semi-group and P x the law of starting at x 2 E. We consider a branching mechanism : R +! R + of the form (u) = au + bu 2 + n(dr)[e ru 1 + ru]; (;1) w

Transcription

1 M.S.R.I. October Superprocesses and nonlinear partial dierential equations We would like to describe some links between superprocesses and some semi-linear partial dierential equations. Let L be a second order dierential operator dened on R d and a general function from R + to R +. We will consider the nonnegative solutions of the parabolic equation: 8 < = Lu in [; 1) Rd u(; ) = f (); where f is a bounded nonnegative function dened on R d. We will mostly be interested with L = 1 2 and (u) = u2, >. We will also consider the Dirichlet problem associated with the elliptic equation. Let D be an open set of R d, we want to study the nonnegative solutions of: ( Lu = (u) in D (2) u j@d = '; where ' is a bounded nonnegative function dened on the Let me stress that we are only interested in the nonnegative solutions of (1) or (2). Those solutions can be related to the Laplace transform of measure valued Markov processes called superprocesses. In a rst part we will recall some basic results on superprocesses which have already been introduced by Alison Etheridge and Robert Adler in September. In a second part, we will present one of the most important tool associated with superprocesses: the so-called exit measures. They are closely related to (2). From there we will focus on the particular case L = 1 2 and (u) = u2. We will also describe the solutions of the Dirichlet problem with blow-up condition at the boundary. Then we will look at the maximal and minimal solutions of (2) with innite boundary condition. In a third part we will describe the polar sets for the superprocesses and give a characterization in terms of removable singularities for solutions of (2). In the last part we will describe the trace of the solutions of u = u 2 in a planar domain and present a representation formula. We will also give some extensions in higher dimension. (1) 1 Superprocesses 1.1 Construction via the semi-group We consider an homogeneous c dl g Markov process ( t ; t ) taking values in a polish space (E; d). (Typically is a diusion in R d.) Let (P t ; t ) denote its transition 1

2 semi-group and P x the law of starting at x 2 E. We consider a branching mechanism : R +! R + of the form (u) = au + bu 2 + n(dr)[e ru 1 + ru]; (;1) where a, b and n is a Radon measure on (; 1) satisfying R (;1) (r^r2 ) n(dr) < 1. Notice this includes the following cases: - (u) = bu 2 (take a = ; n = ); - (u) = cu 1+ (take a = ; b = and n(dr) = c r 2 dr) for 2 (; 1). The function is nonnegative, convex and locally Lipschitz. Let B + (E) be the set of bounded nonnegative measurable functions dened on E. We then consider the following integral equation u(t; x) + E x t (u(t s; s ))ds = E x f ( t ) = P t f (x); where f 2 B + (E). This equation is the mild form of (1), with L replaced by the innitesimal generator of. Using the method of Picard iteration, we can prove there exists a unique measurable (jointly in (t; x) 2 R + R d ) nonnegative function solution to the above integral equation. We denote by V t f (x) this solution. Using the Markov property of, it is easy to show that (f! V t f; t ) forms a nonlinear semi-group of operator on B + (E). Let M f (E) be the set of all nite measures on E endowed with the topology of the weak convergence. Fitzsimmons [11, 12] proved there exists a Markov process ((X t ; t ); (P; 2 M f (E))) taking values in M f (E), such that for every f 2 B + (E), t, i E he (Xt;f ) = e (;Vtf ) ; R where (; f ) = f (x)(dx). We called this process the (; )-superprocess. Computing P[X t = ] from R the Laplace transform, we can see the process X dies out 1 in nite time if and only if (u) 1 du < 1. We can easily deduce from the Laplace 1 transform the rst moment for X: E [(X t ; f )] = (; P t f ) e at : In the case (u) = u 2 we get for the second moment formula: E [(X t ; f ) 2 ] = (; P t f ) (dx) E x t P t s f ( s ) 2 ds : (3) It can also be proved that in this case, the total mass process ((X t ; 1); t ) is a Feller diusion, whose Laplace transform is given by E [e (Xt;1) ] = e (;1)=[1+(;1)t] : 2

