A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution
aa r X i v : . [ m a t h . P R ] J u l A nonlinear stochastic heat equation: H¨older continuityand smoothness of the density of the solution
Yaozhong Hu ∗ , David Nualart † and Jian Song Abstract
In this paper, we establish a version of the Feynman-Kac formula for multidimen-sional stochastic heat equation driven by a general semimartingale. This Feynman-Kacformula is then applied to study some nonlinear stochastic heat equations driven bynonhomogenous Gaussian noise: First, it is obtained an explicit expression for theMalliavin derivatives of the solutions. Based on the representation we obtain thesmooth property of the density of the law of the solution. On the other hand, we alsoobtain the H¨older continuity of the solutions.
In this paper we consider the following nonlinear stochastic heat equation: ∂u∂t = 12 △ u + b ( u ) + σ ( u ) ˙ W ( t, x ) , t ≥ , x ∈ R d u (0 , x ) = u ( x ) , (1.1)where ∆ = P di =1 ∂ ∂x i is the Laplace operator, b and σ are globally Lipschitz continuousfunctions, and W is a zero mean Gaussian random field, which is a Brownian motion in thetime variable and it has a nonhomogeneous spatial covariance with density q ( x, y ) (see (2.1)for the precise definition). Here ˙ W ( t, x ) denotes the generalized random field ∂ d +1 W∂t∂x · · · ∂x d .The case of an homogeneous covariance kernel q ( x, y ) = q ( x − y ) has been studiedin the seminal paper by Dalang [3]. In this case, the existence, uniqueness and H¨oldercontinuity of u ( t, x ) with respect to both parameters t and x is obtained in [15] underintegrability conditions on the spectral measure µ of the noise. We extend these results tothe nonhomogeneous case in Section 4.On the other hand, using the techniques of Malliavin calculus, and assuming suitablenondegeneracy conditions, one can show that for a fixed ( t, x ), t >
0, the random variable u ( t, x ), solution to (1.1), has an absolutely continuous probability law and the density issmooth. The results that have been obtained so far along this direction can be summarizedas follows. ∗ Y. Hu is partially supported by a grant from the Simons Foundation † D. Nualart is supported by the NSF grant DMS0904538.Keywords: fractional noise, stochastic heat equations, Feynman-Kac formula, exponential integrability,absolute continuity, H¨older continuity, chaos expansion. x is in the interval (0 , W is a space-time white noise.In this case, if the coefficients are Lipschitz, then u ( t, x ) has an absolutely continuousdistribution for any t > σ ( u ( x )) = 0 for some x ∈ (0 , | σ ( x ) | ≥ c >
0, and smoothcoefficients, Bally and Pardoux [1] proved that the law of any vector of the form( u ( t, x ) , . . . , u ( t, x n )), 0 ≤ x < · · · < x n ≤ t >
0, has an infinitely differentiabledensity, assuming Neumann boundary conditions on (0 , d -dimensional heat equation with an homogeneous spatial covariance, Nu-alart and Quer-Sardanyons have provided sufficient conditions for the existence andsmoothness of the density of u ( t, x ) for t > x ∈ R d , assuming | σ ( x ) | ≥ c >
0, inthe paper [12] (see also [4]).An open problem for the stochastic heat equation with colored spatial covariance is toderive the existence and smoothness of the density under a nondegeneracy condition of theform σ ( u ( x )) = 0 for some x ∈ R d . The main purpose of this paper is obtain new resultsin this direction. To prove such results we need to show that the norm of the Malliavinderivative of the solution R t k D s u ( t, x ) k H ds is either strictly positive almost surely (for theabsolute continuity) or it has negative moments of all orders (for the smoothness of thedensity), where H is the Hilbert space associated with the spatial covariance.We develop a new approach to prove these results based on the Feynman-Kac represen-tation for the solution to the heat equation with multiplicative noise driven by a generalcontinuous semimartingale. The main idea is to express k D s u ( t, x ) k H as the norm in L ( R d )of a function V s,ξ ( t, x ) given by V s,ξ ( t, x ) = R R d c ( ξ, y ) D s,y u ( t, x ) dy , where c is the squareroot of the kernel q as an operator. Then for any fixed ( s, ξ ), V s,ξ ( t, x ) satisfies the linearstochastic heat equation with random coefficients ∂V s,ξ ∂t = 12 △ V s,ξ + b ′ ( u ) V s,ξ + σ ′ ( u ) V s,ξ ˙ W ( t, x ) , t ≥ s, x ∈ R d , (1.2)with initial condition V s,ξ ( s, x ) = c ( ξ, x ) σ ( u ( s, x )).In order to establish a Feynman-Kac representation for the solution to Equation (1.2)we need to assume that the covariance kernel q ( x, y ) is non-singular and this implies theexistence of a random field W ( t, x ) such that ˙ W ( t, x ) = ∂W ∂t ( t, x ). Then, Equation (1.2)is a particular case of a more general stochastic heat equation of the form ∂V∂t ( t, x ) = 12 △ V ( t, x ) + V ∂F∂t ( t, x ) , (1.3)where { F ( t, x ) , t ≥ , x ∈ R d } is a continuous semimartingale in the sense of Kunita [9],with local characteristic b ( t, x ) = b ′ ( u ( t, x )) and a ( t, x, y ) = σ ′ ( u ( t, x )) σ ′ ( u ( t, y )) q ( x, y ). InSection 3 (see Theorem 3.1) we derive a Feynman-Kac formula for the solution of (1.3)assuming that the functions b and a are bounded by C (1 + | x | β ), and C (1 + | x | β + | y | β ) forsome 0 ≤ β <
2. This result has its own interest. The proof is based on a generalized Itˆoformula proved in [9]. 2here have been other papers on the Feynman-Kac formula for the stochastic heatequation. We can mention the recent works [6] and [7] on the stochastic heat equationdriven by fractional white noise. We refer to the references in these papers for relatedworks.In Section 4 we show the existence and uniqueness of a solution for the general stochasticheat equation (1.1) with a nonhomogeneous spatial covariance and we deduce the H¨oldercontinuity of the solution. This result is an extension of the results proved in [15]. Finally,in Section 5, assuming that the covariance kernel is continuous and under a nondegeneracycondition of the form q ( x , x ) > σ ( u ( x )) = 0 for some x ∈ R , we establish theabsolute continuity of the law of the solution and the smoothness of the density if thecoefficients are smooth.To simplify the presentation we have assumed that the functions b and σ depend only onthe variable u . All the results of this paper could be extended without difficulty to the caseof coefficients b ( t, x, u ) and σ ( t, x, u ) such that they are Lipschitz and with linear growth in u , uniformly in ( t, x ) ∈ [0 , T ] × R d for any T >
