DIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS
CHAPTER INTRODUCTION PART PARABOLIC EQUATIONS AND BRANCHING EXIT MARKOV SYSTEMS LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS BRANCHING EXIT MARKOV SYSTEMS SUPERPROCESSES SEMILINEAR PARABOLIC EQUATIONS AND SUPERDIFFUSIONS ELLIPTIC EQUATIONS AND DIFFUSIONS LINEAR ELLIPTIC EQUATIONS AND DIFFUSIONS POSITIVE HARMONIC FUNCTIONS MODERATE SOLUTIONS OF LU = (U) CONTENTS CHAPTER STOCHASTIC BOUNDARY VALUES OF SOLUTIONS ROUGH TRACE FINE TRACE MARTIN CAPACITY AND CLASSES N1 AND N0 NULL SETS AND POLAR SETS SURVEY OF RELATED RESULTS INTRODUCTION 1 BROWNIAN AND SUPER-BROWNIAN MOTIONS AND DI?ERENTIAL EQUATIONS 1 1 BROWNIAN MOTION AND LAPLACE EQUATION LET D BE A BOUNDED DO- MAIN IN RD WITH SMOOTH BOUNDARY ?D AND LET F BE A CONTINUOUS FUNCTION ON ?D THEN THERE EXISTS A UNIQUE FUNCTION U OF CLASS C2 SUCH THAT ?U = IN D, U = F ON ?D (1 1) IT IS CALLED THE SOLUTION OF THE DIRICHLET PROBLEM FOR THE LAPLACE EQUATION IN D WITH THE BOUNDARY VALUE F A PROBABILISTIC APPROACH TO THIS PROBLEM CAN BE TRACED TO THE CLASSICAL WORK [CFL28] OF COURANT, FRIEDRICHS AND LEWY PUBLISHED IN 1928 THE AUTHORS REPLACED THE LAPLACIAN ?BY ITS LATTICE APPROXIMATION AND THEY REPRESENTED THE SOLUTION OF THE CORRESPONDING BOUNDARY VALUE PROBLEM IN TERMS OF THE RANDOM WALK ON THE LATTICE SUPPOSE THAT A PARTICLE STARTS FROM A SITE X IN D AND MOVES IN ONE STEP FROM A SITE X TO ANY OF 2D NEAREST NEIGHBOR SITES WITH EQUAL PROBABILITIES LET ? BE ITS ?RST EXIT TIME FROM D AND ?? BE ITS LOCATION AT TIME ? THEN THE SOLUTION OF THE DIRICHLET PROBLEM ON THE LATTICE IS GIVEN BY THE FORMULA (1 2) U(X) = ?XF(??) = Z F(??(?)(?))?X(D?), WHERE ?X IS THE PROBABILITY DISTRIBUTION IN THE PATH SPACE ?CORRESPONDING TO THE INITIAL POINT X THE SOLUTION OF THE PROBLEM (1 1) CAN BE OBTAINED BY THE PASSAGE TO THE LIMIT AS THE LATTICE MESH AND THE DURATION OF EACH STEP TEND TO 0 IN A CERTAIN RELATION IN FACT, THIS PASSAGE TO THE LIMIT YIELDS A MEASURE ?X ON THE SPACE OF CONTINUOUS PATHS THE STOCHASTIC PROCESS ? = (?T, ?X) IS CALLED THE BROWNIAN MOTION AND FORMULA (1 2) GIVES AN EXPLICIT SOLUTION OF THE PROBLEM (1 1) IN TERMS OF THE BROWNIAN MOTION ? THIS RESULT IS DUE TO KAKUTANI [KAK44A], [KAK44B] 1 2 SEMILINEAR EQUATIONS PARTIAL DI?ERENTIAL EQUATIONS INVOLVING A NONLIN- EAR OPERATOR ?U ??