3 1.2 Construction via the branching particle system Although we have a direct denition for superprocesses, they can be viewed as limit of branching particle systems (cf [5]). Let be a continuous Markov process. Let us assume for simplicity that (u) = 1 2 u2. Let n be an integer which will tend to innity. We consider N (n) particles starting at time at points fx i ; 1 i N (n) g. Each particle evolves, independently of the others, according to the law of. At independent exponential times of parameter n, each particle dies and give birth to two children with probability 1=2 or to none with probability 1=2. This is a critical binary branching mechanism. Conditionally on the fact that each newborn particle starts from the death point of its parent, the particles evolve independently from the past and from the other particles. The law of their trajectory is the law of. They will also die at random independent exponential times (of parameter n), and the branching occurs again. We repeat again and again this procedure. Since we have a critical branching mechanism, the particle system dies out in nite time. At time t we have N (n) t particles located at points fx i t ; 1 i N (n) t g. To study this system we look at the measure valued process X (n) t = 1 n where x stands for the Dirac measure at point x. (Notice we consider particles of weight 1=n.) The process (X (n) t ; t ) is a Markov process taking values in M f (E) and starting at P X (n) = n 1 (n) N (n) i=1 x i. Let us assume that X converges, as n goes to innity, to 2 M f (E) (for the weak topology). Then the nite dimensional distribution of X (n) converges in law to the nite dimensional distribution of the (; 1 2 u2 )-superprocess X starting at (see also [2] and [3] for a stronger convergence and general results on superprocesses). 2 Exit measure The exit measures have been introduced by Dynkin [6]. Let D be a given domain in E. Intuitively the exit measure of D describes, in the particle setting, the repartition of the particles frozen when they leave the domain D for the rst time. There are three ways to construct the exit measures. - We can dene a semi-group of operator on B + (E) indexed not by the time but by the open sets of E. - We can consider the historical superprocess, that is the superprocess with underlying (inhomogeneous) Markov process ~ t = ( s ; s 2 [; t]) and then consider the paths which just end when they go out of D for the rst time. - In the particular case where (u) = u 2, we can use the Brownian snake approach introduced by Le Gall. It will enable us to have some nice proofs and representations for further results. 2.1 The Brownian snake We assume from now on that is a nice continuous Markov process in E (for example the d-dimensional Brownian motion). Let ~ E denotes the set of all continuous stopped paths 3 N tx (n) i=1 x i t ;

4 in E. An element w 2 ~ E is a continuous path w : [; ]! E, is called the lifetime of the path w. We denote by x the trivial path of lifetime such that x() = x. The space ( ~ E; ~ d) is a polish space for the distance ~d(w; w ) = + sup t d(w(t ^ ); w (t ^ )): The Brownian snake W = (W s ; s ) is a strong continuous Markov process with values in ~ E. We denote by s the lifetime of the path W s. The law of W can be characterized as follows: - The lifetime process = ( s ; s ) is a Brownian motion reecting at. - Conditionally on ( s ; s ), W is an inhomogeneous Markov process. Its transition kernel is characterized by: for s < s, W s (t) = W s (t) for t m(s; s ), where m(s; s ) = inf r2[s;s ] r. Conditionally on W s (m(s; s )), the path (W s (t); m(s; s ) t s ) is independent of W s and has the same distribution as started at time m(s; s ) at point W s (m(s; s )). We denote by Px the law of W started at time at the trivial path x. Let ^W s = W s ( s ) denotes the end point of the path W s. It can be proved that the function s 7! ^W s is a.s. continuous. Notice that for s 2 [s ; s ] the paths (W s (t); t s ) coincide for t 2 [; m(s ; s )]. We will refer later to this property as the snake property. We also introduce (L r s ; s ) the local time of at level r. Let = inf s ; L s > 1. Then the process (L t ; t ) is a Feller diusion and its Laplace transform is given by: E h e Lt j = i = e =[1+2t] : We can now give a construction of superprocesses via the Brownian snake. Proposition 1. The process dened under Px by: for t, for f 2 B + (E), (X t ; f ) = is the (; 2u 2 )-superprocess started at x. Notice that for all t, a.s. supp X t = n f ( ^W s )dl t s; o ^W s ; s 2 [; ]; s = t. Remark. It is also easy to built the so-called historical process ( ~ X t ; t ) a M f ( ~ E)-valued process: for F 2 B + ( ~ E), ( ~ X t ; F ) = F (W s )dl t s: 2.2 Exit measures for the Brownian snake For simplicity, let us assume from now on that is the d-dimensional Brownian motion. We are now ready to build the exit measure of a connected open set D R d for the Brownian snake. 4