0. In this case, the nondegeneracy conditionwould be σ (0 , x , u ( x )) = 0, for some x ∈ R d .The results of this paper can be extended to the stochastic heat equation on an open andbounded set A ⊂ R d , with Dirichlet boundary conditions. In this case, the Feynman-Kacformula involves a d -dimensional Brownian motion starting form a point x ∈ A , and killedwhen it leaves the set A . On the other hand, the existence and smoothness of the densityhave been deduced, applying techniques of Malliavin calculus, for stochastic differentialequations of the form Lu = b ( u ) + σ ( u ) ˙ W , where L is a differential operator more generalthan ∂ t − ∆ (see, for instance, [8, 16] where L is a pseudodifferential operator and [12] where L is a general parabolic or hyperbolic operator). In all these examples, one assumes that σ is bounded away from the origin. Our approach to handle a nondegeneracy of the form σ (0 , x , u ( x )) only works if a Feynman-Kac representation is available for the correspondingstochastic linear equation satisfied by the Malliavin derivative. This happens, for instance,for parabolic operators of the form L = ∂ t − P i b i ∂ x i − P i,j a i,j ∂ x i ,x j . The methodologydeveloped in this paper could be extended to these operators, replacing the Brownian motionby the diffusion process with generator P i b i ∂ x i − P i,j a i,j ∂ x i ,x j . Let (Ω , F , P ) be a complete probability space. Consider a family of zero mean Gaussianrandom variables W = { W t ( ϕ ) , ϕ ∈ C ∞ ( R d ) , t ≥ } , where C ∞ ( R d ) denotes the space ofinfinitely differentiable functions on R d with compact support, with covariance E [ W t ( ϕ ) W s ( ψ )] = ( t ∧ s ) Z R d ϕ ( x ) ψ ( y ) q ( x, y ) dxdy, (2.1)where q is a nonnegative definite and locally integrable function.Let H be the Hilbert space defined as the completion of C ∞ ( R d ) by the inner product h ϕ, ψ i H := Z R d ϕ ( x ) ψ ( y ) q ( x, y ) dxdy. [0 ,t ] ϕ W t ( ϕ ) can be extended to a linear isometry between H ∞ := L ([0 , ∞ ); H ) and the L space spanned by W . Then { W ( h ) , h ∈ H ∞ } is an isonormalGaussian process associated with the Hilbert space H ∞ .We will denote by D the derivative operator in the sense of Malliavin calculus. That is,if F is a smooth and cylindrical random variable of the form F = f ( W ( h ) , . . . , W ( h n )) ,h i ∈ H ∞ , f ∈ C ∞ p ( R n ) ( f and all its partial derivatives have polynomial growth), then DF is the H ∞ -valued random variable defined by DF = n X j =1 ∂f∂x j ( W ( h ) , . . . , W ( h n )) h j . The operator D is closable from L (Ω) into L (Ω; H ∞ ) and we define the Sobolev space D , as the closure of the space of smooth and cylindrical random variables under the norm k DF k , = q E ( F ) + E ( k DF k H ∞ ) . We denote by δ the adjoint of the derivative operator, given by duality formula E ( δ ( u ) F ) = E (cid:0) h DF, u i H ∞ (cid:1) , (2.2)for any F ∈ D , and any element u ∈ L (Ω; H ∞ ) in the domain of δ . The operator δ is alsocalled the Skorohod integral. The higher Malliavin derivatives can be defined in similar wayand we can define D k,p for any integer k ≥ p ≥
1. Set D ∞ = \ k ≥ ,p ≥ D k,p .To obtain the existence and smoothness of the density, we make use of the following criteria. Theorem 2.1
Let F : Ω → R be a random variable. If F ∈ D , and k DF k H ∞ > almostsurely, then the probability law of F is absolutely continuous with respect to the Lebesguemeasure. Moreover, if F ∈ D ∞ and E (cid:2) k DF k − p H ∞ (cid:3) < ∞ for all p ≥ , then the density of F is infinitely differentiable. For the proof of this result and a detailed presentation of the Malliavin calculus we referto [11] and the references therein.
In this section we introduce some preliminaries on continuous semimartingales dependingon a parameter and the corresponding generalized Itˆo formula. We refer to [9] for moredetails.Fix a time interval [0 , T ], a complete probability space (Ω , F , P ) and a filtration {F t , ≤ t ≤ T } satisfying the usual conditions (increasing, right-continuous, and F contains all thenull sets). Let { F ( t, x ) , ≤ t ≤ T, x ∈ O } be a family of real valued processes withparameter x ∈ O , where O is a domain in R d . We can regard it as random field withdouble parameters x and t . If F ( t, x ) is m -times continuously differentiable with respect to x a.s. for any t , it can be regarded as stochastic process with values in C m or a C m -process.4ere we denote by C m = C m ( O, R ) the set of all real valued functions on O which are m times continuously differentiable. If furthermore, for each multi-index α ∈ { , . . . , d } k with | α | = k ≤ m , { D αx F ( t, x ) , x ∈ O } is a family of continuous semimartingales, then F ( t, x ) iscalled a C m -semimartingale. Here we have used the notation D αx = ∂ | α | ∂x α ··· ∂x αk .We denote by C , the set of all functions on a : [0 , T ] × O × O → R such that thepartial derivatives ∂a∂x i ( t, x, y ), ∂a∂y j ( t, x, y ) and ∂ a∂x i ∂y j ( t, x, y ) exist for any 1 ≤ i, j ≤ d and arecontinuous in ( x, y ), and for any compact set K ⊂ O and 1 ≤ i, j ≤ d Z T sup x,y ∈ K (cid:18) | a ( t, x, y ) | + | ∂a∂x i ( t, x, y ) | + | ∂a∂y j ( t, x, y ) | + | ∂ a∂x i ∂y j ( t, x, y ) | (cid:19) dt < ∞ . We also denote C the set of all functions on b : [0 , T ] × O → R which are continuouslydifferentiable in x , and for any compact set K ⊂ O and any 1 ≤ i ≤ d Z T sup x ∈ K (cid:18) | b ( t, x ) | + | ∂b∂x i ( t, x ) | (cid:19) dt < ∞ . Let { F ( t, x ) , x ∈ O } be a family of continuous semimartingales decomposed as F ( t, x ) = M ( t, x ) + B ( t, x ), where M ( t, x ) is a continuous local martingale and B ( t, x ) is a continuousprocess of bounded variation. Let A ( t, x, y ) be the joint quadratic variation of M ( t, x )and M ( t, y ) and assume that A ( t, x, y ) = R t a ( s, x, y ) ds and B ( t, x ) = R t b ( s, x ) ds , where a ( t, x, y ) and b ( t, x ) are predictable processes. Then ( a ( t, x, y ) , b ( t, x )) is called the localcharacteristic of the family of semimartingales { F ( t, x ) , x ∈ O } . Following Section 3.2 of[9], we say that the local characteristic ( a, b ) belongs to the class B , if a ( t, x, y ) and b ( t, x )are predictable processes with values in C , and C , respectively.Now let { F ( t, x ) , x ∈ O } be a continuous semimartingale with local characteristic ( a, b ).Let { f t , ≤ t ≤ T } be a predictable process with values in O satisfying Z T a ( s, f s , f s ) ds < ∞ , Z T | b ( s, f s ) | ds < ∞ a.s. (2.3)Then, the Itˆo stochastic integral of f t based on the kernel F ( dt, x ) is defined as the followinglimit in probability if it exists Z t F ( ds, f s ) = lim | ∆ |→ n − X k =0 { F ( t k +1 ∧ t, f t k ∧ t ) − F ( t k ∧ t, f t k ∧ t ) } , where ∆ = { t < · · · < t n = T } , and | ∆ | = max ≤ i ≤ n ( t i − t i − ).The joint quadratic variation of the Itˆo integrals R t F ( ds, f s ) and R t F ( ds, g s ) satisfies h Z · F ( ds, f s ) , Z · F ( ds, g s ) i t = Z t a ( s, f s , g s ) ds. The following is the generalized Itˆo formula (see Theorem 3.3.1 in [9]).
