\(\newcommand{\footnotename}{footnote}\)
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\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
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\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
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\(\let \symbbit \mathbb \)
\(\let \symcal \mathcal \)
\(\let \symscr \mathscr \)
\(\let \symfrak \mathfrak \)
\(\let \symsfup \mathsf \)
\(\let \symsfit \mathit \)
\(\let \symbfsf \mathbf \)
\(\let \symbfup \mathbf \)
\(\newcommand {\symbfit }[1]{\boldsymbol {#1}}\)
\(\let \symbfcal \mathcal \)
\(\let \symbfscr \mathscr \)
\(\let \symbffrak \mathfrak \)
\(\let \symbfsfup \mathbf \)
\(\newcommand {\symbfsfit }[1]{\boldsymbol {#1}}\)
\(\let \symup \mathrm \)
\(\let \symbf \mathbf \)
\(\let \symit \mathit \)
\(\let \symtt \mathtt \)
\(\let \symbffrac \mathbffrac \)
\(\newcommand {\mathfence }[1]{\mathord {#1}}\)
\(\newcommand {\mathover }[1]{#1}\)
\(\newcommand {\mathunder }[1]{#1}\)
\(\newcommand {\mathaccent }[1]{#1}\)
\(\newcommand {\mathbotaccent }[1]{#1}\)
\(\newcommand {\mathalpha }[1]{\mathord {#1}}\)
\(\def\upAlpha{\unicode{x0391}}\)
\(\def\upBeta{\unicode{x0392}}\)
\(\def\upGamma{\unicode{x0393}}\)
\(\def\upDigamma{\unicode{x03DC}}\)
\(\def\upDelta{\unicode{x0394}}\)
\(\def\upEpsilon{\unicode{x0395}}\)
\(\def\upZeta{\unicode{x0396}}\)
\(\def\upEta{\unicode{x0397}}\)
\(\def\upTheta{\unicode{x0398}}\)
\(\def\upVartheta{\unicode{x03F4}}\)
\(\def\upIota{\unicode{x0399}}\)
\(\def\upKappa{\unicode{x039A}}\)
\(\def\upLambda{\unicode{x039B}}\)
\(\def\upMu{\unicode{x039C}}\)
\(\def\upNu{\unicode{x039D}}\)
\(\def\upXi{\unicode{x039E}}\)
\(\def\upOmicron{\unicode{x039F}}\)
\(\def\upPi{\unicode{x03A0}}\)
\(\def\upVarpi{\unicode{x03D6}}\)
\(\def\upRho{\unicode{x03A1}}\)
\(\def\upSigma{\unicode{x03A3}}\)
\(\def\upTau{\unicode{x03A4}}\)
\(\def\upUpsilon{\unicode{x03A5}}\)
\(\def\upPhi{\unicode{x03A6}}\)
\(\def\upChi{\unicode{x03A7}}\)
\(\def\upPsi{\unicode{x03A8}}\)
\(\def\upOmega{\unicode{x03A9}}\)
\(\def\itAlpha{\unicode{x1D6E2}}\)
\(\def\itBeta{\unicode{x1D6E3}}\)
\(\def\itGamma{\unicode{x1D6E4}}\)
\(\def\itDigamma{\mathit{\unicode{x03DC}}}\)
\(\def\itDelta{\unicode{x1D6E5}}\)
\(\def\itEpsilon{\unicode{x1D6E6}}\)
\(\def\itZeta{\unicode{x1D6E7}}\)
\(\def\itEta{\unicode{x1D6E8}}\)
\(\def\itTheta{\unicode{x1D6E9}}\)
\(\def\itVartheta{\unicode{x1D6F3}}\)
\(\def\itIota{\unicode{x1D6EA}}\)
\(\def\itKappa{\unicode{x1D6EB}}\)
\(\def\itLambda{\unicode{x1D6EC}}\)
\(\def\itMu{\unicode{x1D6ED}}\)
\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
\(\def\itPi{\unicode{x1D6F1}}\)
\(\def\itRho{\unicode{x1D6F2}}\)
\(\def\itSigma{\unicode{x1D6F4}}\)
\(\def\itTau{\unicode{x1D6F5}}\)
\(\def\itUpsilon{\unicode{x1D6F6}}\)
\(\def\itPhi{\unicode{x1D6F7}}\)
\(\def\itChi{\unicode{x1D6F8}}\)
\(\def\itPsi{\unicode{x1D6F9}}\)
\(\def\itOmega{\unicode{x1D6FA}}\)
\(\def\upalpha{\unicode{x03B1}}\)
\(\def\upbeta{\unicode{x03B2}}\)
\(\def\upvarbeta{\unicode{x03D0}}\)
\(\def\upgamma{\unicode{x03B3}}\)
\(\def\updigamma{\unicode{x03DD}}\)
\(\def\updelta{\unicode{x03B4}}\)
\(\def\upepsilon{\unicode{x03F5}}\)
\(\def\upvarepsilon{\unicode{x03B5}}\)
\(\def\upzeta{\unicode{x03B6}}\)
\(\def\upeta{\unicode{x03B7}}\)
\(\def\uptheta{\unicode{x03B8}}\)
\(\def\upvartheta{\unicode{x03D1}}\)
\(\def\upiota{\unicode{x03B9}}\)
\(\def\upkappa{\unicode{x03BA}}\)
\(\def\upvarkappa{\unicode{x03F0}}\)
\(\def\uplambda{\unicode{x03BB}}\)
\(\def\upmu{\unicode{x03BC}}\)
\(\def\upnu{\unicode{x03BD}}\)
\(\def\upxi{\unicode{x03BE}}\)
\(\def\upomicron{\unicode{x03BF}}\)
\(\def\uppi{\unicode{x03C0}}\)
\(\def\upvarpi{\unicode{x03D6}}\)
\(\def\uprho{\unicode{x03C1}}\)
\(\def\upvarrho{\unicode{x03F1}}\)