5 For w 2 ~ R d, let D (w) = inf ft ; w(t) 62 Dg with the convention that inf ; = +1. Let us assume that for x 2 D, P x [ D () < 1] >. For " >, we consider the following measures on R d : for f 2 B + (R d ), where (X " D; f ) = 1 " L D;" s = 1 " s 1 fd (W s)<s< D (W s)+"g f ( ^W s )ds = 1 fd (W r)<r< D (W r)+"g dr = 1 " s f ( ^W s )dl D;" s ; 1 f<r<"gdr; R s and s = ( s D (W s )) +. We introduce the functions K s = 1 f r>gdr and A t = inf fs ; K s > tg. And we consider the time change process s = As. We have Ks = s. Using the snake property, we can prove that the process ( s ; s ) is a reecting Brownian motion in R +. Furthermore we have L D;" s = 1 " s 1 f<r<"gdr = 1 " Ks 1 f< r<"gdr: Clearly this quantity converges a.s. to the local time of at level up to time K s. This implies that a.s. the measure dl D;" s converges to a measure, which we denote by dl D s. Since the local time of at level increases only when r =, we deduce that the measure dl D s increases only when s = D (W s ). The measure dened by: for f 2 B + (R d ), (X D ; f ) = f ( ^W s )dl D s ; is the exit measure of D. Notice that supp X since ^W s is continuous. We can compute the Laplace transform of the exit measure. Theorem 2. Let ' be a bounded nonnegative measurable function dened Then we have E x h i e (X D;') = e u(x) ; x 2 D; where the function u, dened on D, solves the equation D u(x) + 2E x u( s ) 2 ds = E x ['( D )]: Remarks. i) The above integral equation is the mild form of (2) with L = 1 2 and (u) = 2u 2. The exit measure has been built in the case (u) = 2u 2, this restriction is due to the Brownian snake approach. The approach of Dynkin [8] yields the general. ii) We have been looking at elliptic equation in domain D R d, but using the process (t; t ) instead of ( t ) leads to parabolic equation in domains of R + R d. 5

