar{E})$ emanating from some
$p_{0}\in\mathcal{P}(\bar{E})$ as in (3.13), the solution $q_{t}(x)$ from
Proposition 3.6 coincides with the $\mathsf{EVI}$ solution, i-e

$q_{t}=\mathsf{S}_{t}q_{0},\hskip 28.45274ptt\geq 0.$

  3. 3.

For any $q_{0}^{1},q_{0}^{2}\in\mathcal{P}(\bar{E})$ we have contraction in
the Wasserstein-Shahshahani distance,

${\mathcal{W}}_{\theta}(q^{1}_{t},q^{2}_{t})\leq e^{-\lambda
t}{\mathcal{W}}_{\theta}(q^{1}_{0},q^{2}_{0}),\hskip 28.45274ptt\geq 0.$

  4. 4.

Long time behaviour and logarithmic Sobolev inequality

    1. (a)

If $\lambda>0$ then

$\int_{E}\sigma\log\sigma\,\pi={\mathcal{H}}_{\pi}(q)\leq\frac{1}{2\lambda}\mathcal{I}_{\pi}(q)=\frac{1}{2\lambda}\int_{E}{\theta}\left|\frac{\nabla\sigma}{\sigma}\right|^{2}q,$
($LSI_{\lambda}$)

for all $q=\sigma\cdot\pi\in D(\mathcal{H}_{\pi})$, with moreover

${\mathcal{W}}_{\theta}(q_{t},\pi)\leq e^{-\lambda
t}{\mathcal{W}}_{\theta}(q_{0},\pi)\qquad\mbox{and}\qquad\|q_{t}-\pi\|_{\mathrm{TV}}\leq
e^{-\lambda t}\sqrt{2{\mathcal{H}}_{\pi}(q_{0})}$ (5.4)

for all $t\geq 0$. In particular $q_{t}$ converges exponentially fast both in
the Wasserstein distance and in total variation to the stationary distribution
(provided ${\mathcal{H}}_{\pi}(q_{0})>0$, but otherwise the argument can be
applied starting from ${\mathcal{H}}_{\pi}(q_{t_{0}})<\infty$ for small
$t_{0}>0$).

    2. (b)

If $\lambda=0$ then

${\mathcal{W}}_{\theta}(q_{t},\pi)\xrightarrow[t\to\infty]{}0.$

In our opinion the main point here is really 2: Although the unconditioned
process is _not_ variational (as discussed in Section 2), the $Q$-process
automatically inherits after conditioning a (rigorous) variational structure,
once the right geometry is used (that of ${\mathcal{W}}_{\theta}$). This
significantly strengthens previous results from [8], where the Wasserstein-
Shahshahani gradient flow structure was identified only formally for the
original Kimura PDE (3.5) satisfied by $p$ and without any real probabilistic
interpretation of the change of variables from $p$ to $q$.

It is not clear what the biological interpretation might be for initial data
$q_{0}$ in 1 at least partially supported on the boundary. Indeed, in our
probabilistic construction (3.13) prevented $q_{0}$ from having atoms on the
boundary (and rightly so, since $q_{0}$ precisely represents $p_{0}$
conditioned to non absorption). We still chose to include this possibility for
the sake of generality.

As regards ($LSI_{\lambda}$), we point out that very similar weighted
logarithmic Sobolev inequalities were obtained independently in [21] from a
purely PDE and less geometric point of view (their proof is actually a hands-
on change of variables, corresponding to our isometry argument in Section
4.3).

###### Proof.

  1. 1.

We already proved in Proposition 4.3 that ${\mathcal{H}}_{\pi}$ is
$\lambda$-geodesically convex. Since in dimension $d=1$ there is no Ricci
curvature, it is well-known that
$(\mathcal{P}_{\theta}(\bar{E}),{\mathcal{W}}_{\theta})$ is a non-positively
curved $\mathsf{CAT}(0)$ space in the Alexandrov sense. As a consequence the
generation of the $\mathsf{EVI}_{\lambda}$-flow can be found verbatim in [38,
Theorem 3.14], see also [2, Theorem 4.0.4] (in which Assumption 4.0.1 holds
along geodesics in dimension 1 only).

  2. 2.

