James NOLEN Jean-Michel ROQUEJOFFRE Lenya RYZHIK

(Dedicated to Haim Brezis,with admiration and respect)

1 Introduction

We consider the Fisher-KPP equation

with an initial condition u(0,x)=u0(x)which is a compact perturbation of a step function,in the sense that there exist x1and x2,so that u0(x)=1 for all x≤x1,and u0(x)=0 for all x≥x2.

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Each solution φ(ξ)of(1.2)is a shift of a fixed profile φ∗(ξ): φ(ξ)= φ∗(ξ+s),with some fixed s ∈ R.The profile φ∗(ξ)satisfies the asymptotics

with two universal constantsω0>0,k∈ R.

The large time behaviour of the solutions of this problem has a long history,starting with a striking paper of Fisher[10],which identifies the spreading velocity c∗=2 via numerical computations and other arguments.In the same year,the pioneering KPP paper[15]proved that the solution of(1.1),starting from a step function:u0(x)=1 for x≤0,u0(x)=0 for x>0,converges toφ∗in the following sense:There is a function

such that

Theorem 1.1(see[5–6])There is a constant x∞,depending on u0,such that

Models with temporal variation in the branching process have also been considered.In[9],Fang and Zeitouni studied the extremal particle of such a spatially homogeneous BBM where the branching particles satisfy

A natural question is to prove Theorem 1.1 with purely PDE arguments.In that spirit,a weaker version,precise up to the O(1)term(but valid also for a much more difficult case of the periodic in space coefficients),is the main result of[11–12],

Here,we will give a simple and robust proof of Theorem 1.1.These ideas are further developed to study the refined asymptotics of the solutions in[21].

2 Probabilistic Links and Some Related M odels

We regard(3.3)as a perturbation of this equation,and expect that v(t,x)→ exφ(x−x∞)as t→ ∞ for some x∞∈R.

Lemma 4.2There exists a constantα∞>0 with the following property.For anyγ>0 and allε>0,we can find Tε,so that for all t>Tεwe have

with the initial dataTherefore,Theorem 1.1 is about the median location of the maximal particle XNt.Building on the work of Lalley and Sellke[16],recent probabilistic analyses(see[1–3,7–8])of this particle system have identified a decorated Poissontype point process which is the limit of the particle distribution “seen from the tip”:There is a random variable Z>0,such that the point process defined by the shifted particleswith

has a well-defined limit process as t→∞.Furthermore,Z is the limit of the martingale

and

As we have mentioned,the logarithmic term in Theorem 1.1 arises also in inhomogeneous variants of this model.For example,consider the Fisher-KPP equation in a periodic medium

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where λ∗>0 is the rate of exponential decay(as x → ∞)of the minimal front Uc∗,which depends on μ(x)but not on s or on u0.This implies the convergence of u(t,x− σs(t))to a closed subset of the family of minimal fronts.It is an open problem to determine whether convergence to a single front holds,not to mention the rate of this convergence.Whenμ(x)>0 everywhere,the solution u of the related model may be interpreted in terms of the extremal particle in a BBM with a spatially-varying branching rate(see[12]).

TrustSVD协同过滤算法对于冷启动问题做了较大改善[6,7]，本文的DPTrustSVD算法也不会因为引入了差分隐私保护而明显破坏算法对冷启动问题的改善程度.图5～图8给出了从冷启动角度，本文的DPTrustSVD与无隐私保护的TrustSVD和无社会化关系的DPSVD在两个数据集上的预测准确率的比较情况.

Theorem 1.1 was proved through elaborate probabilistic arguments.Bramson also gave necessary and sufficient conditions on the decay of the initial data to zero(as x→ +∞)in order that the solution converges to φ∗(x)in some moving frame.Lau[17]also proved those necessary and sufficient conditions(for a more general nonlinear term)using a PDE approach based on the decrease in the number of the intersection points for a pair of solutions of the parabolic Cauchy problem.The asymptotics ofσ∞(t)were not identified by that approach.

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They proved that ifκis increasing,and f is of the Fisher-KPP type,the shift is algebraic and not logarithmic in time:There exists C>0,such that

In[20],we proved the asymptotics

Here,β<0 is the first zero of the Airy function.Maillard and Zeitouni[18]refined the asymptotics further,proving a logarithmic correction to(2.3),and convergence of u(T)to a traveling wave.

3 Strategy of the Proof of Theorem 1.1

3.1 Why converge to a traveling wave?

We first provide an informal argument for the convergence of the solution of the initial value problem to a traveling wave.Consider the Cauchy problem(1.1),starting at t=1 for the convenience of the notation

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and proceed with a standard sequence of changes of variables.We first go into the moving frame

leading to

Next,we take out the exponential factor:Set

so that v satisfies

We postpone the proof of this lemma for the moment,and show how it is used.A consequence of Lemma 4.2 is that the problem for the moment is to understand,for a givenα>0,the behavior of the solutions of

Observe that for any shift x∞∈ R,the function V(x)=exφ(x−x∞)is a steady solution of

followed by a change of the unknown

3.2 The self-similar variables

We note that for x→ +∞,the term e−xv2in(3.3)is negligible,while for x→ −∞ the same term will create a large absorption and force the solution to be close to zero.For this reason,the linear Dirichlet problem

This transformation strengthens the reason why the Dirichlet problem(3.4)appears naturally:For

the last term in the left-hand side of(3.6)becomes exponentially large,which forces w to be almost 0 in this region.On the other hand,for

this term is very small,so it should not play any role in the dynamics of w in that region.The transition region has width of the order

3.3 The choice of the shift

Also,through this change of variables,we can see how a particular translation of the wave will be chosen.Considering(3.4)in the self-similar variables,one can show(see[11,14])that,asτ→+∞,we have

with someα∞>0.Therefore,taking(3.4)as an approximation to(3.3),we should expect that

Comparing this with(3.8),we infer that

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