6 2.3 Dirichlet problem in D Let D be an open subset of R d. A point a is said to be regular if P a [ D () = ] = 1. We say that D is regular if all points a are regular. Let us recall some well-known results. a) If ' is a bounded measurable function dened then the function h(x) = E x ['( D )] is in C 2 (D) and h = in D. Furthermore if ' is continuous at a and if a is regular, then lim x2d;x!a h(x) = '(a). b) If is a bounded measurable R function dened in a bounded domain D, then the function F (x) = E x ( D s ) is in C 1 (D). Let a be regular, then we have lim x2d;x!a F (x) =. Furthermore if is H lder continuous in D, then F 2 C 2 (D) and 1 F = in D. 2 We then deduce from a) and b) that if D is a bounded regular domain and if ' is nonnegative continuous dened then the function u dened in theorem 2 is in C 2 (D)\C(D) and solves ( u = 4u 2 in D (4) u j@d = ': We also will need the following comparison principle for elliptic dierential equations. Comparison principle. Let D be a bounded open set of R d. Let L be an elliptic differential operator. Let be a nondecreasing function from R + to R +. Let u, v in C 2 (D) such that then u(x) v(x) in D. Lu (u) Lv (v) in D; lim sup [u(x) v(x)] x2d;x!a for all a We deduce from this comparison principle that the solution of (4) is unique. 2.4 Dirichlet problem in D with blows-up boundary condition We keep the same S hypothesis as in (2.2). We dene the range of the superprocess as the closed set R = t supp X t. Recall that a.s. supp X t = ^W s ; s 2 [; ]; s = t. It is then clear, using the continuity of the path s 7! ^W s, that a.s. R = ^W s ; s 2 [; ]. Notice the range is a compact set. Let us now consider the two functions dened on D: u D (x) = log Px [X D = ] v D (x) = log Px [R D] : For any nonnegative function u dened in theorem 2, with ' bounded nonnegative, we have u u D in D. Since supp X D R we get that u D h v D in D. Since s 7! ^W s is continuous and is nite, for every > the quantity Px sup ^W s2[;] s i x is 6 n n o o

7 positive. It is even independent of x by space translation invariance. Let K be a compact set. Hence inf x2k P x [R B(x; )] > ; where B(x; ) is the ball with center x and radius. Thus we deduce that the function v D is uniformly bounded on every compact subset of D. We can now give a result on the maximal and minimal solutions of u = 4u 2 with blow-up boundary condition. Proposition Let D R d be a bounded regular open set, then the function u D is the minimal solution of 8 < : u = 4u 2 in D lim u(x) = 1: (5) x2d;x!@d 2. Let D R d be an open set, then the function v D is the maximal nonnegative solution of u = 4u 2 in D. This means that if u 2 C 2 (D) is nonnegative and solves u = 4u 2 then u v D. We deduce from this proposition that if D is regular and bounded, then v D blows up at the boundary. Before going into the proof, let us prove a nonlinear mean-value property. Let D R d be an open set. Let v 2 C 2 (D) be a solution of u = 4u 2 in D. Consider O a regular bounded open set such that O D. The function u(x) = log E x e (X O ;v) dened in O is a nonnegative solution of (4) in O with ' = v. But so is v. We deduce from the comparison principle that u = v in O, that is v(x) = log E x h e (X O;v) i e n(x D ;1). for x 2 O: Proof 1. Let D R d be a regular bounded open set. Consider the following increasing sequence of functions u n (x) = log E x This sequence converges to u D in D. Recall that the function v D is uniformly bounded on every compact subset K D. Since u D v D, we deduce that u D is bounded on K. Let O D be a regular bounded open set such that O D. By the non-linear mean-value property, we have u n (x) = log E x e (X O ;u n) for x 2 O. By monotone convergence, we get that the nonlinear meanvalue property also holds for u D. This implies that u D 2 C 2 (O) and solves u = 4u 2 in O and thus in D. Since u n (x) converges to n as x goes we deduce that u D (x) blows up as x goes Let v a solution of (5). By the comparison principle, we have v u n in D. Thus we get v u D in D. Proof 2. A.s. we have fr Dg fx D = g R D : (6) 7