Theorem 2.2 (Generalized Itˆo formula)
Let { F ( t, x ) , x ∈ O } be a continuous C -processand a continuous C -semimartingale with local characteristic belonging to the class B , and et { X t , ≤ t ≤ T } be a continuous semimartingale with values in O . Then { F ( t, X t ) , ≤ t ≤ T } is a continuous semimartingale and satisfies F ( t, X t ) = F (0 , X ) + Z t F ( dr, X r ) + d X i =1 Z t ∂F∂x i ( r, X r ) dX ir + 12 d X i,j =1 Z t ∂ F∂x i ∂x j ( r, X r ) d h X i , X j i r + d X i =1 h Z · ∂F∂x i ( dr, X r ) , X i i t , (2.4) for any t ∈ [0 , T ] . In this section we establish a general Feynman-Kac formula for the d -dimensional heatequation driven by a continuous semimartingale. Suppose that F = { F ( t, x ) , ≤ t ≤ T, x ∈ R d } is a continuous semimartingale with local characteristic ( a, b ). We are going to imposethe following condition. (H1) Assume that a ( t, x, y ) and b ( t, x ) are continuous and satisfy | a ( t, x, y ) | ≤ C (1 + | x | β + | y | β ) , (3.1) | b ( t, x ) | ≤ C (1 + | x | β ) , (3.2)for t ∈ [0 , T ], with 0 ≤ β < ( ∂V∂t ( t, x ) = 12 △ V ( t, x ) + V ∂F∂t ( t, x ) V ( x,
0) = h ( x ) . (3.3)An adapted random field { V ( t, x ) , ≤ t ≤ T, x ∈ R d } is called a mild solution to the aboveequation if V ( t, x ) satisfies the following integral equation V ( t, x ) = Z R d p t ( x − z ) h ( z ) dz + Z R d (cid:18)Z t p t − r ( x − z ) V ( r, z ) F ( dr, z ) (cid:19) dz, (3.4)where p t ( x ) = (2 πt ) − d exp( −| x | / t ). Theorem 3.1 (Feynman-Kac Formula)
Let h ( x ) be continuous and with polynomial growth.Then the process V ( t, x ) = E B (cid:18) h ( x + B t ) exp (cid:18)Z t F ( dr, x + B t − B r ) − Z t ¯ a ( r, x + B t − B r ) dr (cid:19)(cid:19) , (3.5) where B is a d -dimensional standard Brownian motion independent of F , E B denotes themathematical expectation with respect to B , and ¯ a ( t, x ) = a ( t, x, x ) , is a mild solution toEquation (3.3). roof. We divide the proof into three steps.
Step 1 . First we show that the process (3.5) is well defined. In the sequel we denote by E the mathematical expectation in the probability space where F is defined, and E B denotesthe expectation with respect to the independent Brownian motion B . Set Y t = Z t F ( dr, x + B t − B r ) − Z t ¯ a ( r, x + B t − B r ) dr. Notice that R t F ( dr, x + B t − B r ) is a well defined Itˆo stochastic integral, because theprocess { B t − B r , ≤ r ≤ t } is independent of the semimartingale F , and conditions (2.3)are satisfied. Then V ( t, x ) = E B ( h ( x + B t ) exp( Y t )). We claim that this expectation existsand V ( t, x ) satisfies the following condition for any x ∈ R d and p ≥ ≤ t ≤ T E | V ( t, x ) | p ≤ K exp (cid:0) K | x | β (cid:1) , (3.6)where the constants K and K depend on p and T . In particular, this implies that thestochastic integral in (3.4) is well defined. We can write E | V ( t, x ) | p ≤ (cid:0) E B | h ( x + B t | p EE B exp(2 pY t ) (cid:1) . Let us denote by M ( t, x ) the martingale part of F ( t, x ). Then we make the decomposition Y t = Y (1) t + Y (2) t , where Y (1) t = Z t M ( dr, x + B t − B r ) − p Z t ¯ a ( r, x + B t − B r ) dr, and Y (2) t = Z t (cid:20) b ( r, x + B t − B r ) + (cid:18) p − (cid:19) ¯ a ( r, x + B t − B r ) (cid:21) dr. Using conditions (3.1) and (3.2) and taking into account that β <
2, we obtain for all t ∈ [0 , T ] E B exp(2 pY (2) t ) ≤ E B (cid:18) exp (cid:18) C Z t (1 + | x + B t − B r | β ) dr (cid:19)(cid:19) ≤ K exp( K | x | β ) . On the other hand, taking into account that exp(2 pY (1) t ) is a martingale, we can write EE B exp(2 pY (1) t ) = E B E exp(2 pY (1) t ) ≤ , which completes the proof of (3.6). Step 2 . We now show that the process (3.5) is a solution to Equation (3.3) under someadditional regularity assumptions on the semimartingale F ( t, x ). Suppose that F ( t, x ) isa C -semimartingale, such that the local characteristic ( a, b ) satisfies (3.1) and (3.2). Wealso assume that the functions D αx D αy a ( t, x, y ) satisfy the estimate (3.1) for all multi-index7 with 1 ≤ | α | ≤
2, and the functions D αx b ( t, x ) and D αx ¯ a ( t, x ) satisfy the estimate (3.2) forall multi-index α with 1 ≤ | α | ≤