(U) APPEARED IN METEOROLOGY (EMDEN, 1897), THEORY OF ATOMIC SPECTRA (THOMAS-FERMI, 1920S) AND ASTROPHYSICS (CHANDRASEKHAR, 1937) SINCE THE 1960S, GEOMETERS HAVE BEEN INTERESTED IN THESE EQUATIONS IN CONNECTION WITH THE YAMABE PROBLEM: WHICH TWO FUNCTIONS REPRESENT SCALAR CURVATURE OF TWO RIEMANNIAN METRICS RELATED BY A CONFORMAL MAPPING THE EQUATION (1 3) ?U = ?(U) 1SEE THE BIBLIOGRAPHY IN [V´ER96] AND IT IS EQUAL TO Q | X |?2/(??1) WITH Q = [2(? ?1)?1(?? ?D)]1/(??1) IF D < ?? 1 3 SUPER-BROWNIAN APPROACH TO SEMILINEAR EQUATIONS A PROBABILISTIC FORMULA (1 2) FOR SOLVING THE PROBLEM (1 1) INVOLVES THE VALUE OF F AT A RANDOM POINT ?? ON THE BOUNDARY THE PROBLEM (1 5) CAN BE APPROACHED BY INTRODUCING, INSTEAD, A RANDOM MEASURE XD ON ?D AND BY TAKING THE INTEGRAL ?F, XD?OF F WITH RESPECT TO XD THE PROBABILITY LAW Pµ OF XD DEPENDS ON AN INITIAL MEASURE µ AND THE ROLE SIMILAR TO THAT OF (1 2) IS PLAYED BY THE FORMULA (1 6) U(X) = ?LOGPXE??F,XD? HERE PX STANDS FOR Pµ CORRESPONDING TO THE INITIAL STATE µ = ?X (UNIT MASS CON- CENTRATED AT X) WE CALL (XD, Pµ) THE EXIT MEASURE FROM D HEURISTICALLY, WE CAN THINK OF A RANDOM CLOUD FOR WHICH XD IS THE MASS DISTRIBUTION ON AN ABSORBING BARRIER PLACED ON ?D WE CONSIDER FAMILIES OF EXIT MEASURES WHICH WE CALL BRANCHING EXIT MARKOV (SHORTLY, BEM) SYSTEMS BECAUSE THEIR PRINCIPAL CHARACTERISTICS ARE A BRANCHING PROP- ERTY AND A MARKOV PROPERTY THE BEM SYSTEM USED IN FORMULA (1 6) IS CALLED THE SUPER-BROWNIAN MOTION IN THE NEXT SECTION WE EXPLAIN HOW IT CAN BE OBTAINED BY A PASSAGE TO THE LIMIT FROM DISCRETE BEM SYSTEMS BEFORE THAT, WE GIVE, AS THE ?RST APPLICATION OF (1 6), AN EXPRESSION FOR A SOLUTION EXPLODING ON THE BOUNDARY NOTE THAT, IF XD ?= 0, THEN E?C?1,XD??0 AS C ?+?AND, IF XD = 0, THEN E?C?1,XD?= D?2 DIFFUSIONS, SUPERDIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS FIGURE THE INITIAL AND EXIT MEASURES ARE GIVEN BY THE FORMULAE µ = X ?XI XD = X ?YI WHERE ?C IS THE UNIT MASS CONCENTRATED AT C THIS WAY WE ARRIVE AT A FAMILY X OF INTEGER-VALUED RANDOM MEASURES (XD, Pµ) WHERE D IS AN ARBITRARY BOUNDED OPEN SET AND µ IS AN ARBITRARY INTEGER-VALUED MEASURE SINCE PARTICLES DO NOT INTERACT, WE HAVE (1 8) PµE??F,XD ?= E??U,µ? WHERE (1 9) U(X) = ?LOGPXE??F,XD? WE CALL THIS RELATION THE BRANCHING PROPERTY WE ALSO HAVE THE FOLLOWING MARKOV PROPERTY: FOR EVERY C ?F?D AND FOR EVERY µ, (1 10) Pµ{C | F?D} = PXD(C) Pµ-A S HERE F?D AND F?D ARE THE ?