8 The last inclusion is non trivial and will be admitted here. We now consider an increasing sequence of bounded regular domains D n such that D n D n+1 D and D = S n1 D n. According to the inclusions (6), we have for x 2 D n Px [R D n ] P [X Dn = ] Px R D n P x [R D n+1 ] : Thus we have v Dn (x) u Dn (x) v Dn+1 (x) in D n. The sequences (v Dn ; n 1) and (u Dn ; n 1) are nonincreasing. They converge to the same limit v D (x) = log Px [R D] which is dened on D (we used the fact that D n+1 " D and that R is compact). By the nonlinear mean value property, we have for every open bounded regular set O such that O D n, 8x 2 O, 8n n, u Dn (x) = log E x e (X O ;u Dn ). By dominated convergence, e (X O ;v D ). Since v D is bounded on O, we get that it we get 8x 2 O, v D (x) = log E x solves u = 4u 2 in O, and thus in D. It remains to prove that v D is the maximal solution in D. Let g a nonnegative solution of u = 4u 2 in D. The function g is at least bounded on every D n. By the comparison principle, we get that on D n, g u Dn. This implies that g v D in D. We are now led to two natural questions: 1. On what condition on the open set D do we have the existence of a solution with blow-up at the boundary? 2. On what condition on D, do we have u D = v D? The rst question has been answered in the case u = 4u 2 by Dhersin and Le Gall [4] (see also [18] for a more general setting). A point x is said to be super-regular if Px[ = ] = 1, where = inf ft > ; supp X t \ D c 6= ;g. We now dene the following capacities. Let. Let A be a Borel set of R d. cap (A) = inf (A)=1 where h (x y) = (dx)(dy)h (x y) 1 ; ( 1 + log + [1= jx yj] if = ; jx yj if > ; with log + x = (log x) _. Then we have the next result which includes the Wiener's test for the (; 2u 2 )-superprocess. Theorem 4. Let D R d be a domain. Let x For n 1 let F n (x) = y 2 D c ; 2 n jx yj < 2 n+1 : Then the next three properties are equivalent. 1. The point x is super-regular. 2. Either d 3, or d 4 and X 2 n(d 2) cap d 4 (F n (x)) = 1: n1 3. There exists a solution of u = 4u 2 in D such that lim y2d;y!x u(y) = 1. Furthermore, if every point x is super regular, then there exists a solution to (5). The answer to the second question does not seem optimal yet (see [15] and [18]). 8

9 3 Polar sets We still assume L = 1 2 and (u) = 2u2. Let K be a compact set of R d. We say that the set K is polar if 8x 2 R d nk; Px[R \ K 6= ;] = : Since the function v K c = log Px[R \ K = ;] is the maximal solution of u = 4u 2 in K c. We deduce the next three statements are equivalent: - K is polar. - v K c =. - There is no non trivial solution u of u = 4u 2 in K c (i.e. K is a removable singularity). We now give a characterization for polar sets. Theorem 5. The set K is a polar set (for the super Brownian motion) if and only if - K = ; if d 3, - cap d 4 (K) = if d 4. This theorem is due to Perkins [19] ()) and Dynkin [7] (() (see also [1]). Those results have been extended by Dynkin to the general equation Lu = u 1+, 2 (; 1]. We will only give the proof of ()). Proof. Let us RR assume d 4 and cap d 4 K >. Then there exists a probability on K such that (dx)(dy)h d 4 (x y) is nite. We consider a continuous nonnegative function f on d with support in B(; 1), which is radial (i.e. f (y) = f (x) if jyj = jxj) and such R R that f (y)dy = 1. And we set f " (y) = " d f (y=") (thus f " (y)dy R ) ). Let 1 g " (x) = f " (x y)(dy). We consider the occupation measure (dx) = dt X t (dx), and we compute the rst two moments of ( ; g " ). We have for x 2 K c : E x [( ; g " )] = G(x y)g " (y)dy; where G is the Green function G(x) = R d jxj 2 d. Since the right hand side converges to G(x y)(dy) which is nite positive, we deduce there exists a constant c1, depending on x; K; " >, such that for every " 2 (; " ], E x [( ; g " )] c 1 > : The formula (3) for the second moment gives E x ( ; g" ) 2 = G(x y)g " (y)dy dz G(x; z) dydy G(z y)g(z y )g " (y)g " (y ): 9