2. Clearly this implies that the local characteristic belongsto the class B , . Suppose also that the functions D αx h ( t, x ) have polynomial growth for allmulti-index α with 1 ≤ | α | ≤ x , letΦ( t, y ) = Z t F ( dr, x + y − B r ) − Z t ¯ a ( r, x + y − B r ) dr. According to Theorem 3.3.3 in [9], Φ( t, y ) is a C -semimartingale with local characteristicbelonging to the class B , . We can apply the generalized Itˆo formula (2.4) to the process Y t = Φ( t, B t ), and we obtain dY t = F ( dt, x ) −
12 ¯ a ( t, x ) dt + d X i =1 (cid:20)Z t ∂F∂x i ( dr, x + B t − B r ) (cid:21) dB it − d X i =1 (cid:20)Z t ∂ ¯ a∂x i ( r, x + B t − B r ) dr (cid:21) dB it + 12 d X i =1 (cid:20)Z t ∂ F∂x i ( dr, x + B t − B r ) (cid:21) dt − d X i =1 (cid:20)Z t ∂ ¯ a∂x i ( r, x + B t − B r ) dr (cid:21) dt. The terms h R · ∂F∂x i ( dr, x ) , B i · i t = h R · ∂M∂x i ( dr, x ) , B i · i t vanish since M and B are independent.The quadratic variation of the semimartingale Y is given by d h Y i t = ¯ a ( t, x ) dt + d X i =1 (cid:2) Z t ∂F∂x i ( dr, x + B t − B r ) − Z t ∂ ¯ a∂x i ( r, x + B t − B r ) dr (cid:3) dt . (3.7)Consider the process Z ( t, x ) = h ( x + B t ) e Y t . Applying Itˆo’s formula to h ( x + B t ) e Y t yields Z ( t, x ) = h ( x ) + Z t Z ( s, x ) dY s + d X i =1 Z t ∂h∂x i ( x + B s ) e Y s dB is + 12 d X i =1 Z t ∂ h∂x i ( x + B s ) e Y s ds + 12 Z t ˆ V ( s, x ) d h Y i s + d X i =1 Z t ∂h∂x i ( x + B s ) e Y s d h B i , Y i s . (3.8)We claim that the stochastic integrals with respect to B i in the above expression have zeroexpectation with respect to B . This is a consequence of the following properties Z T E B Z ( t, x ) E B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 Z t ∂F∂x i ( dr, x + B t − B r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt < ∞ , (3.9) Z T E B Z ( t, x ) E B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 Z t ∂ ¯ a∂x i ( r, x + B t − B r ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt < ∞ , (3.10)8nd Z T E B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 Z t ∂h∂x i ( x + B s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Y s ds < ∞ . (3.11)These properties follow from our additional assumptions. For instance, to show (3.9) forthe martingale component of F , we take the expectation in the probability space where F is defined and we use the fact that for any p ≥ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 Z t ∂M∂x i ( dr, x + B t − B r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ c p E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j =1 Z t ∂ a∂x i ∂y j ( r, x + B t − B r , x + B t − B r ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ CE Z t (1 + | B t − B r | β ) ds < ∞ . Then, taking the expectation with respect to B in (3.8) yields V ( t, x ) = h ( x ) + Z t V ( s, x ) F ( ds, x )+ 12 d X i =1 E B (cid:18) Z t V ( s, x ) (cid:26) Z s ∂ F∂x i ( dr, x + B s − B r ) − Z s ∂ ¯ a∂x i ( r, x + B s − B r ) dr + (cid:20) Z s ∂F∂x i ( dr, x + B s − B r ) − Z s ∂ ¯ a∂x i ( r, x + B s − B r ) dr (cid:21) (cid:27) ds + Z t ∂ h∂x i ( x + B s ) e Y s ds + 2 Z t ∂h∂x i ( x + B s ) e Y s [ Z s ∂F∂x i ( dr, x + B s − B r ) − Z s ∂ ¯ a∂x i ( r, x + B s − B r ) dr ] ds (cid:19) Using that ∂Y s ∂x i = Z s ∂F∂x i ( dr, x + B s − B r ) − Z s ∂ ¯ a∂x i ( r, x + B s − B r ) dr , we obtain easily V ( t, x ) = h ( x ) + Z t V ( s, x ) F ( ds, x ) + 12 d X i =1 Z t ∂ V∂x i ( s, x ) ds . This shows that under some the additional regularity conditions on F and h the process u defined by (3.5) is a strong solution to Equation (3.3), and also a mild solution. Step 3 . Consider now the case of a general semimartingale F . For any ǫ > M ǫ ( t, x ) = Z R d M ( t, y ) p ǫ ( x − y ) dy,B ǫ ( t, x ) = Z R d B ( t, y ) p ǫ ( x − y ) dy, h ǫ ( x ) = R R d h ( y ) p ǫ ( x − y ) dy . It is easy to check that h ǫ is infinitely differentiableand it has polynomial growth together with all its partial derivatives. Also F ǫ ( t, x ) = M ǫ ( t, x ) + B ǫ ( t, x ) is a C -semimartingale with local characteristic given by b ǫ ( t, x ) = Z R d b ( t, y ) p ǫ ( x − y ) dy, and a ǫ ( t, x, y ) = Z t Z R d a ( s, x − z , y − z ) p ǫ ( z ) p ǫ ( z ) dz dz ds. It easy to check that a ǫ and b ǫ satisfy the estimates (3.1) and (3.2) respectively, the partialderivatives D αx D αy a ǫ ( t, x, y ) satisfy the estimate (3.1) for all multi-index α with 1 ≤ | α | ≤ D αx b ǫ ( t, x ) satisfy the estimate (3.2) for all multi-index α with 1 ≤ | α | ≤ V ǫ ( t, x ) = E B (cid:8) h ǫ ( x + B t ) exp (cid:0) Z t F ǫ ( dr, x + B t − B r ) − Z t ¯ a ǫ ( dr, x + B t − B r ) (cid:1)(cid:9) is the strong solution to ( ∂V ǫ ∂t ( t, x ) = 12 △ V ǫ ( t, x ) + V ǫ ∂F ǫ ∂t ( t, x ) V ǫ ( x,
