Growth-fragmentation and quasi-stationary methods

Denis Villemonais

Alex Watson

17 September 2021

A model of growth-fragmentation

(Diagram of cell division. Left to right: a single cell of size x(0) = x. Cell grows to x(t), with arrow x.(t) = tau(x(t)). Cell divides to cells of size px(t), (1-p)x(t), with label 'rate B(x(t))' and
'chosen by kappa(x(t),dp)'. The cell of size px(t) continues to grow and divide. The cell of size (1-p)x(t) dies, with label 'rate D(.)'.)

  • • List sizes at time \(t\): \(\Zb (t) = (Z_u(t): u \in U)\)

Equilibrium behaviour

(-tikz- Diagram of simulated cell division. Top row, left to right: single cell; arrow; collection of coloured cells with 't=4.71, 100 cells'; arrow; collection of cells with 't=6.32, 500 cells';
arrow; collection of cells with 't=7.02, 1000 cells'. Middle row: histograms beneath the cell collections, with overlaid density estimate. The histograms appear to be converging loosely to a density. Bottom left:
plot of log <Z(t),1> against t, roughly linear, with linear fit line overplotted. Bottom right: '<Z(t),f> ~ C exp(lambda t)<nu, f>'. )

Mean measures

Look at \(T_tf(x) = \EE _x\left [\sum _u f(Z_u(t))\right ]\) (formally)

\begin{align*} \partial _t T_tf(x) &= T_t \symcal {A}f(x) \\ \symcal {A}f(x) &= \tau (x)f'(x) + \int _0^x f(y) \, k(x,\mathrm {d}y) - K(x)f(x), \quad \text { for suitable } f \end{align*} ...where \(k(x,\dd y) = 2B(x)\frac {\kappa (x,\dd y) + \kappa (x,x-\dd y)}{2}\), and \(K(x) = B(x)+D(x)\).

Questions

  • • Existence and uniqueness of such \(T_t\)? (For which coefficients; for which \(f\)?)

  • • Long term behaviour: \(T_tf(x) \sim e^{\lambda t} h(x) \int f(y)\nu (\dd y)\)? Rate?

Existing approaches

  • • Spectral: find \(\Aa h = \lambda h\), \(\nu \Aa = \lambda \nu \) and use entropy methods

  • • When \(h\) is known, make connection with an Markov process and use its stationary distribution

  • • ‘Harris-type theorem for non-conservative semigroups’: Lyapunov function approach, Bansaye et al. (2019+)

Our approach

  • • Try to link to a killed Markov process

  • • Study the quasi-stationary distribution (QSD) (‘stationary after conditioning on survival’)

  • • Find conditions for existence of the process and its QSD, and link back to desired semigroup \(T\)

1 Existence and uniqueness

Finding a killed Markov process spine

  • • Fix \(\alpha ,\beta \in \RR \) and let

    \[ V(x) = \exp \left ( -\symbb {1}_{\{x \le 1\}} \alpha \int _x^1 \protect \frac {\mathrm {d}y}{\tau (y)} + \symbb {1}_{\{x > 1\}} \beta \int _1^x \protect \frac {\mathrm {d}y}{\tau (y)} \right ) \]

  • • Let \(\Ll f = \frac {1}{V} \Aa (fV) - bf\) where \(b=\sup _{x>0}\bigl ( \frac {1}{V(x)} \Aa V(x)\bigr )\)

  • • \(\Ll 1 \le 0\); it generates a killed Markov process

Finding a killed Markov process spine

  • • \(\Ll f(x) = \tau (x) f'(x) + \int _0^x \bigl [ f(y)-f(x) \bigr ] k_V(x,\dd y) - q(x)f(x)\),

  • • ...where \(\tau \) is growth rate, \(k_V\) is jump rate, \(q\) is killing rate, and...