10 The rst right hand side term is bounded (use the above remark on E x [( ; g " )]). Now using the properties of f " and the fact that the function G(y) is super-harmonic on R d, we get dz G(x; z) dy G(z y)f " (y a) dy G(z y )f " (y a ) dz G(x; z)g(z a)g(z a ) c 2 h d 4 (a a ): Thanks to the assumption on, we deduce there exists a constant c 3, depending on x; K; " >, such that for every " 2 (; " ], E x ( ; g" ) 2 c 3 : Now, using Cauchy-Schwarz inequality, we get for " 2 (; " ] Px [( ; g " ) > ] E x [( ; g " )] 2 E x [( ; g c2 1 > : " ) 2 ] c 3 But since the support of f is in B(; 1), we get that f( ; g " ) > g fr \ K " 6= ;g ; where K " = y 2 R d ; d(y; K) ". Letting " goes to, we get that Px[R \ K 6= ;] >. Thus K is not polar. Let d 3. In fact it is sucient to consider the case d = 3. Now using the denition of the capacity, it is easy to check that a segment is not polar in dimension d = 4. Thus by projection, we deduce that points are not polar in dimension d = 3. Remark. The points are polar if and only if d 4. In the case u = u 1+ for 2 (; 1] the points are polar if and only if d 2(1 + )=. 4 Representation theorems We use the Brownian snake approach. We want to describe all the solutions of u = 4u 2 in D R d, where D is a smooth open set. Let us consider the set of exit points of D: n o E D = ^W s ; s ; D (W s ) = s : Notice that supp X D E D a.s. 4.1 The critical case d = 2 The next theorem is due to Le Gall [17]. Theorem 6. Let D be a domain of class C 2 (not necessarily bounded). There is a oneto-one correspondence between nonnegative solution of u = 4u 2 in D and pairs (K; ), where K is a closed subset and is a Radon measure The correspondence is characterized as follows: 1

11 - On one hand, for x 2 D, u(x) = log E x h R i 1 fed \K=;g e Y D (y)(dy) ; where (Y D (y); y is the continuous density of X D with respect to the Lebesgue measure (dy). - On the other hand K =fz lim sup 2 u(x) > g; x2d;x!z (; ') = lim u(z + rn z )'(z)(dz); where ' is continuous with compact support and N z denotes the inward pointing vector normal at z. The pair (K; ) will be called the trace of the solution u. Remarks. i) If K = ;, D bounded, and (dy) = '(y)(dy), where the function ' is continuous, then we recover the fact that the function h i u(x) = log E x e (X D;') ; is the only solution of the Dirichlet problem (4). ii) If K D bounded, then we recover the fact that the function u(x) = log Px [E D \ K = ;] = log Px [R \ D c = ;] is the maximal solution of u = 4u 2 in D. 4.2 sets The above formula cannot be extended in higher dimension the same way. The density of the exit measure does not exist if d 3. Furthermore, there exist compact sets such that Px[E D \ K 6= ;] = : Such sets are sets (boundary polar sets). This means that a.s. no path W s will exit D through K. We can prove the following result (see [14]). Proposition 7. Let D be a bounded Lipschitz domain. Let K be a compact subset The function u(x) = log Px [E D \ K = ;] is the maximal nonnegative solution of ( u = 4u 2 in D lim x2d;x!y u(x) = 8y (7) 11