0) = h ǫ ( x ) (3.12)As a consequence, it is also a mild solution to (3.12), namely, V ǫ ( t, x ) = Z R d p t ( x − z ) h ǫ ( z ) dz + Z R d (cid:18)Z t p t − r ( x − z ) V ǫ ( r, z ) F ǫ ( dr, z ) (cid:19) dz. Finally, we are going to take the limit as ǫ tends to zero in each term of the above expressionin order to deduce the Feynman-Kac formula of V . The estimate (3.1) impliessup ǫ> E exp (cid:18) p Z t | ¯ a ǫ ( r, x + B t − B r ) | dr (cid:19) < ∞ , . for all p ≥ V ǫ ( t, x ) converges to V ( t, x ) in L p for all p ≥
1. Clearly,lim ǫ ↓ Z R d p t ( x − z ) h ǫ ( z ) dz = Z R d p t ( x − z ) h ( z ) dz, and lim ǫ ↓ Z t Z R d p t − r ( x − z ) V ǫ ( r, z ) b ǫ ( r, z ) dzdr = Z t Z R d p t − r ( x − z ) V ( r, z ) b ( r, z ) dzdr, also in L p for all p ≥
1. The following limits in L p are also easy to check:lim ǫ ↓ Z R d (cid:18)Z t p t − r ( x − z ) V ǫ ( r, z )[ M ǫ ( dr, z ) − M ( dr, z )] (cid:19) dz = 0and lim ǫ ↓ Z R d (cid:18)Z t p t − r ( x − z )[ V ǫ ( r, z ) − V ( r, z )] M ( dr, z ) (cid:19) dz = 0 . This completes the proof of the theorem. 10
Stochastic heat equation: H¨older continuity of thesolution
Consider the following nonlinear stochastic partial differential equation: ∂u∂t = 12 △ u + b ( u ) + σ ( u ) ˙ W ( t, x ) , t ≥ , x ∈ R d u (0 , x ) = u ( x ) . (4.1)where W is the Gaussian family introduced in Section 2.1 with covariance function given by(2.1). Let us recall that an adapted random field { u ( t, x ) , t ≥ , x ∈ R d } is called a mildsolution to Equation (4.1) if u satisfies the following integral equation. u ( t, x ) = Z R d p t ( x − z ) u ( z ) dz + Z t Z R d p t − r ( x − z ) b ( u ( r, z )) dzdr + Z t Z R d p t − r ( x − z ) σ ( u ( r, z )) W ( dr, dz ) , (4.2)where the stochastic integral is defined as the integral of an H -valued predictable process.We are going to impose the following condition on the covariance function.( H1 ) For each t ≥ x ∈ R d Z t Z R d p t − s ( x − z ) p t − s ( x − z ) | q ( z , z ) | dz dz ds < ∞ . Theorem 4.1
Suppose that b and σ are globally Lipschitz continuous functions and supposethat the covariance function q satisfies ( H1 ). Let u ( x ) be a bounded function in R d . Thenthere exists a unique adapted process u = { u ( t, x ) , t ∈ [0 , T ] , x ∈ R d } satisfying (4.2).Moreover, sup t ∈ [0 ,T ] ,x ∈ R d E | u ( t, x ) | p < ∞ , ∀ p ≥ . (4.3) Proof.
Fix p ≥
2. Let B p be the Banach space of all adapted random fields u such that k u k p < ∞ , where k u k pp = sup t ∈ [0 ,T ] ,x ∈ R d E | u ( t, x ) | p . On B p , define the following mappingΨ( u )( t, x ) := Z R d p t ( x − z ) u ( z ) dz + Z t Z R d p t − r ( x − z ) b ( u ( r, z )) dzdr + Z t Z R d p t − r ( x − z ) σ ( u ( r, z )) W ( dr, dz ) . It is straightforward to obtain E | Ψ( u ) − Ψ( v ) | p ( t, x ) ≤ C (cid:20) E (cid:18)Z t Z R d p t − s ( x − z ) | u ( s, z ) − v ( s, z ) | dzds (cid:19) p + E ( Z t Z R d p t − s ( x − z ) | u ( s, z ) − v ( s, z ) | p t − s ( x − z ) | u ( s, z ) − v ( s, z ) | | q ( z , z ) | dz dz ds ) p/ (cid:21) . Taking the supremum with respect to t and x , we have k Ψ( u ) − Ψ( v ) k pp ≤ C Z T k u − v k pp ds ≤ CT k u − v k pp . Consequently, Ψ is a contraction mapping on B p when T sufficiently small. This proves theexistence and uniqueness of the solution for some small T . From the above argument itis clear that the T such that Ψ is a contraction is independent of the initial value of thesolution. This can be used to to show the existence and uniqueness of the solution for any T . The inequality (4.3) follows in a similar way.Now we apply the factorization method to obtain the H¨older continuity of u . Fix anarbitrary α ∈ (0 ,
1) and denote Y α ( r, z ) = Z r Z R d p r − s ( z − y ) σ ( u ( s, y ))( r − s ) − α W ( ds, dy ) . (4.4)The semigroup property of the heat kernel and the stochastic Fubini’s theorem yield Z t Z R d p t − s ( x − y ) σ ( u ( s, y )) W ( ds, dy )= sin( πα ) π Z t Z R d p t − r ( x − z )( t − r ) α − Y α ( r, z ) dzdr. (4.5)Consider the following stronger condition on the covariance function.( H1a ) There exists γ > − t ≥ x ∈ R d Z R d p t ( x − z ) p t ( x − z ) | q ( z , z ) | dz dz < Ct γ . Lemma 4.2
Let the assumptions of Theorem 4.1 be satisfied. Assume the covariance func-tion q satisfies ( H1a ). Then for any fixed
T > , p ≥ , α ∈ (0 , γ , we have sup r ∈ [0 ,T ] ,z ∈ R d E ( | Y α ( r, z ) | p ) < ∞ . Proof.