  • • ...where \(k_V(x,\dd y) = \frac {V(y)}{V(x)} k(x,\dd y)\)

  • • \(e^{-bt} \frac {1}{V(x)} T_t (fV)(x) = \EE _x[f(X_t)]\)

(A plot representing the evolution of cell sizes. It starts with a single line, increasing, which bifurcates. The two new lines have the same behaviour which is repeated a few times. Some of the subsequent
lines stop with an 'x'. One line is drawn bold throughout. The initial line is bold; at its jump time, the first of the two new lines is bold; at this line's jump time, the second of the new lines is bold. At the
first jump time, the initial position of the first new line is y1, and that of the second is y2. The time is labelled with 'P = V(y1)/(V(y1)+V(y2))'.)

Lemma

Assume, for all \(M>0\),

\[ \sup _{x\in (0,M)} k_V(x,(0,x]) < \infty \qquad \text {and} \qquad \limsup _{x\to \infty } \protect \bigl [ k_V(x,(0,x]) - K(x) \protect \bigr ] < \infty . \]

Then there is a Markov process \(X\) on \(E = (0,\infty )\cup \{\partial \}\) with

\begin{align*} Q_tf(x) &\coloneqq \symbb {E}_x[f(X_t)] = f(x) + \int _0^t \symbb {E}_{x}[\symcal {L}f(X_s)]\,\mathrm {d}s \\ \symcal {L}f(x) &= \tau (x) f'(x) + \int _0^x \protect \bigl [ f(y) - f(x) \protect \bigr ] k_V(x,\mathrm {d}y) + \protect \bigl [ f(\partial ) - f(x) \protect \bigr ] q(x), \quad \symcal {L}f(\partial ) = 0, \end{align*} for \(f\colon E \to \RR \) such that \(f\vert _{(0,\infty )}\) compactly supported and suitably differentiable.

Moreover, \(Q\) is the unique semigroup with these properties.

Proof ideas

  • • Construction: follow the ODE \(\dot {x}(t) = \tau (x(t))\), jump at rate \(k_V\), follow ODE from new position...

  • • Show no accumulation of jumps: uses \(\sup _{x\in (0,M)}k_V(x,(0,x]) < \infty \), no build-up of jumps toward zero

  • • A bit of legwork yields \(X\), unique solution of martingale problem

  • • Most difficult part: uniqueness of the semigroup

    • – Show any solution does not approach \(\infty \) or \(0\) (supermartingale argument)

    • – Compare solutions with solutions of martingale problem (a priori not necessarily the same!)

Theorem

Let

\begin{align*} \symcal {A}f(x) &= \tau (x) f'(x) + \int _0^x f(y) k(x,\mathrm {d}y) - K(x)f(x) \\ \symcal {D}(\symcal {A}) &= \{ f \colon (0,\infty ) \to \symbb {R}\ \text {suitably differentiable, compactly supported} \} \cup \{ V \}. \end{align*}

Then there exists a unique semigroup \(T\) such that

\[ \partial _t T_t f(x) = T_t\symcal {A}f(x), \quad f \in \symcal {D}(\symcal {A}), \]

and

\[T_t f(x) = e^{bt} V(x) \symbb {E}_x[f(X_t)/V(X_t)]. \]

‘Unbias the spine motion and add the branching back in’.

2 Long-term behaviour

Quasi-stationary distributions

  • • If \(X\) is a Markov process killed at \(T_\partial \), Champagnat and Villemonais (2018+) give criteria for

    \[ \symbb {P}_x(X_t \in \mathrm {d}y \mid T_\partial > t) \to \nu ^X(\mathrm {d}y), \]

    at exponential rate.

  • • \(\nu ^X\) is the quasi-stationary distribution.

  • • \(X\) is killed at random rate, our \(T\) has branching at random rate...