12 We will now give a characterization of sets. The next theorem is due to Le Gall [16] and has been extended by Dynkin and Kuznetsov [9] in the general case. Recall the capacity dened in the previous section. Theorem 8. Let D R d be an open bounded set, suciently smooth (C 5 ). A compact set K if and only if - K 6= ; if d 2, - cap d 3 (K) = if d 3. Thus we deduce that (7) has a non trivial nonnegative solution if and only if d 2 and K 6= ; or d 3 and cap d 3 (K) >. 4.3 Moderate solutions We would like to characterize all the nonnegative solutions of u = 4u 2 in D, bounded by harmonic function. Such a solution will be called a moderate solution. For example assume D is a bounded regular domain, and consider a continuous nonnegative function ' dened We know that the function u(x) = log E x e (X D ;') is the only solution of (4). And we have seen in section 2.2 that D u(x) + 2E x u( s ) 2 ds = E x ['( D )]; where D = inf ft > ; t 2 D c g. Now the function h(x) = E x ['( D )] is harmonic in D and u h. Furthermore u is the maximal solution of u = 4u 2 in D, bounded by h. Indeed, let v be an other nonnegative solution bounded by h. And assume that u v h in D. By continuity we get v j@d = '. Since the nonnegative solution of (4) is unique, we get that v = u. We now give a complete description of all the moderate solutions. Let D be a bounded domain in R d, d 2. Proposition 9. Let u be a moderate solution of u = 4u 2 in D. Then there exists a unique harmonic function h such that for all x 2 D, D u(x) + 2E x u( s ) 2 ds = h(x): (8) Furthermore the function h is the smallest harmonic function dominating u. Proof. In order to prove this proposition, we consider S an increasing sequence of regular open set D n such that D n D n+1 D and D = D n1 n. Let u be a moderate solution. Since u solves (4) in D n with boundary condition ' = u, we deduce that for all x 2 D n, Dn u(x) + 2E x u( s ) 2 ds = E x [u( Dn )]: (9) Let h denotes the limit of the nondecreasing sequence of functions E x [u( Dn )]. (h is dened in D.) Since u is bounded by an harmonic function let say g, we get for x 2 D: h(x) = lim n!1 E x[u( Dn )] lim n!1 E x[g( Dn )] = g(x): 12

13 Now by dominated convergence, h is clearly harmonic in D. Letting n goes to innity in (9), we get (8). And by construction h is clearly the smallest harmonic function dominating u. The next step is to characterize all the harmonic function h such that there exists a function u satisfying (8). Let us assume that D is bounded and suciently smooth (at least C 5 ). Theorem 1. There exists a one-to-one correspondence between the moderate solutions of u = 4u 2 in D and nite measures that does not sets. The correspondence is given by u(x) + 2 G D (x; y)u(y) 2 dy P D (x; y)(dy); where G D and P D are respectively the Green function on D and the associated Poisson kernel. Let N be the set of nite measures which doesn't set. We will write u for the moderate solution corresponding to the measure 2 N. In fact we have a probabilistic representation of u. Let D n be an increasing sequence of R bounded open sets such that D = [ n1 D n. Consider the harmonic function h(x) P D(x; y)(dy). We can built a continuous additive functional (A h s ; s ) of the Brownian snake in the following way: for all s, A h s = lim n!1 s h( ^W s )dl Dn s : We have A h 1 = lim n!1 < X Dn ; h >. This convergence can be proved by martingale techniques using the so called special Markov property. The function u is then dened by: u (x) = log E x h e Ah 1 i x 2 D: 4.4 Trace of solutions in the super critical case The following results are recent work from Dynkin and Kuznetsov [1] and Kuznetsov [13]. They have been presented by their author in September. I will only present them for the nonnegative solutions of u = 4u 2 in D R d, a smooth bounded domain. We rst dene the Borel set of singular points of a nonnegative solution u: SG(u) = y P D x!ya:s: u( s )ds = 1 ; where P D x!y is the law of the h-transform of the Brownian motion started at point x 2 D with respect to the Poisson kernel P D (; y). Intuitively P D x!y is the law of the Brownian motion started at x conditionally on going out of D at point y. In dimension d = 2, the set of singular points of the solution with trace (K; ) is SG(u) = K. In higher dimension, if u is a moderate solution then SG(u) is set. 13