We just established in Corollary 5.2 that, if $q_{0}$ originates from some
$p_{0}$ in the sense of (3.13), the solution $q_{t}(x)$ obtained by pure PDE
methods is actually an $\mathsf{EVI}_{t_{0}}$-solution for any $t_{0}>0$.
Since $q_{t}\to q_{0}$ narrowly (Proposition 3.6), and given that we just
proved that ${\mathcal{H}}_{\pi}$ generates an $\mathsf{EVI}_{\lambda}$-flow
on the whole $\mathcal{P}_{\theta}(\bar{E})$, we can simply appeal to Theorem
4 and conclude that $q_{t}=\mathsf{S}_{t}q_{0}$.

  3. 3.

This is one of the classical properties of $\mathsf{EVI}$ solutions in Theorem
3.

  4. 4.

For $\lambda>0$ the statement (in particular the exponential convergence and
the logarithmic Sobolev inequality) can be found e.g. in [38, Theorem 3.5] and
[2, Lemma 2.4.13] (Assumption 2.4.5 and Assumption 4.0.1 therein are both
satisfied in our one-dimensional $\mathsf{CAT}(0)$ setting). For the
asymptotic convergence for $\lambda=0$ the result can be found in [2,
Corollary 4.0.6]. The convergence in Total variation comes from the celebrated
Csiszár-Kullback-Pinsker inequality
$\|q-\pi\|_{\mathrm{TV}}\leq\sqrt{2{\mathcal{H}}_{\pi}(q)}$ as well as the
decay in entropy ${\mathcal{H}}_{\pi}(q_{t})\leq e^{-2\lambda
t}{\mathcal{H}}_{\pi}(q_{0})$.

∎

Let us finish with a brief discussion on convergence rates. Note first that no
explicit rate of convergence can be derived for $\lambda=0$. In the better
case $\lambda>0$, one may wonder what additional information (5.4) brings,
compared to any possible spectral analysis one may try carrying out directly
for $\tilde{\mathcal{L}}$. From (3.17) clearly the spectrum of
$\tilde{\mathcal{L}}$ is shifted by $-\lambda_{0}$ w.r.t. that of
${\mathcal{L}}$, and one should expect that a spectral analysis for
$\tilde{\mathcal{L}}$ leads to exponential convergence $q_{t}\to\pi$ with rate
$\lambda_{1}-\lambda_{0}>0$ (see also the proof of Theorem 1 for a hint in
this direction). However, already in the Euclidean case it is known that the
modulus of displacement convexity $\lambda<\lambda_{1}-\lambda_{0}$ predicts a
slower convergence rate. In other words, our analysis should predict
suboptimal rates with respect to more direct spectral considerations. This can
be confirmed in the prototypical case $U\equiv 0$ (neutral genetic preference)
with ${\theta}(x)=x(1-x)$, where one can check that $\alpha_{0}(x)=1$ and
$\alpha_{1}(x)=x-\frac{1}{2}$ are the first eigenfunctions of
$\Delta(x(1-x)\,\cdot\,)$ with eigenvalues $\lambda_{0}=6,\lambda_{1}=2$ and
gap $\lambda_{1}-\lambda_{0}=4$, while our best predicted rate in this case
can be tracked down to $\lambda=3$ by explicitly evaluating (4.16). We are not
aware of a general statement that $\lambda<\lambda_{1}-\lambda_{0}$ always, in
particular the ${\theta}$-degeneracy in our framework does not fit the usual
irreducible setting and might induce new behaviours. Despite this
suboptimality, one possible advantage of our approach is that the long-time
convergence towards the stationary distribution $q_{t}\to\pi$ can be
explicitly quantified by accessing only the lowest spectral level
$\lambda_{0}$ of the unconditioned operator ${\mathcal{L}}$, which may be
interesting for applications in biology.

## Acknowledgments

J.-B.C. was supported by FCT - Fundação para a Ciência e a Tecnologia, under
the project UIDB/04561/2020 L.M. was funded by the Portuguese Science
Foundation through a personal grant 2020/00162/CEECIND as well as the FCT
project PTDC/MAT-STA/28812/2017, and wishes to thank F. Chalub and N.
Champagnat for fruitful discussions.

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