Theorem

In addition to our assumption about \(k_V\), assume

\[ \int _0^\infty \symbb {1}_{\{k(y,(0,x]) > 0\}}\,\mathrm {d}y > 0, \quad \text {for } x > 0, \]

that there is a measure \(\mu \) and a nonempty interval \(I\) with

\[ k(x,\cdot ) \ge \mu , \quad \text {for } x\in I, \]

and the existence of Lyapunov functions \(\psi ,\phi \) such that

\begin{align*} \symcal {A}\psi (x) &\le \lambda _1 \psi (x) + C\symbb {1}_{L}(x), \\ \symcal {A}\phi (x) &\ge \lambda _2 \phi (x), \end{align*} with \(\lambda _2 < \lambda _1\) and \(L\) compact, (plus boundary behaviour).

Then...

Theorem

...there exist \(\lambda \in \RR \), \(\nu \) a measure, \(h\) a function and \(\gamma >0\), such that

\[ \left \lVert e^{-\lambda t} T_t f(x) - h(x) {\textstyle \int } f\mathrm {d}\nu \right \rVert _{TV} \le C e^{-\gamma t} \psi (x) \]

with \(T_t h = e^{\lambda t} h\) and \(\nu T_t = e^{\lambda t} \nu \).

(Collation of images from earlier slides. There is a population of coloured cells of various sizes, a histogram of the sizes with overlaid density, and a scatter plot showing linear relationship between
log <Z(t),1> and t.)

   

“\(\EE \Zb (t) \sim e^{\lambda t} h(x) \nu \)”

OK, but can you actually prove anything?

  • • Assume \(\int f(y) k(x,\dd y) = K(x) \int f(xr) p(\dd r)\) (‘self-similarity’), \(\int r p(\dd r) = 1\) (conservation of mass), \(\int _0^1 \frac {\dd y}{\tau (y)} <\infty \) (entrance from mass \(0\))

  • • Can take \(\phi (x) = x\), then \(\Aa \phi (x) = \frac {\tau (x)}{x} \phi (x)\)

  • • Can take \(\psi (x) = V(x)\) and put \(\alpha = 0\)

  • • Very specific coefficients: if

    • – \(p(\dd r) = 2\dd r\) (uniform binary repartition of mass),

    • – \(\tau (x) = O(x)\) as \(x\to \infty \),

    • – and \((3+\sqrt {8})\limsup _{x\to \infty } \frac {\tau (x)}{x} < \liminf _{x\to \infty } K(x)\),

    then result holds.

3 Perspectives

Perspectives: computation

  • • Fleming-Viot process (Pierre Del Moral’s talk)

  • • How to find \(h\) and \(\lambda \)?

    • – Analogy with Bertoin and Watson (2018) suggests that if

      \[ L(p) = \symbb {E}_x e^{\int _0^{T_x} (p-q(X_s))\, \mathrm {d}s}, \]

      where \(T_x\) is hitting time of \(x\), then \(\lambda -b\) is unique solution to \(L(p) = 1\)

    • – The naive Monte Carlo estimator (Cornett 2021) has very high variance

    • – How to handle this?

Perspectives: extensions

  • • Say something about \(\Zb (t)\) itself (as \(t\to \infty \))

    • – Bertoin and Watson (2020): more restrictive conditions

    • – Horton and Watson (2021+): perturbed Lévy-type coefficients

  • • Replace deterministic growth with diffusion

    • – Existence and uniqueness get easier!

    • – Need to handle behaviour at zero carefully...

    • – cf. Laurençot and Walker (2021)

Perspectives: related models – typed cells

  • • Old and new pole cells – Cloez, da Saporta and Roget (2020+)

    • – At division, one daughter cell is ‘old’, one is ‘new’ (after E. coli)

    • – Type influences growth rate

    • – It is preferable (for \(\lambda \)) to have distinct growth rates for old and new cells

    • – Could one approach this via spine and optimal control of Lévy-type processes?

  • • Parasite branching process inside a growth-fragmentation – Marguet and Smadi (2020+)

    • – Embed CSBP (parasite population) and divide it when cell divides (growth-fragmentation)

  • • Spatially dependent fragmentation process – Callegaro and Roberts (2021+)

Further reading

  • [1]  D. Villemonais and A. R. Watson Asymptotic behaviour of growth-fragmentations via quasi-stationarity of the spine In preparation (working title)

Thank you!