14 Let N be the set of all measures obtained as increasing limits of measures of N. (Notice the measures of N are not necessarily -nite.) A solution u is called -moderate, if it is an increasing limit of moderate solutions u n. Let 2 N be the increasing limit of n. Since the limit u is independent of the sequence ( n ) which converge to we write u for the limit. For every Borel we put u (x) = sup fu (x); ( c ) = ; 2 N g. The function u is -moderate. We say that is nely closed if SG(u ). This dene a ner topology than the induced topology. We write u ; for the maximal solution dominated by u + u. The trace of a nonnegative solution u is the pair ( ; ) dened by = SG(u) and is a function dened on B(@D) by: for every Borel set (B) = sup f(b); 2 N ; ( ) = ; u ug : Theorem 11. The trace ( ; ) of a nonnegative solution of u = 4u 2 in D has the following properties i) is nely closed. ii) is a -nite measure, 2 N, ( ) = and SG(u ). Moreover, u ; is the maximal -moderate solution dominated by u. We say that two pairs ( ; ) and ( ; ) are equivalent if = and 4 Clearly u ; = u ; if the two pairs ( ; ) and ( ; ) are equivalent. Theorem 12. Let ( ; ) satisfy i) and ii) of the above theorem, then the trace of u ; is equivalent to ( ; ). Furthermore, u ; is the minimal solution with this property and the only -moderate one. There is still an open question: Are the -moderate solutions the only nonnegative solutions of u = 4u 2 in D? (The answer is yes if the dimension is less or equal to 2 (cf section 4.1).) References [1] P. BARAS and M. PIERRE. Singularit s liminables pour les quations semi-lin aires. Ann. Inst. Four., 34:18526, [2] D. A. DAWSON. Measure-valued markov processes. In cole d' t de probabilit de Saint Flour 1991, volume 1541 of Lect. Notes Math., pages 126. Springer Verlag, Berlin, [3] D. A. DAWSON and E. PERKINS. Historical processes. Memoirs of the Amer. Math. Soc., 93(454), [4] J.-S. DHERSIN and J.-F. LE GALL. Wiener's test for super-brownian motion and the Brownian snake. Probab. Th. Rel. Fields, 18(1):13129,

15 [5] E. DYNKIN. Branching particle systems and superprocesses. Ann. Probab., 19: , [6] E. DYNKIN. Path processes and historical superprocesses. Probab. Th. Rel. Fields, 9:136, [7] E. DYNKIN. A probabilistic approach to one class of nonlinear dierential equations. Probab. Th. Rel. Fields, 89:89115, [8] E. DYNKIN. Superprocesses and partial dierential equations. Ann. Probab., 21: , [9] S. DYNKIN, E.B. KUNETSOV. Superdiusions and removable singularities for quasilinear partial dierential equations. Comm. Pure Appl. Math., 49:125176, [1] S. DYNKIN, E.B. KUNETSOV. Trace on the boundary for solutions of nonlinear dierential equations. Trans. Amer. Math. Soc., to appear. [11] P. FITSIMMONS. Construction and regularity of measure-valued Markov branching processes. Israel J. Math., 64:337361, [12] P. FITSIMMONS. On the martingale problem for measure-valued Markov branching processes. In Seminar on Stoch. Proc. 1991, volume 29, pages 3951, Boston, Birkh user. [13] S. KUNETSOV. -moderate solutions of Lu = u and ne trace on the boundary. [14] J.-F. LE GALL. Hitting probabilities and potential theory for the Brownian pathvalued process. Ann. Inst. Four., 44:27736, [15] J.-F. LE GALL. A path-valued Markov process and its connections with partial dierential equations. In Proceedings in First European Congress of Mathematics, volume II, pages Birkh user, Boston, [16] J.-F. LE GALL. The Brownian snake and solutions of u = u 2 in a domain. Probab. Th. Rel. Fields, 12:393432, [17] J.-F. LE GALL. A probabilistic Poisson representation for positive solutions of u = u 2 in a planar domain. Comm. Pure Appl. Math., 5(1):6913, [18] M. MARCUS and L. V RON. Uniqueness and asymptotic behavior of solutions with boundary blow up for a class of nonlinear elliptic equations. Ann. Inst. Henri Poincar, 14(2):237274, [19] E. PERKINS. Polar sets and multiple points for super-brownian motion. Ann. Probab., 18:453491,