\(\def\upsigma{\unicode{x03C3}}\)
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\(\def\upupsilon{\unicode{x03C5}}\)
\(\def\upphi{\unicode{x03D5}}\)
\(\def\upvarphi{\unicode{x03C6}}\)
\(\def\upchi{\unicode{x03C7}}\)
\(\def\uppsi{\unicode{x03C8}}\)
\(\def\upomega{\unicode{x03C9}}\)
\(\def\italpha{\unicode{x1D6FC}}\)
\(\def\itbeta{\unicode{x1D6FD}}\)
\(\def\itvarbeta{\unicode{x03D0}}\)
\(\def\itgamma{\unicode{x1D6FE}}\)
\(\def\itdigamma{\mathit{\unicode{x03DD}}}\)
\(\def\itdelta{\unicode{x1D6FF}}\)
\(\def\itepsilon{\unicode{x1D716}}\)
\(\def\itvarepsilon{\unicode{x1D700}}\)
\(\def\itzeta{\unicode{x1D701}}\)
\(\def\iteta{\unicode{x1D702}}\)
\(\def\ittheta{\unicode{x1D703}}\)
\(\def\itvartheta{\unicode{x1D717}}\)
\(\def\itiota{\unicode{x1D704}}\)
\(\def\itkappa{\unicode{x1D705}}\)
\(\def\itvarkappa{\unicode{x1D718}}\)
\(\def\itlambda{\unicode{x1D706}}\)
\(\def\itmu{\unicode{x1D707}}\)
\(\def\itnu{\unicode{x1D708}}\)
\(\def\itxi{\unicode{x1D709}}\)
\(\def\itomicron{\unicode{x1D70A}}\)
\(\def\itpi{\unicode{x1D70B}}\)
\(\def\itvarpi{\unicode{x1D71B}}\)
\(\def\itrho{\unicode{x1D70C}}\)
\(\def\itvarrho{\unicode{x1D71A}}\)
\(\def\itsigma{\unicode{x1D70E}}\)
\(\def\itvarsigma{\unicode{x1D70D}}\)
\(\def\ittau{\unicode{x1D70F}}\)
\(\def\itupsilon{\unicode{x1D710}}\)
\(\def\itphi{\unicode{x1D719}}\)
\(\def\itvarphi{\unicode{x1D711}}\)
\(\def\itchi{\unicode{x1D712}}\)
\(\def\itpsi{\unicode{x1D713}}\)
\(\def\itomega{\unicode{x1D714}}\)
\(\let \lparen (\)
\(\let \rparen )\)
\(\newcommand {\cuberoot }[1]{\,{}^3\!\!\sqrt {#1}}\,\)
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\(\newcommand {\intclockwise }{\mathop {\unicode {x2231}}\limits }\)
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\(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\)
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\(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\)
\(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\)
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\(\newcommand {\zproject }{\mathop {\unicode {x2A21}}\limits }\)
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\(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\)
\(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\)
\( \newcommand {\PP }{\mathbb {P}} \newcommand {\EE }{\mathbb {E}} \newcommand {\RR }{\mathbb {R}} \newcommand {\FF }{\mathcal {F}} \newcommand \Aa {\mathcal {A}} \newcommand \Ll {\mathcal {L}} \newcommand \Dd
{\mathcal {D}} \newcommand \ip [2]{\langle #1,#2\rangle } \newcommand \Ind {\mathbb {1}} \newcommand {\Indic }[1]{\Ind _{\{#1\}}} \)
\( \definecolor {tPrim}{RGB}{0,95,107} \definecolor {TolDarkBlueAcc}{RGB}{43,87,130} \definecolor {TolDarkPink}{RGB}{136,34,85} \newcommand \alert [1]{{\color {tPrim}#1}} \newcommand \Zb {\mathbf {Z}} \newcommand
\dd {\mathrm {d}} \)
\(\let \symsf \symsfup \)
\(\def\Alpha{\unicode{x0391}}\)
\(\def\Beta{\unicode{x0392}}\)
\(\def\Gamma{\unicode{x0393}}\)
\(\def\Digamma{\unicode{x03DC}}\)
\(\def\Delta{\unicode{x0394}}\)
\(\def\Epsilon{\unicode{x0395}}\)
\(\def\Zeta{\unicode{x0396}}\)
\(\def\Eta{\unicode{x0397}}\)
\(\def\Theta{\unicode{x0398}}\)
\(\def\Vartheta{\unicode{x03F4}}\)
\(\def\Iota{\unicode{x0399}}\)
\(\def\Kappa{\unicode{x039A}}\)
\(\def\Lambda{\unicode{x039B}}\)
\(\def\Mu{\unicode{x039C}}\)
\(\def\Nu{\unicode{x039D}}\)
\(\def\Xi{\unicode{x039E}}\)
\(\def\Omicron{\unicode{x039F}}\)
\(\def\Pi{\unicode{x03A0}}\)
\(\def\Varpi{\unicode{x03D6}}\)
\(\def\Rho{\unicode{x03A1}}\)
\(\def\Sigma{\unicode{x03A3}}\)
\(\def\Tau{\unicode{x03A4}}\)
\(\def\Upsilon{\unicode{x03A5}}\)
\(\def\Phi{\unicode{x03A6}}\)
\(\def\Chi{\unicode{x03A7}}\)
\(\def\Psi{\unicode{x03A8}}\)
\(\def\Omega{\unicode{x03A9}}\)