Since sup r ∈ [0 ,T ] ,z ∈ R d E ( | u ( r, z ) | p ) < ∞ from Theorem 4.1, and σ is Lipschitz continuous,we have sup r ∈ [0 ,T ] ,z ∈ R d E ( | σ ( u ( r, z )) | p ) < ∞ . Then we can write E | Y α ( r, z ) | p ≤ C (cid:18) E Z r Z R d p r − s ( z − y ) p r − s ( z − y ) × σ ( u ( s, y )) σ ( u ( s, y ))( r − s ) − α q ( y , y ) dy dy ds (cid:19) p C sup r ∈ [0 ,T ] ,z ∈ R d E ( | σ ( u ( r, z )) | p ) × (cid:18) Z r Z R d p r − s ( z − y ) p r − s ( z − y )( r − s ) − α | q ( y , y ) | dy dy ds (cid:19) p ≤ C (cid:18)Z r ( r − s ) γ − α ds (cid:19) p < ∞ . Equation (4.5) and Lemma 4.2 constitute the main ingredients to prove the followingtheorem concerning the H¨older continuity of the solution u . Theorem 4.3
Suppose that b and σ are globally Lipschitz continuous. Assume ( H1a ) andsuppose that u ( x ) is bounded and ρ -H¨older continuous. Then the solution u to the equation(4.1) is a.s. β -H¨older continuous in the time variable t and β -H¨older continuous in thespace variable x for any β ∈ (0 ,
12 [ ρ ∧ (1 + γ )]) and β ∈ (0 , ρ ∧ (1 + γ )) , respectively. Proof.
It suffices to follow the idea of the proof of Theorem 2.1 in [15], and we provide asketch of the proof for the reader’s convenience. The proof contains two parts for time andspace variables, respectively.
Part I
Fix
T, h > p ∈ [2 , ∞ ) . First we show thatsup ≤ t ≤ T sup x ∈ R d E ( | u ( t + h, x ) − u ( t, x ) | p ) ≤ C ( p, T ) h ηp , (4.6)for any η ∈ (0 , [ ρ ∧ (1 + γ )]). Let Y α be as defined in (4.4) with α ∈ (0 , γ ) and denote P t f ( x ) = R R d p t ( x − z ) f ( z ) dz . We have E ( u ( t + h, x ) − u ( t, x ) | p ) ≤ C ( p, α ) X i =1 I i ( t, h, x ) , where I ( t, h, x ) = | P t + h u ( x ) − P t u ( x ) | p ,I ( t, h, x ) = E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d [ p t + h − r ( x − z )( t + h − r ) α − − p t − r ( x − z )( t − r ) α − ] Y α ( r, z ) (cid:12)(cid:12)(cid:12)(cid:12) p dzdr (cid:19) ,I ( t, h, x ) = E (cid:12)(cid:12)(cid:12)(cid:12)Z t + ht Z R d p t + h − r ( x − z )( t + h − r ) α − Y α ( r, z ) (cid:12)(cid:12)(cid:12)(cid:12) p dzdr ! ,I ( t, h, x ) = E (cid:12)(cid:12)(cid:12)(cid:12)Z t + h Z R d p t + h − r ( x − z ) b ( u ( r, z )) dzdr − Z t Z R d p t − r ( x − z ) b ( u ( r, z )) (cid:12)(cid:12)(cid:12)(cid:12) p dzdr ! . For the term I ( t, h, x ), using the fact that u is ρ -H¨older continuous we have I ( t, h, x ) ≤ Ch ρp . For any α ∈ (0 , γ ) set ψ α ( t, x ) = p t ( x ) t α − . By H¨older’s inequality and Lemma4.2, we have I ( t, h, x ) ≤ C (cid:18)Z t Z R d | ψ α ( t + h − r, x − z ) − ψ α ( t − r, x − z ) | dzdr (cid:19) p . I , ( t, h, x ) = Z t Z R d exp (cid:18) − | x − z | t − r ) (cid:19) | ( t + h − r ) α − − d − ( t − r ) α − − d | dzdr, and I , ( t, h, x ) = Z t Z R d ( t + h − r ) α − − d (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − | x − z | t + h − r ) (cid:19) − exp (cid:18) − | x − z | t − r ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dzdr. Then I ( t, h, x ) ≤ C ( I , ( t, h, x ) p + I , ( t, h, x ) p ), and using the same arguments as in theproof of Theorem 2.1 in [15] to estimate the terms I , ( t, h, x ) p and I , ( t, h, x ) p , we obtainfor η ∈ (0 , α ) that I ( t, h, x ) ≤ Ch ηp . By H¨older’s inequality and Lemma 4.2, we have I ( t, h, x ) ≤ C (cid:18)Z t + ht Z R d p t + h − r ( x − z )( t + h − r ) α − dzdr (cid:19) p ≤ Ch αp . A change of variable yields I ( t, h, x ) ≤ C ( I , ( t, h, x ) + I , ( t, h, x )) , with I , ( t, h, x ) = E (cid:12)(cid:12)(cid:12)(cid:12)Z h Z R d P t + h − r ( x − z ) b ( u ( r, z )) dzdr (cid:12)(cid:12)(cid:12)(cid:12) p ! ,I , ( t, h, x ) = E (cid:12)(cid:12)(cid:12)(cid:12)Z h Z R d P t − r ( x − z )[ b ( u ( r + h, z )) − b ( u ( r, z ))] dzdr (cid:12)(cid:12)(cid:12)(cid:12) p ! , Since b is Lipschitz, using H¨older’s inequality and Equation (4.3), we have I , ( t, h, x ) ≤ Ch p . The Lipschitz property of b also implies I , ( t, h, x ) ≤ Z t sup z ∈ R d E ( | u ( r + h, z ) − u ( r, z ) | p ) dr. Putting together all estimations for I i , i = 1 , . . . ,
4, we obtainsup x ∈ R d E ( | u ( t + h, x ) − u ( t, x ) | p ) ≤ C (cid:18) h p min( ρ ,η,α ) + Z t sup x ∈ R d E ( | u ( r + h, x ) − u ( r, x ) | p ) dr (cid:19) . Since 0 < η < α < γ , the estimate (4.6) follows by Gronwall’s Lemma. Part II
Now consider the increments in the space variable. We want to show that for any