-ALGEBRAS GENERATED BY XD?, D? ?D AND BY XD??, D?? ? D IF THE MASS OF EACH PARTICLE IS EQUAL TO ?, THEN THE INITIAL MEASURE AND THE EXIT MEASURES TAKE VALUES 0, ?, 2?, WE PASS TO THE LIMIT AS ? AND THE EXPECTED LIFE TIME OF PARTICLES TEND TO 0 AND THE INITIAL NUMBER OF PARTICLES TENDS TO IN?NITY IN 3OF COURSE, THIS IS ONLY A SCHEME PATH OF THE BROWNIAN MOTION ARE VERY IRREGULAR WHICH IS NOT RE?ECTED IN OUR PICTURE DIFFUSIONS, SUPERDIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS WHICH CONTAINS ?(U) = U? WITH 1 < ? ?2 BUT NOT WITH ? > 2 THE PROBABILITY DISTRIBUTION OF THE RANDOM MEASURE (XD, Pµ) IS DESCRIBED BY (1 8)–(1 9) AND U IS A SOLUTION OF THE INTEGRAL EQUATION (1 11) U(X) + ?X Z ? ?[U(?S)]DS = ?XF(??) IF ?D IS SMOOTH AND F IS CONTINUOUS, THEN (1 11) IMPLIES (1 5) HENCE, (1 6) IS A SOLUTION OF THE PROBLEM (1 5) FORMULAE (1 8) AND (1 11) DETERMINE THE PROBABILITY DISTRIBUTION OF XD FOR A ?XED D JOINT PROBABILITY DISTRIBUTIONS OF XD1, , XDN CAN BE DE?NED RECURSIVELY FOR EVERY N BY USING THE BRANCHING AND MARKOV PROPERTIES THE FOLLOWING EQUATIONS, SIMILAR TO (1 8) AND (1 11), DESCRIBE THE MASS DISTRI- BUTION XT AT TIME T: Pµ EXP??F, XT?= EXP??UT, µ?, (1 12) UT(X) + ?X Z T ?[UT?S(?S)]DS = ?XF(?T) (1 13) WE COVER BOTH SETS OF EQUATIONS, (1 8), (1 11) AND (1 12), (1 13), BY CONSIDERING EXIT MEASURES (XQ, Pµ) FOR OPEN SUBSETS Q OF THE TIME-SPACE S = R × RD AND MEASURES µ ON S THEY SATISFY THE EQUATIONS Pµ EXP??F, XQ?= EXP??U,µ?, (1 14) U(R, X) + ?R,X Z ? R ?[U(S, ?S)]DS = ?R,XF(?, ?? ) (1 15) WHERE ? = INF{T : (T, ?T) /?Q} IS THE ?RST EXIT TIME FROM Q NOTE THAT XT = XS<T WHERE S<T = (??, T) × RD IF Q = (??, T) × D WHERE D IS A BOUNDED SMOOTH DOMAIN 4 AND IF F IS BOUNDED AND CONTINUOUS, THEN U IS A SOLUTION OF A PARABOLIC EQUATION (1 16) ?U + 2?U = ?(U) IN Q SUCH THAT U = F ON ?Q THE MAXIMAL SOLUTION OF THE EQUATION (1 3) CAN BE DESCRIBED THROUGH THE RANGE OF X THIS IS THE MINIMAL CLOSED SET R WHICH CONTAINS THE SUPPORT SUPP XT FOR ALL T (IT CONTAINS, A S , SUPP XD FOR EACH D ) 5 FOR EVERY OPEN SET D, A MAXIMAL SOLUTION IN D IS GIVEN BY (1 17) U(X) = ?LOG PX{R ?D} 4THE NAME “SMOOTH” IS USED FOR DOMAINS OF CLASS C2,? (SEE SECTION 6 1 3) 5WRITING “A S ” MEANS Pµ-A S FOR ALL µ [OR ?