\(\def\alpha{\unicode{x1D6FC}}\)
\(\def\beta{\unicode{x1D6FD}}\)
\(\def\varbeta{\unicode{x03D0}}\)
\(\def\gamma{\unicode{x1D6FE}}\)
\(\def\digamma{\mathit{\unicode{x03DD}}}\)
\(\def\delta{\unicode{x1D6FF}}\)
\(\def\epsilon{\unicode{x1D716}}\)
\(\def\varepsilon{\unicode{x1D700}}\)
\(\def\zeta{\unicode{x1D701}}\)
\(\def\eta{\unicode{x1D702}}\)
\(\def\theta{\unicode{x1D703}}\)
\(\def\vartheta{\unicode{x1D717}}\)
\(\def\iota{\unicode{x1D704}}\)
\(\def\kappa{\unicode{x1D705}}\)
\(\def\varkappa{\unicode{x1D718}}\)
\(\def\lambda{\unicode{x1D706}}\)
\(\def\mu{\unicode{x1D707}}\)
\(\def\nu{\unicode{x1D708}}\)
\(\def\xi{\unicode{x1D709}}\)
\(\def\omicron{\unicode{x1D70A}}\)
\(\def\pi{\unicode{x1D70B}}\)
\(\def\varpi{\unicode{x1D71B}}\)
\(\def\rho{\unicode{x1D70C}}\)
\(\def\varrho{\unicode{x1D71A}}\)
\(\def\sigma{\unicode{x1D70E}}\)
\(\def\varsigma{\unicode{x1D70D}}\)
\(\def\tau{\unicode{x1D70F}}\)
\(\def\upsilon{\unicode{x1D710}}\)
\(\def\phi{\unicode{x1D719}}\)
\(\def\varphi{\unicode{x1D711}}\)
\(\def\chi{\unicode{x1D712}}\)
\(\def\psi{\unicode{x1D713}}\)
\(\def\omega{\unicode{x1D714}}\)
Growth-fragmentation and quasi-stationary methods
A model of growth-fragmentation
Equilibrium behaviour
Mean measures
Look at \(T_tf(x) = \EE _x\left [\sum _u f(Z_u(t))\right ]\) (formally)
\(\seteqnumber{0}{}{0}\)
\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
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 \(k_V(x,\dd y) = \frac {V(y)}{V(x)} k(x,\dd y)\)
-
• \(e^{-bt} \frac {1}{V(x)} T_t (fV)(x) = \EE _x[f(X_t)]\)
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
\(\seteqnumber{0}{}{0}\)
\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
Theorem
Let
\(\seteqnumber{0}{}{0}\)
\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
\(\seteqnumber{0}{}{0}\)
\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 \(\nu T_t = e^{\lambda t} \nu \).
“\(\EE \Zb (t) \sim e^{\lambda t} h(x) \nu \)”
OK, but can you actually prove anything?
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• 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 \(0\))
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• Can take \(\phi (x) = x\), then \(\Aa \phi (x) = \frac {\tau (x)}{x} \phi (x)\)
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• Can take \(\psi (x) = V(x)\) and put \(\alpha = 0\)
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• Very specific coefficients: if
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– \(p(\dd r) = 2\dd r\) (uniform binary repartition of mass),
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– \(\tau (x) = O(x)\) as \(x\to \infty \),
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– 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
Perspectives: extensions
Perspectives: related models – typed cells
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• Old and new pole cells – Cloez, da Saporta and Roget (2020+)
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– At division, one daughter cell is ‘old’, one is ‘new’ (after E. coli)
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– Type influences growth rate
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– It is preferable (for \(\lambda \)) to have distinct growth rates for old and new cells
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– Could one approach this via spine and optimal control of Lévy-type processes?
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• Parasite branching process inside a growth-fragmentation – Marguet and Smadi (2020+)
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• Spatially dependent fragmentation process – Callegaro and Roberts (2021+)
Further reading