T > , p ∈ [2 , ∞ ) , x, a ∈ R d and η ∈ (0 , ρ ∧ (1 + γ )) , sup ≤ t ≤ T sup x ∈ R d E ( | u ( t, x + a ) − u ( t, x ) | p ) ≤ C | a | ηp . (4.7)14ix α ∈ (0 , γ ). Ee have E ( | u ( t, x + a ) − u ( t, x ) | p ) ≤ C X i =1 J i ( t, x, a ) , with J ( t, x, a ) = | P t u ( x + a ) − P t u ( x ) | p J ( t, x, a ) = E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d [ ψ α ( t − r, x + a − z ) − ψ α ( t − r, x − z )] Y α ( r, z ) dzdr (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) ,J ( t, x, a ) = E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d [ p t − r ( x + a − z ) − p t − r ( x − z )] b ( u ( r, y )) dzdr (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) . It is easy to show that J ( t, x, a ) ≤ C | a | ρp . For the term J ( t, x, a ), first we have using themean value theorem, Z R d | ψ α ( t − r, x + a − z ) − ψ α ( t − r, x − z ) | dz ≤ C ( t − r ) α − − η | a | η , where η ∈ (0 , α ∈ (0 , γ ) , η ∈ (0 , α ∧ , we deduce J ( t, x, a ) ≤ C (cid:18)Z t Z R d | ψ α ( t − r, x + a − z ) − ψ α ( t − r, x − z ) | dzdr (cid:19) p ≤ C | a | ηp . Finally, by a change of variable, the Lipschitz property of b, and H¨older’s inequality, J ( t, x, a ) ≤ E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d p t − r ( x − y )[ b ( u ( r, z + a ) − b ( u ( r, z )] dzdr (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) ≤ C Z t sup z ∈ R d E ( | u ( r, a + z ) − u ( r, z ) | p ) dr. Then (4.7) follows from the Gronwall’s lemma and the estimates of J i , i = 1 , , Example 4.4
A similar H¨older continuity result was obtained in [15] in the case of anhomogeneous covariance function q ( x, y ) = q ( x − y ) ≥ , where q is a nonnegative continuousfunction on R d \{ } such that it is the Fourier transform of a non-negative definite temperedmeasure µ on R d , and for some η ∈ (0 , we have Z R d µ ( dξ )(1 + | ξ | ) η < ∞ . This condition implies (
H1a ). In fact, we can write Z R d p t ( x − z ) p t ( x − z ) q ( z − z ) dz dz = Z R d p t ( x − z − z ) p t ( x − z ) q ( z ) dz dz = Z R d p t ( z ) q ( z ) dz = Z R d e − tξ µ ( dξ ) ≤ Z R d e − t ( ξ +1) µ ( dξ ) ≤ Ct − η Z R d µ ( dξ )(1 + | ξ | ) η . Example 4.5
Consider the case where d = 1 and the covariance structure in space is thatof a bifractional Brownian motion with parameters H ∈ (0 , , K ∈ (0 , , that is, q ( x, y ) = 2 − K ∂ ∂x∂y (( | x | H + | y | H ) K − | x − y | HK ) , where KH > . Then, B H,K ( t, x ) = W ( [0 ,t ] × [0 ,x ] ) , is a bifractional Brownian motion in x ≥ for each fixed t , and formally, W ( t, x ) = ∂∂x B H,K ( t, x ) . Then | q ( x, y ) | ≤ C [ | x | HK − + | y | HK − + | x − y | HK − ] and γ = HK − ∈ ( − , . Thus, Theorem 4.3 can be applied to this case.
In this section we consider again the solution u ( t, x ) to (4.1), and we will impose the followingcondition on the covariance q ( x, y ).( H2 ) q is γ -H¨older continuous for some γ >
0, and for some β ∈ [0 , | q ( x , x ) | ≤ C (1 + | x | β + | x | β ) . In this case we can assume that the random field { W t ( ϕ ) } has a density with respect to theLebesgue measure on R d . That means, we suppose that there exists a zero mean Gaussianrandom field { W ( t, x ) , t ≥ , x ∈ R d } with covariance E ( W ( t, x ) W ( s, y )) = ( s ∧ t ) q ( x, y ) , such that W t ( ϕ ) = R R d ϕ ( x ) W ( t, x ) dx , for any ϕ ∈ C ( R d ), where q ( x, y ) is positive definite,namely, R R d q ( x, y ) f ( x ) f ( y ) dxdy ≥ f ∈ L ( R d , dx ). The additional regularityconditions imposed on q have allowed us to introduce the density process W ( t, x ), which isa Brownian motion in the time variable and it has the spacial covariance q .From a Theorem of Mercer’s type (section 98 on page 245 in [14]) we know that if R R d | q ( x, y ) | dxdy < ∞ , then q ( x, y ) = ∞ X n =1 λ n e n ( x ) e n ( y ) , where e n , n = 1 , , · · · is an orthonormal basis of L ( R d , dx ) and P ∞ n =1 λ n < ∞ . The positivedefinite property of q ( x, y ) implies λ n ≥
0. If we take C ( x, y ) = P ∞ n =1 √ λ n e n ( x ) e n ( y ),then q ( x, y ) = R R d c ( ξ, x ) c ( ξ, y ) dξ . Thus it is without loss of generality for us to assumethat q ( y , y ) = R R d c ( ξ, y ) c ( ξ, y ) dξ for some c ( ξ, y ). Furthermore, we assume c ( x, y ) haspolynomial growth.The following is the main result of this section.16 heorem 5.1 Assume that q is γ -H¨older continuous for some γ > and satisfies ( H1a ).Suppose q ( x , x ) ≤ C (1 + | x | β + | x | β ) for some β ∈ [0 , . (5.1) Let u be bounded and ρ -H¨older continuous for some ρ > . Suppose that there is a x ∈ R d such that q ( x , x ) > and σ ( u ( x )) = 0 . Then,(1) If b and σ are continuous differentiable functions with bounded first order derivatives,for any t > and x ∈ R d , the probability law of u ( t, x ) is absolutely continuous withrespect to the Lebesgue measure.(2) If b and σ be infinitely differentiable with bounded derivatives of all orders, then forany t > and x ∈ R d , the probability law of u ( t, x ) has a smooth density with respectto Lebesgue measure Proof.