µ-A S FOR ALL µ IN THE CASE OF A BROWNIAN MOTION] DIFFUSIONS, SUPERDIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS 2 EXCEPTIONAL SETS IN ANALYSIS AND PROBABILITY 2 1 CAPACITIES THE MOST IMPORTANT CLASS OF EXCEPTIONAL SETS IN ANALYSIS ARE SETS OF LEBESGUE MEASURE 0 THE NEXT IMPORTANT CLASS ARE SETS OF CAPACITY 0 A CAPACITY IS A FUNCTION C(B) ?0 DE?NED ON ALL BOREL SETS 6 IT IS NOT NECESSARILY ADDITIVE BUT IT IS MONOTONE INCREASING AND CONTINUOUS WITH RESPECT TO THE MONOTONE INCREASING LIMITS FOR EVERY B, C(B) IS EQUAL TO THE SUPREMUM OF C(K) OVER ALL COMPACT SETS K ?B AND IT IS EQUAL TO THE IN?MUM OF C(O) OVER ALL OPEN SETS O ?B [A MORE SYSTEMATIC PRESENTATION OF CHOQUET’S CAPACITIES IS GIVEN IN SECTION 10 3 2] TO EVERY RANDOM CLOSED SET (F(?), P) THERE CORRESPONDS A CAPACITY (2 1) C(B) = P{F ?B ?= ?} ANOTHER REMARKABLE CLASS OF CAPACITIES CORRESPOND TO PAIRS (K, ? ?) WHERE K(X, Y) IS A FUNCTION ON THE PRODUCT SPACE E × ˜E AND ? ?IS A NORM IN A SPACE OF FUNCTIONS ON E THE MOST IMPORTANT ARE THE UNIFORM NORM (2 2) ?F?= SUP X |F(X)| AND THE L?(M)-NORMS (2 3) ?F?? = Z |F(X)|? M(DX) 1/? WHERE 1 ?? < ?AND M IS A MEASURE ON E WE ASSUME THAT E AND ˜E ARE NICE METRIC SPACES AND THAT K(X, Y) IS POSITIVE VALUED, LOWER SEMICONTINUOUS IN X AND MEASURABLE IN Y TO EVERY MEASURE ? ON ˜E THERE CORRESPONDS A FUNCTION (2 4) K?(X) = Z ˜ E K(X, Y) ?(DY) ON E THE CAPACITY CORRESPONDING TO (K, ? ?) IS DE?NED ON SUBSETS B OF ˜E BY THE FORMULA (2 5) C(B) = SUP{?(B) : ? IS CONCENTRATED ON B AND ?K???1} OUR PRIMARY INTEREST IS NOT IN CAPACITIES THEMSELVES BUT RATHER IN THE CLASSES OF SETS ON WHICH THEY VANISH, AND WE SAY THAT TWO CAPACITIES ARE OF THE SAME TYPE IF THESE CLASSES COINCIDE 2 2 EXCEPTIONAL SETS FOR THE BROWNIAN MOTION THE BROWNIAN MOTION ? IN A DOMAIN D KILLED AT THE ?RST EXIT TIME ? FROM D HAS A TRANSITION DENSITY PT(X, Y) IF D IS BOUNDED, THEN (2 6) G(X, Y) = Z ? PT(X, Y)DT IS ?NITE FOR X ?= Y WE CALL G(X, Y) GREEN’S FUNCTION THE GREEN’S CAPACITY CORRE- SPONDS TO THE KERNEL G(X, Y) AND THE UNIFORM NORM (2 2) 7 IT IS OF THE SAME TYPE AS THE CAPACITY CORRESPONDING TO (2 7) G1(X, Y) = Z ? E?TPT(X, Y) DT 6AND EVEN ON A LARGER CLASS OF ANALYTIC SETS 7IF D > 1, THEN G(X,X) = ? THEREFORE GREEN’S CAPACITY OF A SINGLE POINT IS EQUAL TO 0 DIFFUSIONS, SUPERDIFFUSIONS AND PARTIAL DIFFERENTIAL EQUATIONS