First we claim that for all ( t, x ) the random variable u ( t, x ) belongs to theSobolev space D , under condition (1), and to the space D ∞ under condition (2). Thisfollows from standard arguments and we omit the proof (see, for instance [11], Proposition2.4.4 in the case of the stochastic heat equation). On the other hand, the Malliavin derivative D s,y u ( t, x ) satisfies the linear stochastic evolution equation D s,y u ( t, x ) = p t − s ( x − y ) σ ( u ( s, y )) + Z R d Z t p t − r ( x − z ) b ′ ( u ( r, z )) D s,y u ( r, z ) drdz + Z R d Z t p t − r ( x − z ) σ ′ ( u ( r, z )) D s,y u ( r, z ) W ( dr, z ) dz . Denote F := k Du ( t, x ) k H ∞ = Z t Z R d D s,y u ( t, x ) D s,y u ( t, x ) q ( y , y ) dy dy ds , where k · k H ∞ is the Hilbert norm introduced in Section 2. We are going to show only thestatement (2), and the first one follows from similar arguments. It suffices to show that E [ F − p ] < ∞ for any p ≥
1. We divide the proof into two steps.
Step 1 . Introduce V s,ξ ( t, x ) = Z R d c ( ξ, y ) D s,y u ( t, x ) dy . Then we can write F = Z t Z R d V s,ξ ( t, x ) dξds . For any fixed ( s, ξ ), the random field { V s,ξ ( t, x ) , t ≥ s, x ∈ R d } satisfies the following linearstochastic heat equation for t ≥ s , and x ∈ R d , ∂V s,ξ ∂t = 12 △ V s,ξ + b ′ ( u ) V s,ξ + σ ′ ( u ) V s,ξ ∂W ∂t ( t, x ) ,V s,ξ ( s, x ) = c ( ξ, x ) σ ( u ( s, x )) . Consider the continuous semimartingale { F ( t, x ) , t ≥ , x ∈ R d } given by F ( t, x ) = Z t b ′ ( u ( r, x )) dr + Z t σ ′ ( u ( r, x )) W ( dr, x ) . b ( t, x ) = b ′ ( u ( t, x )) and a ( t, x, y ) = Z t σ ′ ( u ( r, x )) σ ′ ( u ( r, y )) q ( x, y ) dr . Notice that conditions (3.1) and (3.2) hold because b ′ and σ ′ are bounded and q satisfies(5.1). Then, Theorem 3.1 gives an explicit Feynman-Kac formula for the above equation.This means that we have V s,ξ ( t, x ) = E B (cid:20) c ( ξ, x + B t − s ) σ ( u ( s, x + B t − s )) × exp (cid:26)Z ts F ( dr, x + B t − s − B r − s ) − Z ts ¯ a ( r + x, B t − s − B r − s ) dr (cid:27) (cid:21) . Step 2 . Let Y ( s, t ; B ) = Z ts F ( dr, x + B t − s − B r − s ) − Z ts ¯ a ( r + x, B t − s − B r − s ) dr. Then Z t Z R d | V s,ξ ( t, x ) | dξds = Z t Z R d E B, ˜ B (cid:20) c ( ξ, x + B t − s ) c ( ξ, x + ˜ B t − s ) × σ ( u ( s, x + B t − s )) σ ( u ( s, x + ˜ B t − s )) exp { Y ( s, t ; B ) + Y ( s, t ; ˜ B ) } (cid:21) dξds = Z t E B, ˜ B (cid:20) q ( x + B t − s , x + ˜ B t − s ) × σ ( u ( s, x + B t − s )) σ ( u ( s, x + ˜ B t − s )) exp { Y ( s, t ; B ) + Y ( s, t ; ˜ B ) } (cid:21) ds = Z t H ( s ) ds, where ˜ B is a standard Brownian motion independent of B . If we can show that E ( H (0) − p ) < ∞ for all p >
1, and H ( s ) is H¨older continuous, then by Lemma 5.2 below we deduce E (cid:18)Z t Z R d | V s,ξ ( t, x ) | dξds (cid:19) − p = E (cid:18)Z t H ( s ) ds (cid:19) − p < ∞ for all p ≥
1. The H¨older continuity of H ( s ) can be verified from the following inequality: E | H ( s ) − H ( s ) | p ≤ C | s − s | p min { ρ,γ , γ } , where C is determined bysup s ∈ [0 ,t ] (cid:26) E | q ( x + B t − s , x + ˜ B t − s ) | p , E | σ ( u ( s, x + B t − s )) | p , E exp { pY ( s, t ; B ) } (cid:27) . It remains to show that E ( H (0) − p ) < ∞ . Notice that H (0) = E B, ˜ B (cid:16) G x exp { Y (0 , t ; B ) + Y (0 , t ; ˜ B ) } (cid:17) , G x = q ( x + B t , x + ˜ B t ) σ ( u ( x + B t )) σ ( u ( x + ˜ B t )) . We can write, by Jensen’s inequality, E (cid:18) E B, ˜ B (cid:20) G x exp { Y (0 , t ; B ) + Y (0 , t ; ˜ B ) } (cid:21)(cid:19) − p = E (cid:12)(cid:12)(cid:12)(cid:12) E B, ˜ B (cid:20) | G x | sign( G x ) exp { Y (0 , t ; B ) + Y (0 , t ; ˜ B ) } (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) − p ≤ (cid:20) E B, ˜ B | G x | (cid:21) − p − E (cid:20) | G x | exp {− p ( Y (0 , t ; B ) + Y (0 , t ; ˜ B )) } (cid:21) . Our nondegeneracy hypotheses imply that E B, ˜ B G >
0, and this allows us to conclude theproof.
Lemma 5.2
Let { S t , ≤ t ≤ } be a non-negative stochastic process. If ES − a < ∞ forsome a > , and sup ≤ s ≤ t | S s − S | ≤ Gt γ where G is a positive random variable with EG b < ∞ for some b > , then we have E (cid:12)(cid:12)(cid:12)(cid:12)Z S t dt (cid:12)(cid:12)(cid:12)(cid:12) − p < ∞ , for < p < abγ/ ( a + b + bγ ) . In particular, if a and b can be arbitrarily large, then p can also be chosen arbitrarily large. Proof.
Let α, β > , where α + β < bβγ − bα ≥ aα , and 0 < ǫ < α + β − . We have P (cid:20)Z S t dt < ǫ (cid:21) ≤ P "Z ǫ β S t dt < ǫ, S > ǫ α + P [ S < ǫ α ] ≤ P " sup ≤ t ≤ ǫ β | S t − S | > ǫ α + P (cid:2) S − a > ǫ − aα (cid:3) ≤ b ǫ − bα E sup ≤ t ≤ ǫ β | S t − S | b ! + ǫ aα ES − a ≤ C (cid:0) ǫ bβγ − bα + ǫ aα (cid:1) ≤ Cǫ aα . Then E (cid:12)(cid:12)(cid:12)R S t dt (cid:12)(cid:12)(cid:12) − p < ∞ , for 0 < p < aα. The lemma follows with the choice of α and β such that α < bγ/ ( a + b + bγ ) and β = ( a + b ) / ( a + b + bγ ). References [1] Bally, V.; Pardoux, E. Malliavin calculus for white noise driven parabolic SPDEs.
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